When exploring symmetry symmetry choices, it is very valuable to understand that space groups are linked together by the addition or removal of symmetry. Given a particular space group, the addition of symmetry will give rise to a supergroup, and the removal of a symmetry element will give rise to a subgroup. The relationships between space groups has been worked out via the tabulation of subgroups and supergroups. These relationships branch in that one can have a choice of several symmetry elements that can be added or removed. Likewise, these relationships form chains, since one can follow these links through multiple symmetry addition or removal steps. Note that a unit cell transformation may be involved to relate a subgroup or supergroup to the parent space group. As an example of that, consider that a mirror plane is only possible in a monoclinic or higher space group. If one removes all mirror planes from a C-centered monoclinic space group, the result will be a triclinic space group (either \(P 1\) or \(P \overline {1}\) to be specific) and the triclinic cell will be the primitive subcell of the C-centered monoclinic cell.
The International Tables for Crystallography, Volume A lists the subgroups and supergroups for each space group, but not how the transformations are made, so another volume was created, International Tables Volume A1. Both volumes categorize the subgroup-supergroup relationships into categories with impressive German names, but I don’t think explaining the categories is needed for this discussion. The Bilbao Crystallographic Server (BCS), at web site https://cryst.ehu.es/cryst/, has all the information in Volume A1, but with useful tools for transforming symmetry. GSAS-II performs some symmetry transformations using these tools. The BCS website will be discussed further in §28.4, below.
Another reason why subgroup and supergroup relationships are very important has to do with what happens when a material undergoes a phase transformation. Landau theory (which I know very little about) describes the type of transformations that occur. One type of transformation, which Landau categorizes as first-order, can involve complete transformation of a material. As an example, consider the phase change between graphite, with its sheets of triangularly-bonded carbon atoms and diamond, with its 3D network of tetrahedrally bonded C atoms. Such transitions can be considered as recrystallization. However, the other type of phase transition in Landau theory, a second-order transition, will be a rearrangement in structure where a material’s structure changes into a subgroup or supergroup. Second-order transitions are continuous and are accompanied by what Landau theory calls an order parameter, which describes how far along the transition process the transformation has gone.