Are there examples of $p$-groups satisfying the minimal condition on abelian subgroups but do not satisfying the minimal condition on subgroups?

Obviously such a group cannot be locally finite.

I've already asked it on stackexchange but no answer in a few days. https://math.stackexchange.com/questions/1981177/p-group-satisfying-the-minimal-condition-on-abelian-subgroups