We study the class of all finite directed graphs (digraphs) up to primitive positive constructibility. The resulting order has a unique maximal element, namely the digraph P₁ with one vertex and no edges. The digraph P₁ has a unique maximal lower bound, namely the digraph P₂ with two vertices and one directed edge. Our main result is a complete description of the maximal lower bounds of P₂; we call these digraphs submaximal. We show that every digraph that is not equivalent to P₁ and P₂ is below one of the submaximal digraphs.