  The Galois group of a Schubert problem encodes some structure of its set of
solutions. Galois groups are known for a few infinite families and some special
problems, but what permutation groups may appear as a Galois group of a
Schubert problem is still unknown. We expand the list of Schubert problems with
known Galois groups by fully exploring the Schubert problems on
$\text{Gr}(4,9)$, the smallest Grassmannian for which they are not currently
known. We also discover sets of Schubert conditions for any sufficiently large
Grassmannian that imply the Galois group of a Schubert problem is much smaller
than the full symmetric group.
