  We explore thermalization and quantum dynamics in a one-dimensional
disordered SU(2)-symmetric Floquet model, where a many-body localized phase is
prohibited by the non-abelian symmetry. Despite the absence of localization, we
find an extended nonergodic regime at strong disorder where the system exhibits
nonthermal behaviors. In the strong disorder regime, the level spacing
statistics exhibit neither a Wigner-Dyson nor a Poisson distribution, and the
spectral form factor does not show a linear-in-time growth at early times
characteristic of random matrix theory. The average entanglement entropy of the
Floquet eigenstates is subthermal, although violating an area-law scaling with
system sizes. We further compute the expectation value of local observables and
find strong deviations from the eigenstate thermalization hypothesis. The
infinite temperature spin autocorrelation function decays at long times as
$t^{-\beta}$ with $\beta < 0.5$, indicating subdiffusive transport at strong
disorders.
