A group G is called a BCF-group if there is a positive integer k such that |X:X_G|£ k for each subgroup X of G. The structure of BCF-groups has been studied by Buckley, Lennox, Neumann, Smith and Wiegold; they proved in particular that locally finite groups with the property BCF are abelian-by-finite. As a group lattice version of this concept, we say that a group G is a BMF-group if there is a positive integer k such that every subgroup X of G contains a modular subgroup Y of G for which the index |X:Y| is finite and the number of its prime divisors with multiplicity is bounded by k (it is known that such number can be characterized by purely lattice-theoretic considerations, and so it is invariant under lattice isomorphisms of groups). It is proved here that any locally finite BM-group contains a subgroup of finite index with modular subgroup lattice.