TITLE: On the inverse Galois problem
QUESTION [18 upvotes]: Q: What is the "simplest"  finite group $G$ for which we don't know how to realise it as a Galois group over $\mathbf{Q}$ ?
So here the word simplest might be interpreted in a broad sense. If you want something precise
you may take the group of smallest order but I prefer to leave the question as it is. 
Also since naturally one classifies finite groups into families one may also ask the following
Q: What is the "simplest"  example of a family of finite groups for which the inverse 
Galois problem is unknown?

REPLY [5 votes]: I am not an expert and I might be misrembering some talks I've attended. Anyway, $SL_2(\mathbb{F}_q)$ can be done for prime $q$ by using torsion on non-CM elliptic curves. But I don't think it's been done for general prime powers $q$. Also, what about $SL_3$?