  In this communication, the co-maximal subgroup graph $\Gamma(G)$ of a finite
group $G$ is examined when $G$ is a finite nilpotent group, finite abelian
group, dihedral group $D_n$, dicyclic group $Q_{2^n}$, and $p$-group. We derive
the necessary and sufficient conditions for $\Gamma(G)$ to be a cluster graph,
triangle-free graph, claw-free graph, cograph, chordal graph, threshold graph
and split graph. For the case of finite nilpotent group, we are able to
classify it entirely. Moreover, we derive the complete structure of finite
abelian group $G$ such that $\Gamma(G)$ is a split graph. We leave the readers
with a few unsolved questions.
