  In this paper, we study the structure of the permutability graphs of
subgroups, and the permutability graphs of non-normal subgroups of the
following groups: the dihedral groups $D_n$, the generalized quaternion groups
$Q_n$, the quasi-dihedral groups $QD_{2^n}$ and the modular groups $M_{p^n}$.
Further, we investigate the number of edges, degrees of the vertices,
independence number, dominating number, clique number, chromatic number, weakly
perfectness, Eulerianness, Hamiltonicity of these graphs.
