Any Lie group containing the Poincaré group (in 4d) as a subgroup and containing a maximal internal symmetry group? must be a direct product of those. In addition, must be a semisimple group? with additional (circle group) factors.
Gel’fand and Likhtman showed that with a slight extension of the concept of Lie group, one can get that and combine in a nontrivial way. This happens for example in the supersymmetric case.
Sidney Coleman, Jeffrey Mandula, All Possible Symmetries of the S Matrix, Physical Review 159 (5): 1251–1256 (1967)
I. M. Gel'fand, E. S. Likhtman, JETP Letters 13, 323 (1971)