This paper studies beam training design for a codebook-based beamforming millimeter wave (mmwave) system where multiple antenna arrays are employed and each array is capable of beamforming independently. To reduce the training overhead and the complexity of subsequent beam direction search, we propose a cooperative multi-subarray beam training method. Specifically, from the perspective of excluding non-effective beam direction combinations and thus reducing search space, method and criterion of beam superposition are proposed to construct a wide beam from multiple narrow beams corresponding to multiple subarrays. Then, a cooperative multi-subarray beam training scheme is proposed based on the proposed criterion. Finally, simulation results show that the proposed scheme achieves a spectral efficiency close to that of the optimal exhaustive search scheme, while has greatly reduced training overhead and computational complexity.