  We show that there is a binary subspace code of constant dimension 3 in
ambient dimension 7, having minimum distance 4 and cardinality 333, i.e., $333
\le A_2(7,4;3)$, which improves the previous best known lower bound of 329.
Moreover, if a code with these parameters has at least 333 elements, its
automorphism group is in one of $31$ conjugacy classes. This is achieved by a
more general technique for an exhaustive search in a finite group that does not
depend on the enumeration of all subgroups.
