  Let $G$ be a finite group, $\mu$ be the M\"obius function on the subgroup
lattice of $G$, and $\lambda$ be the M\"obius function on the poset of
conjugacy classes of subgroups of $G$. It was proved by Pahlings that, whenever
$G$ is solvable, the property $\mu(H,G)=[N_{G^\prime}(H):G^{\prime}\cap
H]\cdot\lambda(H,G)$ holds for any subgroup $H$ of $G$. It is known that this
property does not hold in general; for instance it does not hold for every
simple groups, the Mathieu group $M_{12}$ being a counterexample. In this paper
we investigate the relation between $\mu$ and $\lambda$ for some classes of
non-solvable groups; among them, the minimal non-solvable groups. We also
provide several examples of groups not satisfying the property.
