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.