Finite \(p\)-groups in which the number of subgroups of possible order is less than or equal to \(p^3\). (Q606365)

The \(p\)-groups with the title property are investigated. For \(p=2\) the results are not so full as for \(p\) odd. It appears, in particular, that if \(p>2\) and for all positive integers \(k\), the number of subgroups of order \(p^k\) in a \(p\)-group \(G\) is at most \(p^3\) (by Sylow, \(<p^3\)), then \(d(G)\leq 3\) (Theorem 2.5 presents the list of all such groups).