We study the subspace method for system identification, and look at algorithms that rely on nuclear norm regularization for solving this problem. We introduce our own algorithm for solving the problem, based on the alternating direction method of multipliers (ADMM). Our algorithm involves an iterative minimization step, which is solved using line search methods. We demonstrate the effectiveness of our algorithm on a particular real world example, as well as two benchmark examples. In addition, we compare the computational efficiency of our algorithm to that of other existing algorithms for solving the nuclear norm system identification problem, observing that for single-input single-output systems, our algorithm is faster than an existing interior-point method. We also note that our algorithm converges the fastest when we use a gradient descent direction in the iterative minimization step of the ADMM.