  We establish links between countable algebraically closed graphs and the
endomorphisms of the countable universal graph $R$. As a consequence we show
that, for any countable graph $\Gamma$, there are uncountably many maximal
subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism
group of $\Gamma$. Further structural information about End $R$ is established
including that Aut $\Gamma$ arises in uncountably many ways as a
Sch\"{u}tzenberger group. Similar results are proved for the countable
universal directed graph and the countable universal bipartite graph.
