The most faces or vertices an Archimedean or Catalan solid can have is 92: the snub dodecahedron has 92 faces while its dual polyhedron, the pentagonal hexecontahedron, has 92 vertices.
On the other hand, as a simple polyhedron, the final stellation of the icosahedron has 92 vertices.
92 is the total number of objects that are permuted by the series of five finite, simple Mathieu groups
(collectively), as defined by permutations based on elements
, the largest "sporadic" finite simple group.