The phase transitions and critical properties of the Heisenberg antiferromagnetic model on a cubic lattice with nearest and next-nearest-neighbor interactions are investigated by the replica Monte Carlo method. The range of values of the interaction of the next-nearest-neighbor is considered 0.0 ≤ r ≤ 1.0. The phase diagram relating the transition temperature and the magnitude of next-nearest neighbor interactions is constructed. It is shown that a second order phase transition occurs in the r range under study. The values of all the main static critical exponents are calculated by means of the finite-size scaling theory. It is shown that the universality class of the critical behavior of this model is preserved in the range of 0.0 ≤ r ≤ 0.4.