The question is in the title. Variants have been asked here before (e.g. Which small finite simple groups are not yet known to be Galois groups over Q? specifically for simple groups), but I explicitly don't want to restrict to simple groups, since this let's one get to rather lofty orders all while dodging the true difficulty of solving embedding problems. Of course I'm aware that results in the literature are usually not custom-made to answer these kind of "trivia" questions, so here are some thoughts.

My own guess is that the answer may be as small as $SL_2(13)$ (which has order $2184$), so that I'm essentially asking two types of sub-questions:

1. Is there any quotable reference or reasonably short argument for the claim that all groups of order smaller than $2184$ have been realized? (Or alternatively, am I in fact missing some smaller open case?)

Since a counterexample would have to be nonsolvable involving a very small nonsolvable composition factor (concretely, $PSL_2(q)$ with $q\le 11$), I guess I might be able to piece this together from various results in the literature about solvability of central embedding problems, split embedding problems, and embedding problems with (certain) nonabelian simple kernel, although I haven't checked in full and the full list of relevant groups is not that small after all. (A short argument might be of the form "every embedding problem is either of this or that type, and these types are known to be solvable when only the couple of smallest simple groups are involved.")

2. Is $SL_2(13)$ indeed an open case? (And if not, what is the smallest open $SL_2(p)$?)

The reason why $SL_2(p)$ looks difficult is that, in order to solve the embedding problem from $PSL_2(p)$, one necessarily needs to begin with a (suitable) totally real $PSL_2(p)$-extension, which doesn't go too well with the ``classical" realizations a la Shih from rigidity theory (although $p=7$ and $11$ have been solved). I'm not too familiar with the more recent realizations of $PSL_2(p)$ (for all primes $p$) by Zywina, although I would have thought they don't yield totally real extensions either. In short, is there either a recent reference (say, from the last decade) stating that $p=13$ (or similar) is open, or otherwise does someone know a realization?