The behaviour of $F(n)$ varies dramatically with the prime-factorization of $n$. Typically one gets much larger values for $F(n)$ when $n$ is the power of a prime, particularly when that prime is equal to $2$. The key conjecture in this area (I believe) is due to Higman and Sims:
Conjecture Let $p$ be a (fixed) prime number. Define $f(n,p)$ as the number of groups of order $p^n$. Then: $$f(n,p) = p^{(2/27 + o(1))n^3}.$$
(The link above gives a more detailed version of this conjecture.)