In this communication, the co-maximal subgroup graph $Γ(G)$ 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 $Γ(G)$ 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 $Γ(G)$ is a split graph. We leave the readers with a few unsolved questions.