The commuting graph ${Γ(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all the dominant vertices of $Γ(G)$, the resulting subgraphs of $Γ(G)$ are $Γ^*(G)$ and $Γ^{**}(G)$, respectively. In this paper, we classify all the finite groups $G$ such that the graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the line graph of some graph. We also classify all the finite groups $G$ whose graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the complement of line graph.