Using the finite-size scaling renormalization group, we obtain the two-dimensional flow diagram of the Blume-Capel model for S = 1 and S = 3/2. In the first case our results are similar to those of mean-field theory, which predicts the existence of first- and second-order transitions with a tricritical point. In the second case, however, our results are different. While we obtain in the S=1 case a phase diagram presenting a multicritical point, the mean-field approach predicts only a second-order transition and a critical endpoint.