A subfactor $N \subset M $ is maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
Question: are there only finitely many maximal subfactors of a fixed finite index (up to isomorphism)?
(Need to add "finite depth" and "irreducible" ?)
Bonus question: let $\alpha$ the index of a finite depth irreducible subfactor.
Does there exist a maximal finite depth irreducible subfactor of index $\alpha$ ?