On subgroups with non-zero Möbius numbers in the alternating and symmetric groups

For a subgroup H of an alternating or symmetric group G, we consider the Möbius number μ(H,G) of H in the subgroup lattice of G. Let bm(G) be the number of subgroups H of G of index m with μ(H,G)≠0. We prove that there exist two absolute constants α and β such that for any alternating or symmetric group G, any subgroup H of G and any positive integer m we have bm(G)⩽mα and |μ(H,G)|⩽β|G:H|.