The 230 3Dimensional Space Groups 
The majority of the table is reference material.
Space Groups
The number of permutations of Bravais lattices with rotation and screw axes, mirror and glide planes, plus points of inversion is finite: there are only 230 unique combinations for threedimensional symmetry, and these combinations are known as the 230 space groups. Use of powder diffraction for structural studies does not require a knowledge of their derivation, nor does it require you to memorize a list of the 230 combinations. However, you do need to understand some of the properties of space groups. In order to make life simple, the space groups can be classified according to certain symmetry types.
The table at the bottom of this page lists the 230 threedimensional space groups used by crystallographers to describe the symmetry of their crystal structures. The space groups are numbered from 1 to 230 and are classified here according to the 7 crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.
Within each crystal system, the space groups can be ordered by Laue class, crystal class (e.g. 2 < m < 2/m) and, finally, lattice centring (e.g. P < A,B,C < F < I ), as shown in the table below. Further ordering is based largely on the symmetry elements present: In general, rotation axes come before screw axes and mirror planes before glide planes. Note that space groups are not listed here in exact order of their standard space group number, though the order is very similar.
Notation
Each one of the 230 threedimensional space groups is unique; however, they are frequently specified by means of space group symbols that are not unique to a particular space group. The reason for this is very simple: while each space group symmetry is unique, the choice of vectors that defines a unit cell for that symmetry is not unique. It was mentioned earlier, for example, that the unit cell for the centred monoclinic Bravais lattice is conventionally described as Cface centred with the unique symmetry axis direction being parallel to b. Space group symbols all begin with the lattice type as the first character of the symbol, e.g. C2. With a different choice of labelling for the unitcell axes, the same space group (number 5) could easily be called A2; other possible symbols are I2, B2, or even F2. You should therefore be aware that the symbols used in the table below refer to socalled standard settings (and hence standard symbols) for the space groups.
The 230 Crystallographic Space Groups
Space groups possessing a point of inversion are termed centrosymmetric; these are shown in the table in red. Some space groups have no symmetry element that can change the handedness of an object; these are termed enantiomorphic space groups and are shown in magenta.
Crystal System  Laue Class 
Crystal Class 
Lattice Centring 
230 3Dimensional Space Groups 

Triclinic  1  1  P  P1 
1  P1  
Monoclinic  2/m  2  P  P2, P2_{1} 
C  C2  
m  P  Pm, Pc  
C  Cm, Cc  
2/m  P  P2/m, P2_{1}m, P2/c, P2_{1}c  
C  C2/m, C2/c  
Orthorhombic  mmm  222  P  P222, P222_{1}, P2_{1}2_{1}2, P2_{1}2_{1}2_{1} 
C  C222, C222_{1}  
F  F222  
I  I222, I2_{1}2_{1}2_{1}  
mm2  P  Pmm2, Pmc2_{1}, Pcc2,
Pma2, Pca2_{1}, Pnc2, Pmn2_{1}, Pba2, Pna2_{1}, Pnn2 

C or A  Cmm2, Cmc2_{1}, Ccc2, Amm2, Abm2, Ama2, Aba2 

F  Fmm2, Fdd2  
I  Imm2, Iba2, Ima2  
mmm  P 
Pmmm, Pnnn, Pccm, Pban,
Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Pnma 

C  Cmmm, Cmcm, Cmca, Cccm, Cmma, Ccca  
F  Fmmm, Fddd  
I  Immm, Ibam, Ibcm, Imma  
Tetragonal  4/m  4  P  P4, P4_{1}, P4_{2}, P4_{3} 
I  I4, I4_{1}  
4  P  P4  
I  I4  
4/m  P  P4/m, P4_{2}/m, P4/n, P4_{2}/n  
I  I4/m, I4_{1}/a  
4/mmm  422  P  P422,
P42_{1}2, P4_{1}22,
P4_{1}2_{1}2, P4_{2}22, P4_{2}2_{1}2, P4_{3}22, P4_{3}2_{1}2 

I  I422, I42_{1}2  
4mm  P  P4mm, P4bm,
P4_{2}cm, P4_{2}nm,
P4cc, P4nc, P4_{2}mc, P4_{2}bc 

I  I4mm, I4cm, I4_{1}md, I4_{1}cd  
42m  P  P42m, P42c, P42_{1}m, P42_{1}c  
I  I42m, I42d  
4m2  P  P4m2, P4c2, P4b2, P4n2  
I  I4m2, I4c2  
4/mmm  P 
P4/mmm, P4/mcc,
P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P4_{2}/mmc, P4_{2}/mcm, P4_{2}/nbc, P4_{2}/nnm, P4_{2}/mbc, P4_{2}/mcm, P4_{2}/nmc, P4_{2}/ncm 

I  I4/mmm, I4/mcm, I4_{1}/amd, I4_{1}/acd  
Trigonal  3  3  P  P3, P3_{1}, P3_{2} 
R  R3  
3  P  P3  
R  R3  
3m  312  P  P312, P3_{1}12, P3_{2}12  
321  P321, P3_{1}21, P3_{2}21  
R  R32  
31m  P  P31m, P31c  
3m1  P3m1, P3c1  
R  R3m, R3c  
31m  P  P31m, P31c  
3m1  P3m1, P3c1  
R  R3m, R3c  
Hexagonal  6/m  6  P  P6, P6_{1}, P6_{2}, P6_{3}, P6_{4}, P6_{5} 
6  P6  
6/m  P6/m, P6_{3}/m  
6/mmm  622  P622, P6_{1}22, P6_{2}22, P6_{3}22, P6_{4}22, P6_{5}22  
6mm  P6mm, P6cc, P6_{3}cm, P6_{3}mc  
6m2  P6m2, P6c2  
62m  P62m, P62c  
6/mmm  P6/mmm, P6/mcc, P6_{3}/mcm, P6_{3}/mmc  
Cubic  m3  23  P  P23, P2_{1}3 
F  F23  
I  I23, I2_{1}3  
m3  P  Pm3, Pn3, Pa3  
F  Fm3, Fd3  
I  Im3, Ia3  
m3m  432  P  P432, P4_{2}32, P4_{3}32, P4_{1}32  
F  F432, F4_{1}32  
I  I432, I4_{1}32  
43m  P  P43m, P43n  
F  F43m, F43c  
I  I43m, I43d  
m3m  P  Pm3m, Pn3n, Pm3n, Pn3m  
F  Fm3m, Fm3m, Fd3m, Fd3c  
I  Im3m, Ia3d 
© Copyright 19952006. Birkbeck College, University of London.  Author(s): Jeremy Karl Cockcroft 