Let K be an imaginary quadratic field, A(K) its pclass group, and H(K) its pHilbert class field. If rk(A(K))=2, a result due to Arrigoni tells us that p^3 divides the order of the class group of H(K). Are there any explicit nontrivial lower bounds in the case that rk(A(K))>2 ?
If $G$ is the Galois group of the $p$class field tower over $K$, then $A(H(K))=G'/G''$ is a quotient of $G_2/G_4$, where $G_i$ denotes the lower central series. By Arrigoni's calculation that $G_2/G_4$ has $p$rank exactly $\frac{d(d1)(2d+5)}{6}$, this serves as a lower bound for the $p$rank of $A(H(K))$. When $d=2$, you get the bound of $3$ you mention in the question. Note that the calculation is actually much more precise: The size of $A(H(K))$ depends not only on the rank, but on the orders of the generators of the $p$class group. This will give you a better bound for the class number than simply raising $p$ to the rank bound given above. 

