The number of isomorphism classes of groups of order $p^n$ grows so fast $\big (p^{{\frac{2}{27}}n^{3}+O(n^{8/3})} \big)$, that a folklore conjecture asserts that, asymptotically, almost every finite group is a 2-group.

In other words, classifying $p$-groups is a tall order; although special families ought to be appealing.

Are there interesting classes of $p$-groups where the pertinent property is a (specific?) tower of $p$-subgroups of

everyintermediate power $(1,p,p^2,\ldots,p^n)$?