There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point group of the unit cell.

Symbols In Hermann–Mauguin notation, space groups are named by a symbol combining the point group identifier with the uppercase letters describing the lattice type. Translations within the lattice in the form of screw axes and glide planes are also noted, giving a complete crystallographic space group.

These are the Bravais lattices in three dimensions: P primitive I body-centered (from the German Innenzentriert) F face-centered (from the German Flächenzentriert) S base-centered (from the German Seitenflächenzentriert), or specifically: A centered on A faces only B centered on B faces only C centered on C faces only R rhombohedral

A reflection plane m within the point groups can be replaced by a glide plane, labeled as a, b, or c depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along a quarter of either a face or space diagonal of the unit cell. The d glide is often called the diamond glide plane as it features in the diamond structure. , , or : glide translation along half the lattice vector of this face : glide translation along half the diagonal of this face : glide planes with translation along a quarter of a face diagonal (the space diagonal in FCC) : two glides with the same glide plane and translation along two (different) half-lattice vectors.

A gyration point can be replaced by a screw axis denoted by a number, n, where the angle of rotation is . The degree of translation is then added as a subscript showing how far along the axis the translation is, as a portion of the parallel lattice vector. For example, 21 is a 180° (twofold) rotation followed by a translation of of the lattice vector. 31 is a 120° (threefold) rotation followed by a translation of of the lattice vector. The possible screw axes are: 21, 31, 32, 41, 42, 43, 61, 62, 63, 64, and 65.

Wherever there is both a rotation or screw axis n and a mirror or glide plane m along the same crystallographic direction, they are represented as a fraction or n/m. For example, 41/a means that the crystallographic axis in question contains both a 41 screw axis as well as a glide plane along a.

In Schoenflies notation, the symbol of a space group is represented by the symbol of corresponding point group with additional superscript. The superscript doesn't give any additional information about symmetry elements of the space group, but is instead related to the order in which Schoenflies derived the space groups. This is sometimes supplemented with a symbol of the form which specifies the Bravais lattice. Here is the lattice system, and is the centering type.

In Fedorov symbol, the type of space group is denoted as s (symmorphic ), h (hemisymmorphic), or a (asymmorphic). The number is related to the order in which Fedorov derived space groups. There are 73 symmorphic, 54 hemisymmorphic, and 103 asymmorphic space groups.

Symmorphic The 73 symmorphic space groups can be obtained as combination of Bravais lattices with corresponding point group. These groups contain the same symmetry elements as the corresponding point groups. Example for point group 4/mmm (): the symmorphic space groups are P4/mmm (, 36s) and I4/mmm (, 37s).

Hemisymmorphic The 54 hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. Example for point group 4/mmm (): hemisymmorphic space groups contain the axial combination 422, but at least one mirror plane m will be substituted with glide plane, for example P4/mcc (, 35h), P4/nbm (, 36h), P4/nnc (, 37h), and I4/mcm (, 38h).

Asymmorphic The remaining 103 space groups are asymmorphic. Example for point group 4/mmm (): P4/mbm (, 54a), P42/mmc (, 60a), I41/acd (, 58a) - none of these groups contains the axial combination 422.

List of triclinic + Triclinic Bravais lattice

+ Triclinic crystal system Number Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold11P1 P 1 1s 2P P 2s

List of monoclinic + Monoclinic Bravais lattice Simple (P) Base (S)

+ Monoclinic crystal systemNumber Point group Orbifold Short nameFull name(s) Schoenflies Fedorov Shubnikov Fibrifold (primary) Fibrifold (secondary)32P2 P 1 2 1P 1 1 2 3s 4P21P 1 21 1P 1 1 21 1a 5C2 C 1 2 1B 1 1 2 4s , 6mPm P 1 m 1P 1 1 m 5s 7Pc P 1 c 1P 1 1 b 1h , 8Cm C 1 m 1B 1 1 m 6s , 9Cc C 1 c 1B 1 1 b 2h , 102/mP2/mP 1 2/m 1P 1 1 2/m 7s 11P21/mP 1 21/m 1P 1 1 21/m 2a 12C2/mC 1 2/m 1B 1 1 2/m 8s , 13P2/cP 1 2/c 1P 1 1 2/b 3h , 14P21/cP 1 21/c 1P 1 1 21/b 3a , 15C2/cC 1 2/c 1B 1 1 2/b 4h ,

List of orthorhombic + Orthorhombic Bravais lattice Simple (P) Body (I) Face (F) Base (S)

+ Orthorhombic crystal systemNumber Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold (primary) Fibrifold (secondary)16222P222P 2 2 2 9s 17P2221P 2 2 21 4a 18P21212P 21 21 2 7a 19P212121P 21 21 21 8a 20C2221C 2 2 21 5a 21C222C 2 2 2 10s 22F222F 2 2 2 12s 23I222I 2 2 2 11s 24I212121I 21 21 21 6a 25mm2Pmm2P m m 2 13s 26Pmc21P m c 21 9a , 27Pcc2P c c 2 5h 28Pma2P m a 2 6h , 29Pca21P c a 21 11a 30Pnc2P n c 2 7h , 31Pmn21P m n 21 10a , 32Pba2P b a 2 9h 33Pna21P n a 21 12a , 34Pnn2P n n 2 8h 35Cmm2C m m 2 14s 36Cmc21C m c 21 13a , 37Ccc2C c c 2 10h 38Amm2A m m 2 15s , 39Aem2A b m 2 11h , 40Ama2A m a 2 12h , 41Aea2A b a 2 13h , 42Fmm2F m m 2 17s 43Fdd2F d d 2 16h 44Imm2I m m 2 16s 45Iba2I b a 2 15h 46Ima2I m a 2 14h , 472/m 2/m 2/m (mmm)PmmmP 2/m 2/m 2/m 18s 48PnnnP 2/n 2/n 2/n 19h 49PccmP 2/c 2/c 2/m 17h 50PbanP 2/b 2/a 2/n 18h 51PmmaP 21/m 2/m 2/a 14a , 52PnnaP 2/n 21/n 2/a 17a , 53PmnaP 2/m 2/n 21/a 15a , 54PccaP 21/c 2/c 2/a 16a , 55PbamP 21/b 21/a 2/m 22a 56PccnP 21/c 21/c 2/n 27a 57PbcmP 2/b 21/c 21/m 23a , 58PnnmP 21/n 21/n 2/m 25a 59PmmnP 21/m 21/m 2/n 24a 60PbcnP 21/b 2/c 21/n 26a , 61PbcaP 21/b 21/c 21/a 29a 62PnmaP 21/n 21/m 21/a 28a , 63CmcmC 2/m 2/c 21/m 18a , 64CmceC 2/m 2/c 21/a 19a , 65CmmmC 2/m 2/m 2/m 19s 66CccmC 2/c 2/c 2/m 20h 67CmmeC 2/m 2/m 2/e 21h 68CcceC 2/c 2/c 2/e 22h 69FmmmF 2/m 2/m 2/m 21s 70FdddF 2/d 2/d 2/d 24h 71ImmmI 2/m 2/m 2/m 20s 72IbamI 2/b 2/a 2/m 23h 73IbcaI 2/b 2/c 2/a 21a 74ImmaI 2/m 2/m 2/a 20a

List of tetragonal + Tetragonal Bravais lattice Simple (P) Body (I)

+ Tetragonal crystal systemNumber Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold754P4P 4 22s 76P41P 41 30a 77P42P 42 33a 78P43P 43 31a 79I4I 4 23s 80I41I 41 32a 81PP 26s 82II 27s 834/mP4/mP 4/m 28s 84P42/mP 42/m 41a 85P4/nP 4/n 29h 86P42/nP 42/n 42a 87I4/mI 4/m 29s 88I41/aI 41/a 40a 89422P422P 4 2 2 30s 90P4212P4212 43a 91P4122P 41 2 2 44a 92P41212P 41 21 2 48a 93P4222P 42 2 2 47a 94P42212P 42 21 2 50a 95P4322P 43 2 2 45a 96P43212P 43 21 2 49a 97I422I 4 2 2 31s 98I4122I 41 2 2 46a 994mmP4mmP 4 m m 24s 100P4bm P 4 b m 26h 101P42cm P 42 c m 37a 102P42nm P 42 n m 38a 103P4cc P 4 c c 25h 104P4nc P 4 n c 27h 105P42mc P 42 m c 36a 106P42bc P 42 b c 39a 107I4mm I 4 m m 25s 108I4cm I 4 c m 28h 109I41md I 41 m d 34a 110I41cd I 41 c d 35a 1112mP2m P 2 m 32s 112P2c P 2 c 30h 113P21m P 21 m 52a 114P21c P 21 c 53a 115Pm2 P m 2 33s 116Pc2 P c 2 31h 117Pb2 P b 2 32h 118Pn2 P n 2 33h 119Im2 I m 2 35s 120Ic2 I c 2 34h 121I2m I 2 m 34s 122I2d I 2 d 51a 1234/m 2/m 2/m (4/mmm) P4/mmm P 4/m 2/m 2/m 36s 124P4/mcc P 4/m 2/c 2/c 35h 125P4/nbm P 4/n 2/b 2/m 36h 126P4/nnc P 4/n 2/n 2/c 37h 127P4/mbm P 4/m 21/b 2/m 54a 128P4/mnc P 4/m 21/n 2/c 56a 129P4/nmm P 4/n 21/m 2/m 55a 130P4/ncc P 4/n 21/c 2/c 57a 131P42/mmc P 42/m 2/m 2/c 60a 132P42/mcm P 42/m 2/c 2/m 61a 133P42/nbc P 42/n 2/b 2/c 63a 134P42/nnm P 42/n 2/n 2/m 62a 135P42/mbc P 42/m 21/b 2/c 66a 136P42/mnm P 42/m 21/n 2/m 65a 137P42/nmc P 42/n 21/m 2/c 67a 138P42/ncm P 42/n 21/c 2/m 65a 139I4/mmm I 4/m 2/m 2/m 37s 140I4/mcm I 4/m 2/c 2/m 38h 141I41/amd I 41/a 2/m 2/d 59a 142I41/acd I 41/a 2/c 2/d 58a

List of trigonal + Trigonal Bravais lattice Rhombohedral (R) Hexagonal (P)

+ Trigonal crystal systemNumber Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold1433P3 P 3 38s 144P31 P 31 68a 145P32 P 32 69a 146R3 R 3 39s 147P P 51s 148R R 52s 14932P312 P 3 1 2 45s 150P321 P 3 2 1 44s 151P3112 P 31 1 2 72a 152P3121 P 31 2 1 70a 153P3212 P 32 1 2 73a 154P3221 P 32 2 1 71a 155R32 R 3 2 46s 1563mP3m1 P 3 m 1 40s 157P31m P 3 1 m 41s 158P3c1 P 3 c 1 39h 159P31c P 3 1 c 40h 160R3m R 3 m 42s 161R3c R 3 c 41h 162 2/m (m)P1m P 1 2/m 56s 163P1c P 1 2/c 46h 164Pm1 P 2/m 1 55s 165Pc1 P 2/c 1 45h 166Rm R 2/m 57s 167Rc R 2/c 47h

List of hexagonal + Hexagonal Bravais lattice

+ Hexagonal crystal systemNumber Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Fibrifold1686P6 P 6 49s 169P61 P 61 74a 170P65 P 65 75a 171P62 P 62 76a 172P64 P 64 77a 173P63 P 63 78a 174P P 43s 1756/mP6/m P 6/m 53s 176P63/m P 63/m 81a 177622P622 P 6 2 2 54s 178P6122 P 61 2 2 82a 179P6522 P 65 2 2 83a 180P6222 P 62 2 2 84a 181P6422 P 64 2 2 85a 182P6322 P 63 2 2 86a 1836mmP6mm P 6 m m 50s 184P6cc P 6 c c 44h 185P63cm P 63 c m 80a 186P63mc P 63 m c 79a 187m2Pm2 P m 2 48s 188Pc2 P c 2 43h 189P2m P 2 m 47s 190P2c P 2 c 42h 1916/m 2/m 2/m (6/mmm) P6/mmm P 6/m 2/m 2/m 58s 192P6/mcc P 6/m 2/c 2/c 48h 193P63/mcm P 63/m 2/c 2/m 87a 194P63/mmc P 63/m 2/m 2/c 88a

List of cubic + Cubic Bravais lattice Simple (P) Body centered (I) Face centered (F)

+ Cubic crystal systemNumber Point group Orbifold Short name Full name Schoenflies Fedorov Shubnikov Conway Fibrifold (preserving ) Fibrifold (preserving , , )19523P23 P 2 3 59s 196F23 F 2 3 61s 197I23 I 2 3 60s 198P213 P 21 3 89a 199I213 I 21 3 90a 2002/m (m) Pm P 2/m 62s 201Pn P 2/n 49h 202Fm F 2/m 64s 203Fd F 2/d 50h 204Im I 2/m 63s 205Pa P 21/a 91a 206Ia I 21/a 92a 207432P432 P 4 3 2 68s 208P4232 P 42 3 2 98a 209F432 F 4 3 2 70s 210F4132 F 41 3 2 97a 211I432 I 4 3 2 69s 212P4332 P 43 3 2 94a 213P4132 P 41 3 2 95a 214I4132 I 41 3 2 96a 2153mP3m P 3 m 65s 216F3m F 3 m 67s 217I3m I 3 m 66s 218P3n P 3 n 51h 219F3c F 3 c 52h 220I3d I 3 d 93a 2214/m 2/m (mm)Pmm P 4/m 2/m 71s 222Pnn P 4/n 2/n 53h 223Pmn P 42/m 2/n 102a 224Pnm P 42/n 2/m 103a 225Fmm F 4/m 2/m 73s 226Fmc F 4/m 2/c 54h 227Fdm F 41/d 2/m 100a 228Fdc F 41/d 2/c 101a 229Imm I 4/m 2/m 72s 230Iad I 41/a 2/d 99a

Notes

References

External links

International Union of Crystallography Point Groups and Bravais Lattices Full list of 230 crystallographic space groups Conway et al. on fibrifold notation