gr.group theory - The fraction of groups with normal Hall subgroups

Dear all,

I am a student on computer science. So please forgive me if I state some results in a weird way, and I hope I ask an interesting question to you.

The question is related to finite groups with normal Hall subgroups. I want to know for groups of size $n$, what is the fraction of groups with normal Hall subgroups compared to all groups, up to isomorphism.

For example, we know that for given s, the number of non-isomorphic groups of size $n$ is bounded by $n^{O((log n)^2)}$. While I can prove that for certain class of groups with normal Hall subgroup, for a given n, the number of non-isomorphic groups of size n can be $n^{Omega(log n)}$. But I would like to know an upper bound.

Thank you very much.

Jimmy