  We study six-dimensional N=(1,0) supergravity theories with abelian, as well
as non-abelian, gauge group factors. We show that for theories with fewer than
nine tensor multiplets, the number of possible combinations of gauge groups -
including abelian factors - and non-abelian matter representations is finite.
We also identify infinite families of theories with distinct U(1) charges that
cannot be ruled out using known quantum consistency conditions, though only a
finite subset of these can arise from known string constructions.
