Coherent domain structures are theoretically investigated in the case when they arise after transition from a phase with space group $G$ to a less symmetrical phase with space group $H$, i.e., when $H$ is a subgroup of $G$. $G$ can then be decomposed into left and right cosets with respect to $H$. The number of variants, i.e., of different types of domains, is the index of $H$ in $G$, and a one-to-one mapping exists between cosets (right or left) and types of boundaries. If $H$ is an invariant subgroup, both decompositions (into left and right cosets) are identical and the number of types of boundaries is the number of variants minus one. Furthermore, the symmetry elements are in the same position throughout the structure, disregarding the boundaries. If $H$ is not invariant, there exist several conjugate subgroups. It is proved that the type of boundary is independent of which of the two decompositions into left or right cosets---which are now different---is made. But, owing to the existence of several conjugate subgroups, there may be more types of boundaries and this number is given in each case by writing all coset decompositions with respect to all conjugate subgroups. Moreover, the symmetry elements are no more in the same position throughout the crystal. A number of theoretical examples are examined.