We consider latin square graphs Γ=LSG(H) of the Cayley table of a given finite group H. We characterize all pairs (Γ,G), where G is a subgroup of autoparatopisms of the Cayley table of H such that G acts arc-transitively on Γ and all nontrivial G-normal quotient graphs of Γ are complete. We show that H must be elementary abelian and determine the number k of complete normal quotients. This yields new infinite families of diameter two arc-transitive graphs with k=1 or k=2.