The clusters of young stars in massive star-forming regions show a wide range of sizes, morphologies, and numbers of stars. Their highly subclustered structures are revealed by the MYStIX project's sample of 31754 young stars in nearby sites of star formation (regions at distances <3.6kpc that contain at least one O-type star.) In 17 of the regions surveyed by MYStIX, we identify subclusters of young stars using finite mixture models --collections of isothermal ellipsoids that model individual subclusters. Maximum likelihood estimation is used to estimate the model parameters, and the Akaike Information Criterion is used to determine the number of subclusters. This procedure often successfully finds famous subclusters, such as the BN/KL complex behind the Orion Nebula Cluster and the KW-object complex in M17. A catalog of 142 subclusters is presented, with 1-20 subclusters per region. The subcluster core radius distribution for this sample is peaked at 0.17pc with a standard deviation of 0.43dex, and subcluster core radius is negatively correlated with gas/dust absorption of the stars --a possible age effect. Based on the morphological arrangements of subclusters, we identify four classes of spatial structure: long chains of subclusters, clumpy structures, isolated clusters with a core-halo structure, and isolated clusters well fit by a single isothermal ellipsoid.