The determination of $gnu(n)$, the number of groups of order $n$ upto isomorphism, is very hard in general.
But if no large powers are involved, it should be possible for relatively large numbers.
Upto which $p$ can $gnu(2^4\cdot 3^4\cdot...\cdot p^4)$ be calculated in a reasonable time?
The first two values are :
gap> NrSmallGroups(2^4);
14
gap> NrSmallGroups(2^4*3^4);
3609
I currently run GAP for $gnu(2^4\cdot 3^4\cdot 5^4)$
A lower bound for this gnu is :
gap> NrSmallGroups(2^4*3^4)*NrSmallGroups(5^4);
54135