The normal subgroup structure of maximal pro-"p"-subgroups of rational points of algebraic groups over the "p"-adics and their characteristic "p" analogues are investigated. These groups have finite width, i.e. the indices of the sucessive terms of the lower central series are bounded since they become periodic. The richness of the lattice of normal subgroups is studied by the notion of obliquity. All just infinite maximal groups with Lie algebras up to dimension 14 and most Chevalley groups and classical groups in characteristic 0 and "p" are covered. The methods use computers in small cases and are purely theoretical for the infinite series using root systems or orders with involutions.