A \(p\)-subgroup of a group \(G\) is called \(p\)-radical if it is the largest normal \(p\)-subgroup of its normalizer. These subgroups are the important ones for calculating the \(\text{mod }p\) cohomology, and as a starting point for verifying Dade’s conjectures. The present paper uses the list of maximal 3-local subgroups of Conway’s group \(Co_1\) to classify the 12 conjugacy classes of 3-radical subgroups of \(Co_1\), and the list of maximal 2-local subgroups of the Rudvalis simple group \(Rud\) to classify the 8 conjugacy classes of 2-radical subgroups of \(Rud\).