A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of groups are known to be realizable?
Shafarevich proved that solvable groups are realizable. Thus the following results "nontrivially" imply realizability:
If $n$ is odd, $G$ is solvable.
If $n$ has prime factorization $n=\prod p_i$, then $G$ is solvable.
If $n$ is of the form $p^\alpha q^\beta$ where $p,q$ are prime and $\alpha,\beta \geq 0$, $G$ is solvable.
It is worthwhile to point out that:
If $G_1,\ldots,G_m$ are solvable then so is $G_1 \times \cdots \times G_m$.
What other classes are known to be realizable? I've googled on it but can't find a big list or whatever.
(The smallest group I've found through Groupprops not to be immediately identifiable as realizable is $\mathrm{SL}(2,\mathbb{F}_5)$.)
Edit: If no one objects I'll continuously update this question whenever I find something interesting.