Herein, we characterize the quantum phases present in spin-3/2 fermionic chains. This system may be achieved by ultracold gases loaded onto one-dimentional optical lattices in the hard-core regime. The physics of the lowest band in general is described by the Hubbard Hamiltonian that becomes a generalized Heisenberg model in this strongly-correlated regime. By performing calculations of the energy spectrum, correlation functions, level spectroscopy and chirality we analyze the Mott-insulator phases in absence and presence of an external quadratic Zeeman field.