Solids with icosahedral symmetry

Solids with full icosahedral symmetry
Platonic solids - regular polyhedra (all faces of the same type)

Archimedean solids - polyhedra with more than one polygon face type.

Catalan solids - duals of the Archimedean solids.

Chiral Archimedean and Catalan solids
Archimedean solids:

Catalan solids: