I know a Black hole can have angular momentum, described by the Kerr metric. It can also have an electric charge, described by the Kerr-Newmann metric. I read it cannot have a non-abelian charge. I would like to know why.
If the matter fields include nonabelian gauge fields, then black holes actually can have nonabelian charges, in violation of the no-hair theorem: https://arxiv.org/abs/gr-qc/9606008v1. But as far as I know, all known such black holes are unstable.
Thanks to Lawrence B. Crowell, I found a paper by Hawking, Perry and Strominger which explains everything. I haven't had time to write a clean answer but I will shortly, using this Soft Hair on Black holes paper. More soon.
The proof of the no-hair theorem, which says that Black Holes cannot carry any vector field, nor charge other than the electric charge relies on the asymptotic properties of spacetime, far away from the source. It was originally believed spacetime would be flat and the symmetry group would be the Poincare group. The Bondi Mezner Sacks group includes transformations known as super-translations and super-rotations. These are part of the Black Hole's hair.