  We study global symmetry groups of six-dimensional superconformal field
theories (SCFTs). In the Coulomb branch we use field theoretical arguments to
predict an upper bound for the global symmetry of the SCFT. We then analyze
global symmetry groups of F-theory constructions of SCFTs with a
one-dimensional Coulomb branch. While in the vast majority of cases, all of the
global symmetries allowed by our Coulomb branch analysis can be realized in
F-theory, in a handful of cases we find that F-theory models fail to realize
the full symmetry of the theory on the Coulomb branch. In one particularly
mysterious case, F-theory models realize several distinct maximal subgroups of
the predicted group, but not the predicted group itself.
