On representing some lattices as lattices of intermediate subfactors of finite index

We prove that the very simple lattices which consist of a largest, a smallest and 2n pairwise incomparable elements where n is a positive integer can be realized as the lattices of intermediate subfactors of finite index and finite depth. Using the same techniques, we give a necessary and sufficient condition for subfactors coming from Loop groups of type A at generic levels to be maximal.