A subspace code of size 333 in the setting of a binary q-analog of the Fano plane

We show that there is a binary subspace code of constant dimension 3 in ambient dimension 7, having minimum subspace distance 4 and cardinality 333, i.e., \begin{document}$ 333 \le A_2(7, 4;3) $\end{document} , 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.