  The intersection graph $\Delta_G$ of a finite group $G$ is a simple graph
with vertices the non-trivial proper subgroups of $G$, and an edge between two
vertices if their corresponding subgroups intersect non-trivially. These graphs
were introduced by Cs\'ak\'any and Poll\'ak in 1969. In this paper we answer
two long-standing open questions posed by Cs\'ak\'any and Poll\'ak concerning
the diameter of intersection graphs. We prove some necessary conditions for a
non-simple group to have an intersection graph of diameter 4. We also construct
the first examples of non-simple groups and alternating groups whose
intersection graphs have diameter 4.
