In this paper, the magnetic and thermodynamic behaviors of a mixed-spin ferrimagnetic cubic nanoisland, consisting of a core with spin-1 atoms surrounded by shell with spin-1 and spin-3/2 atoms in the external magnetic field have been investigated. We have used the Monte Carlo simulation to examine the effects of the crystal-field, the exchange coupling and the external magnetic field on the magnetization, susceptibility, internal energy and blocking temperature of the nanoisland. Depending on different values of the magnetic parameters, we have found various types of magnetization curves. In particular, multiple hysteresis loop behaviors such as double and triple hysteresis loops have been discovered for certain parameters.