The multi-peak solitons and their stability are investigated for the nonlocal nonlinear system with the sine-oscillation response, including both the cases of positive and negative Kerr coefficients. The Hermite-Gaussian-type multi-peak solitons and the ranges of the degree of nonlocality within which the solitons exist are analytically obtained by the variational approach. This is the first time, to our knowledge at least, to discuss the solution existence range of the multi-peak solitons analytically, although approximately. The variational analytical results are confirmed by the numerical ones. The stability of the multi-peak solitons are addressed by the linear stability analysis. It is found that the upper thresholds of the peak-number of the stable solitons are five and four for the system with negative and positive Kerr coefficients, respectively.