  Let $G$ be a group. The intersection graph of subgroups of $G$, denoted by
$\mathscr{I}(G)$, is a graph with all the proper subgroups of $G$ as its
vertices and two distinct vertices in $\mathscr{I}(G)$ are adjacent if and only
if the corresponding subgroups having a non-trivial intersection in $G$. In
this paper, we classify the finite groups whose intersection graph of subgroups
are toroidal or projective-planar. In addition, we classify the finite groups
whose intersection graph of subgroups are one of bipartite, complete bipartite,
tree, star graph, unicyclic, acyclic, cycle, path or totally disconnected. Also
we classify the finite groups whose intersection graph of subgroups does not
contain one of $K_5$, $K_4$, $C_5$, $C_4$, $P_4$, $P_3$, $P_2$, $K_{1,3}$,
$K_{2,3}$ or $K_{1,4}$ as a subgraph. We estimate the girth of the intersection
graph of subgroups of finite groups. Moreover, we characterize some finite
groups by using their intersection graphs. Finally, we obtain the clique cover
number of the intersection graph of subgroups of groups and show that
intersection graph of subgroups of groups are weakly $\alpha$-perfect.
