We propose an approach to multi-modal grasp detection that jointly predicts the probabilities that several types of grasps succeed at a given grasp pose. Given a partial point cloud of a scene, the algorithm proposes a set of feasible grasp candidates, then estimates the probabilities that a grasp of each type would succeed at each candidate pose. Predicting grasp success probabilities directly from point clouds makes our approach agnostic to the number and placement of depth sensors at execution time. We evaluate our system both in simulation and on a real robot with a Robotiq 3-Finger Adaptive Gripper and compare our network against several baselines that perform fewer types of grasps. Our experiments show that a system that explicitly models grasp type achieves an object retrieval rate 8.5% higher in a complex cluttered environment than our highest-performing baseline.