  We characterise the primitive 2-closed groups $G$ of rank at most four that
are not the automorphism group of a graph or digraph and show that if the
degree is at least 2402 then there are just two infinite families or
$G\leqslant \mathrm{A}\Gamma\mathrm{L}_1(p^d)$, the 1-dimensional affine
semilinear group. These are the first known examples of non-regular 2-closed
groups that are not the automorphism group of a graph or digraph.
