a group-subgroup relation

Every crystal structure has a symmetry — a full set of operations (rotations, mirrors, translations) that leave it looking unchanged, collected into its space group. When a crystal transforms to a lower-symmetry structure, it does not usually lose its symmetry at random: the new structure keeps a SUBSET of the old operations and loses the rest. A group-subgroup relation is exactly this statement — the daughter's space group is a subgroup of the parent's space group. The parent is the more symmetric; the daughter has strictly fewer symmetry operations.

Take a cubic parent that becomes tetragonal on cooling (as in barium titanate's Pm-3m to P4mm). The tetragonal daughter still has a 4-fold axis along one cube edge and some mirrors, but it has LOST the operations that made the three cube axes equivalent — the three-fold body-diagonal rotations are gone. Those retained operations form a subgroup of the cubic group. The index of the subgroup (how many times smaller it is) counts how many equivalent ways the daughter could have formed — here 3, because the tetragonal axis could be x, y, or z.

This is not bookkeeping; it is predictive. Landau's theorem says a continuous (second-order) transition is only possible when parent and product are related by a group-subgroup relation (with the order parameter transforming as a single irreducible representation) — so the symmetry relation tells you which transitions CAN be continuous. And the lost symmetry operations are precisely what generate the domains: each way the daughter can pick its lower symmetry is a distinct domain variant, and the number of variants equals the index of the subgroup. Symmetry lowering and domain formation are two sides of one coin.