Let M be any quasivariety of Abelian groups, $$dom_G^M$$ (H) be the dominion of a subgroup H of a group G in M, and Lq(M) be the lattice of subquasivarieties of M. It is proved that $$dom_G^M$$ (H ) coincides with a least normal subgroup of the group G containing H, the factor group with respect to which is in M. Conditions are specified subject to which the set L(G,H,M) = { $$dom_G^N$$ (H) | N ∈ Lq(M)} forms a lattice under set-theoretic inclusion and the map ϕ : Lq(M) → L(G,H,M) such that ϕ(N) = $$dom_G^N$$ (H) for any quasivariety N ϕ Lq(M)is an antihomomorphism of the lattice L q (M) onto the lattice L(G, H, M).