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Recent results of Qu and Tuarnauceanu describe explicitly the finite p-groupswhich are not elementary abelian and have the property that the number of theirsubgroups is maximal among p-groups of a given order. We complement theseresults from the bottom level up by determining completely the non-cyclicfinite p-groups whose number of subgroups among p-groups of a given order isminimal.

Finite non-cyclic $p$-groups whose number of subgroups is minimal