To overcome the issues of high memory consumption, long computation time, and limited accuracy in point cloud registration algorithms for graph space maximal clique (MAC) analysis, a registration algorithm based on angle constraints and MACs is proposed. First, existing algorithms are used to obtain initial matching point pairs from the point cloud, and a first-order spatial compatibility graph is constructed using distance constraints between points. Thereafter, a second-order spatial compatibility graph is constructed under the constraint of three-point spatial angles. Next, MACs are searched for in the second-order graph and sorted and filtered using a priority queue. Finally, weighted singular value decomposition is used to calculate and evaluate the optimal rotation-translation transformation for the largest cliques in the priority queue. The proposed algorithm effectively eliminates incorrect matching point pairs by introducing spatial three-point angle constraints and a MAC priority queue, thereby reducing computation time, optimizing memory consumption, and improving registration accuracy. The registration experiments conducted on the KITTI, 3DMatch, and 3DLoMatch open-source datasets demonstrate that the proposed algorithm achieves excellent registration accuracy compared with other typical point-based registration algorithms. Moreover, the proposed algorithm outperforms the MAC algorithm based on graph space analysis with respect to registration time and memory consumption. In addition, the effectiveness of spatial three-point angle constraints for improving registration accuracy and reducing map size is further verified through ablation experiments.