  We study the tunable quantum Hall effects in a non-Abelian honeycomb optical
lattice which is a many-Dirac-points system. We find that the quantum Hall
effects present different features as change as relative strengths of several
perturbations. Namely, a gauge-field-dressed next-nearest-neighbor hopping can
induce the quantum spin Hall effect and a Zeeman field can induce a so-called
quantum anomalous valley Hall effect which includes two copies of quantum Hall
states with opposite Chern numbers and counter-propagating edge states. Our
study extends the borders of the field of quantum Hall effects in honeycomb
optical lattice when the internal valley degrees of freedom enlarge.
