On two Möbius functions for a finite non-solvable group

Let G be a finite group, μ be the Möbius function on the subgroup lattice of G, and λ be the Möbius function on the poset of conjugacy classes of subgroups of G. It was proved by Pahlings that, whenever G is solvable, the property (Formula presented.) holds for any subgroup H of G. It is known that this property does not hold in general, the Mathieu group M 12 being a counterexample. In this paper we investigate the relation between μ and λ 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.