In this letter, we propose a novel hybrid precoding architecture for millimeter wave communications named fully-connected subarrays (FC-SA), wherein each radio frequency (RF) chain is connected to all subarrays through a FC phase shifter (PS) network, and each subarray contains several antennas each with an individual PS. Since PSs inside subarrays are shared by all RF chains, the beamforming gain of the entire array is utilized with much lower hardware complexity than the conventional FC structure. Moreover, we propose a hybrid alternating minimization precoding algorithm to handle the additional rank-one constraint on analog precoder. Performance comparisons between the proposed FC-SA architecture and existing FC and partially connected structures are presented in terms of spectral efficiency (SE) and energy efficiency (EE). Simulation results demonstrate that the proposed scheme approaches the SE of FC structure and achieves the highest EE with a moderate number of RF chain.