Non-permutability graph of subgroups

Given a finite group $G$, the permutability graph of non-normal subgroups $\Gamma_{N}(G)$ is given by all proper non--normal subgroups of $G$ as vertex set and two vertices $H$ and $K$ are joined if $HK = KH$. Rajkumar and Devi generalized $\Gamma_{N}(G)$ to the permutability graph of subgroups $\Gamma(G)$, extending the vertex set to all proper subgroups of $G$ (removing the assumption of being non--normal) and keeping the same criterion to join two vertices. We consider a natural counterpart for $\Gamma(G)$ and $\Gamma_{N}(G)$, that is, the subgroups lattice $\mathrm{L}(G)$ of $G$ and introduce the non--permutability graph of subgroups $\Gamma_{\mathrm{L}(G)}$; its vertices are now given by the set $\mathrm{L}(G)-\mathfrak{C}_{\mathrm{L}(G)}(\mathrm{L}(G))$, where $\mathfrak{C}_{\mathrm{L}(G)}(\mathrm{L}(G))$ denotes the smallest sublattice of $\mathrm{L}(G)$ containing all permutable subgroups of $G$, and we join two vertices $H,K$ of $\Gamma_{\mathrm{L}(G)}$ if and only if $HK\neq KH$. We study classical invariants of $\Gamma_{\mathrm{L}(G)}$, showing generalizations of previous results in literature.