In this paper, we study different forbidden subgraph characterizations of the prime-order element graph $Γ(G)$ defined on a finite group $G$. Its set of vertices is the group $G$ and two vertices $x,y \in G$ are adjacent if the order of $xy$ is prime. More specifically, we investigate the conditions when $Γ(G)$ is perfect, cograph, chordal, claw-free, and interval graph.