In this paper, we introduce the concepts of t-fuzzy congruences and t-fuzzy equivalences. Using these ideas, we investigate completely, on one hand, the lattice structures of the set of fuzzy equivalence relations on a group and the set of fuzzy congruences and, on the other hand, the lattice structures of the set of fuzzy subgroups and fuzzy normal subgroups. Our study reveals some finer and interesting facts about these lattices. It is proved, among other results, that the set Ct of all t-fuzzy congruences of a group G forms lattice, and also the set L_nt of all those fuzzy normal subgroups, which assume the same value t at e the identity of G, forms a lattice. As an important result, we prove that the lattices C_t and L_nt are isomorphic. It is also shown that the lattices C_t and L_nt are modular. Moreover, we construct various important sublattices of the lattice C_t and exhibit their relationship by lattice diagrams. In the process, we improve and unify many results of earlier authors on fuzzy congruences.