  We study various Mott insulating phases of interacting spin-3/2 fermionic
ultracold atoms in two-dimensional square optical lattices at half filling.
Using a generalized one-band Hubbard model with hidden SO(5) symmetry, we
identify two distinct symmetry breaking phases: the degenerate
antiferromagnetic spin-dipole/spin-octupole ordering and spin-quadrupole
ordering, depending on the sign of the spin-dependent interaction. These two
competing orders exhibit very different symmetry properties, low energy
excitations and topological characterizations. Near the SU(4) symmetric point,
a quantum critical state with a $\pi $-flux phase may emerge due to strong
quantum fluctuations, leading to spin algebraic correlations and gapless
excitations.
