Abstract. This note was inspired by A. Mann's letter  at June 28, 2009, in which the number of subgroups of given order in a metacyclic p-group for odd primes p was computed. Below we present another proof of that result. The offered proof is extended to so called quasi-regular metacyclic 2-groups. In Sec. 2 we compute the number of cyclic subgroups of given order in metacyclic 2-groups. In Sec. 3 we complete computation of the number of subgroups of given order in metacyclic 2-groups. In Sec. 4 we study the metacyclic p-groups with small minimal nonabelian subgroups or sections.
2000 Mathematics Subject Classification. 20D15.
Key words and phrases. Metacyclic p-groups, quasi-regular metacyclic p-groups, section, Hall's enumeration principle.