On intervals in subgroup lattices of finite groups

We investigate the question of which finite lattices L are isomorphic to the lattice [H,G] of all overgroups of a subgroup H in a finite group G. We show that the structure of G is highly restricted if [H,G] is disconnected. We define the notion of a "signalizer lattice" in H and show for suitable disconnected lattices L, if [H,G] is minimal subject to being isomorphic to L or its dual, then either G is almost simple or H admits a signalizer lattice isomorphic to L or its dual. We use this theory to answer a question in functional analysis raised by Watatani