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(a) Cubic subgroups
The cubic space groups of classes , 432 and have maximal cubic subgroups of class 23 which are found by simple inspection of the full symbol.
The cubic space groups of class have maximal subgroups which belong to classes 432 and .
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(b) Tetragonal subgroups
In the cubic space groups of classes 432 and , the primary and tertiary symmetry elements are relevant for deriving maximal tetragonal subgroups.
For the space groups of class , the full symbols are needed to recognize their tetragonal maximal subgroups of class . The primary symmetry planes of the cubic space group are conserved in the primary and secondary symmetry elements of the tetragonal subgroup: m, n and d remain in the tetragonal symbol; a remains a in the primary and becomes c in the secondary symmetry element of the tetragonal symbol.
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(c) Rhombohedral subgroups5
Here the secondary and tertiary symmetry elements of the cubic space-group symbols are relevant. For space groups of classes 23, , 432, the maximal R subgroups are R3, and R32, respectively. For space groups of class , the maximal R subgroup is when the tertiary symmetry element is m and R3c otherwise. Finally, for space groups of class , the maximal R subgroup is when the tertiary symmetry element is m and otherwise. All subgroups are of index [4].
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(d) Orthorhombic subgroups
Maximal orthorhombic space groups of index [3] are easily derived from the cubic space-group symbols of classes 23 and .5 Thus, P23, F23, I23, , (195–199) have maximal subgroups P222, F222, I222, , , respectively. Likewise, maximal subgroups of , , , , , , (200–206) are Pmmm, Pnnn, Fmmm, Fddd, Immm, Pbca, Ibca, respectively. The lattice type (P, F, I) is conserved and only the primary symmetry element has to be considered.
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