  We study the eigenstate phases of disordered spin chains with on-site finite
non-Abelian symmetry. We develop a general formalism based on standard group
theory to construct local spin Hamiltonians invariant under any on-site
symmetry. We then specialize to the case of the simplest non-Abelian group,
$S_3$, and numerically study a particular two parameter spin-1 Hamiltonian. We
observe a thermal phase and a many-body localized phase with a spontaneous
symmetry breaking (SSB) from $S_3$ to $\mathbb{Z}_3$ in our model Hamiltonian.
We diagnose these phases using full entanglement distributions and level
statistics. We also use a spin-glass diagnostic specialized to detect
spontaneous breaking of the $S_3$ symmetry down to $\mathbb{Z}_3$. Our observed
phases are consistent with the possibilities outlined by Potter and Vasseur
[Phys. Rev. B 94, 224206 (2016)], namely thermal/ ergodic and spin-glass
many-body localized (MBL) phases. We also speculate about the nature of an
intermediate region between the thermal and MBL+SSB regions where full $S_3$
symmetry exists.
