Topology of subgroup lattices of symmetric and alternating groups

We determine the homotopy type of the order complex of the subgroup lattice of the symmetric group Sn when n is a prime or a power of two. (The prime case has been treated previously in unpublished work of G. Ivanyos.) We do the same for alternating groups of prime degree. In addition, we show that, for any n>1, the homology of the order complex of the subgroup lattice of Sn has rank at least n!/2 in dimension n−3.