Intersection graphs of groups

For a group G the intersection graph of G is a simple graph such that its vertex set consists of the non-trivial proper subgroups of G and two distinct vertices are joined by an edge if and only if their intersection is non-trivial. We have classified all finite groups whose intersection graphs are planar and showed that finite abelian groups can almost be
characterized by their intersection graphs. In this talk, I will outline those results and present some speculations on possible further directions.