In his studies of f electrons in atoms, Racah introduced the group SO(7) and its subgroup G₂, with irreducible representations (irreps) W and U. By using a quarklike basis, these groups can be conveniently embedded in SO(8). This larger group, with irreps V, possesses two other SO(7) groups as subgroups that themselves contain G₂ as a common subgroup. One of them, SO(7)' (with irreps W'), has been used to derive new selection rules on operators of physical interest. We describe methods for calculating the overlaps (VWU mod VW' U), the ultimate aim being to facilitate the transformations between SO(7) and SO(7)'. A table of relevant 6-U symbols (the G₂ generalizations of 6-j symbols) is given. When V possesses null triality (that is, when the symbols labelling the open ends of the Dynkin diagram for SO(8) are equal), an undetermined phase in the overlaps can be used to generate matrix representations of S₃, the permutation group on three objects. A brief table of zero overlaps is given. A remarkable factorization of the overlaps ((4310)W(40) I(4310)W'(40)) is noted, where (4310) is the irrep of SO(8) with dimension 25725.