Subgroup S-commutativity degrees of finite groups

The so–called subgroup commutativity degree $sd(G)$ of a finite group $G$ is the number of permuting subgroups $(H,K) \in $\mathrm{L}$(G) \times $\mathrm{L}$(G)$,where$$\mathrm{L}$(G)$isthesubgrouplatticeof$G$, divided by $\mathrm{\mid }\mathrm{L}(G){\mathrm{\mid }}^2$. It allows us to measure how $G$ is far from the celebrated classification of quasihamiltonian groups of K. Iwasawa. Here we generalize $sd(G)$, looking at suitable sublattices of $\mathrm{L}(G)$, and show some new lower bounds.