  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 $\mathrm{L}(G)$ is the subgroup lattice of $G$, divided
by $|\mathrm{L}(G)|^2$. 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.
