On the maximum number of subgroups of a finite group

Given a finite group $R$, we let $\mathrm{S}\mathrm{u}\mathrm{b}(R)$ denote the collection of all subgroups of $R$. We show that $\mathrm{\mid }\mathrm{S}\mathrm{u}\mathrm{b}(R)\mathrm{\mid }<c\cdot \mathrm{\mid }R{\mathrm{\mid }}^{\frac{{\log }_2\mathrm{\mid }R\mathrm{\mid }}{4}}$, where $c<7.372$ is an explicit absolute constant. This result is asymptotically best possible. Indeed, as $\mathrm{\mid }R\mathrm{\mid }$ tends to infinity and $R$ is an elementary abelian $2$-group, the ratio $$\frac{\mathrm{\mid }\mathrm{S}\mathrm{u}\mathrm{b}(R)\mathrm{\mid }}{\mathrm{\mid }R{\mathrm{\mid }}^{\frac{{\log }_2\mathrm{\mid }R\mathrm{\mid }}{4}}}$$ tends to $c$.