Embeddings ofU 3(8),Sz(8) and the Rudvalis group in algebraic groups of typeE 7

We prove that the nearly simple finite group U-3(8):6 is in the adjoint group of type E(7) over any algebraically closed field. We also prove that the group Sz(8) is embeddable in E(7)(K), for an algebraically closed field K, iff char(K)=2, 5; in characteristic 5, we prove that there is such an embedding which extends to an embedding of the slightly larger group Sz(8):3 and to an embedding of the sporadic simple group of Rudvalis. Our methods include p-local analysis in groups of Lie type, nonabelian cohomology, representation theory of finite groups and computer calculations.