On the Automorphism Group of the Substructure Ordering of Finite Directed Graphs

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 $({\mathbb{Z}}_2^4\times S_4){⋊}_{\alpha }{\mathbb{Z}}_2$. 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.