  We give a new computer-assisted proof of the classification of maximal
subgroups of the simple group ${}^2E_6(2)$ and its extensions by any subgroup
of the outer automorphism group $S_3$. This is not a new result, but no earlier
proof exists in the literature. A large part of the proof consists of a
computational analysis of subgroups generated by an element of order 2 and an
element of order 3. This method can be effectively automated, and via
statistical analysis also provides a sanity check on results that may have been
obtained by delicate theoretical arguments.
