On the maximum number of subgroups of a finite group

Abstract

Given a finite group $R$, we let $\mathrm{Sub}(R)$ denote the collection of all subgroups of $R$. We show that $|\mathrm{Sub}(R)|< c\cdot |R|^{\frac{\log_2|R|}{4}}$, 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 $$\frac{|\mathrm{Sub}(R)|}{|R|^{\frac{\log_2|R|}{4}}}$$ tends to $c$.