A subgroup H < G is maximal subgroup if no Z exists for which H < Z < G holds.
The following types of maximal subgroups of space grous are to be distinguished:
- Translationengleiche subgroups
A subgroup H of a space group G is called a translationengleiche subgroup or t-subgroup of G if the set T(G) of translations is retained, i.e. T(H)=T(G), but the order of the point group PG is reduced. They can only be non-isomorphic subgroups because the (finite) point group has been decreased.
- Klassengleiche subgroups
A subgroup H < G of a space group is called a klassengleiche subgroup or k-subgroup if the set T(G) of all translations of G is reduced, i.e. T(H) < T(G), but all linear parts of G are retained, i.e. the order of the point group PH is the same as that of PG.
There are two types of k-subgroups: non-isomorphic and isomorphic subgroups.
- Isomorphic klassengleiche subroups
A klassengleiche subroup H < G is called isomorphic or an isomorphic subgroup if it belongs to the same affine space-group type (isomorphism type) as G does.
- Non-isomorphic klassenglieche subgroups
Two possibilities are distinguished:
(1) the conventional cell remains unchanged, i.e. only centring translations are lost.
(2) the conventional cell is enlarged.
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