We investigate the new quantum phases on the extended Kane-Mele-Hubbard model of honeycomb lattice in the Hofstadter regime. In this regime, the orbital motion of the electrons can induce various topological phases with spontaneously broken symmetries when the spin-orbit coupling and electron correlations coexist. Here, we consider the interaction effects in the Kane-Mele model and discuss possible phases in the presence of magnetic field at integer fillings of electrons. In particular, focusing on the 2 pi/3 magnetic flux per plaquette, the realization of numerous quantum phases are discussed within the mean-field framework: insulator with coplanar magnetic ordering, ferrimagnetic Chern insulator with nematic charge order, ferrimagnetic-ferrielectric Chern insulators, etc. Many of these phase transitions are also accompanied with a change in the topological invariants of the system. Based on our theoretical study, we propose topological multiferroic phases with a scope of realization in 2D van-der Waals materials and optical lattice systems where a significant interplay of magnetic field, spin-orbit coupling, and interactions can be engineered.