In this talk we will discuss our recent result on representing some very simple lattices as lattices of intermediate subfactors of finite index. These lattices were conjectured among the simplest lattices not realized by intervals of group-subgroup lattices in a finite group. The subfactors which realize these lattices are orbifolds of Jones-Wassermann subfactors from conformal field theory. We will sketch a proof of this result, with emphasize on diagrammatic arguments which partially motivated the proof.