On the maximum number of subgroups of a finite group

Abstract

Given a finite group R, we let Sub(R) denote the collection of all subgroups of R. We show that |Sub(R)|<c⋅|R|log2⁡(|R|)4, where c<7.372 is an explicit absolute constant. This result is asymptotically best possible. Indeed, as |R| tends to infinity and R is an elementary abelian 2-group, the ratio|Sub(R)||R|log2⁡(|R|)4 tends to c.