  We investigate the automorphism group of the substructure ordering of finite
directed graphs. The second author conjectured that it is isomorphic to the
768-element group $(\mathbb{Z}_2^4 \times S_4)\rtimes_{\alpha} \mathbb{Z}_2$.
Though unable to prove it, we solidify this conjecture by showing that the
automorphism group behaves as expected by the conjecture on the first few
levels of the poset in question. With the use of computer calculation we
analyze the first four levels holding 3160 directed graphs.
