Lattices of intermediate subfactors

Let N be an irreducible subfactor of a type II1 factor M. If the Jones index CM: N] is finite, then the set L at(N subset of M) of the intermediate subfactors for the inclusion N subset of K forms a Finite lattice. The commuting and co-commuting square conditions for intermediate subfactors are related to the modular identity in the lattice L at(N subset of N). In particular, simplicity of a finite group G is characterized in terms of commuting square conditions of intermediate subfactors for N subset of M = N x G. We investigate the question of which finite lattices can be realized as intermediate subfactor lattices. (C) 1996 Academic Press, Inc.