Lattices of Intermediate Subfactors

LetNbe an irreducible subfactor of a typeII1factorM. If the Jones index [M:N] is finite, then the set Lat(N⊂M) of the intermediate subfactors for the inclusionN⊂Mforms afinitelattice. The commuting and co-commuting square conditions for intermediate subfactors are related to the modular identity in the lattice Lat(N⊂N). In particular, simplicity of a finite groupGis characterized in terms of commuting square conditions of intermediate subfactors forN⊂M=N⋊G. We investigate the question of which finite lattices can be realized as intermediate subfactor lattices.