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The subclasses of polyhedron include icosahedron, prism, square pyramid, Triangular dipyramid and Pentagonal dipyramid.

prism

prism (a polyhedron with two congruent and parallel faces (the bases) and whose lateral faces are parallelograms)

Triangular dipyramid

triangular dipyramid, a first in the infinite set of face-transitive dipyramids

Pentagonal dipyramid

pentagonal dipyramid, third of the infinite set of face-transitive dipyramids

Net

arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron

Deltoidal icositetrahedron

deltoidal icositetrahedron, a Catalan solid which looks a bit like an overinflated cube

Cell

cell, a three-dimensional element that is part of a higher-dimensional object

Disdyakis dodecahedron

disdyakis dodecahedron, or hexakis octahedron, a Catalan solid and the dual to the Archimedean truncated cuboctahedron

Deltoidal hexecontahedron

deltoidal hexecontahedron, a catalan solid which looks a bit like an overinflated dodecahedron

Disdyakis triacontahedron

disdyakis triacontahedron, or hexakis icosahedron, a Catalan solid and the dual to the Archimedean truncated icosidodecahedron

Pentagonal icositetrahedron

pentagonal icositetrahedron, a catalan solid which is the dual of the snub cube

gyroelongated pentagonal pyramid

gyroelongated pentagonal pyramid, one of the Johnson solids

Pentagonal pyramid

pentagonal pyramid, a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point

Gyroelongated square dipyramid

gyroelongated square dipyramid, one of the Johnson solids

Elongated square dipyramid

elongated square dipyramid, one of the Johnson solids

Pentagonal hexecontahedron

pentagonal hexecontahedron, a Catalan solid, dual of the snub dodecahedron

Heronian tetrahedron

Heronian tetrahedron, a tetrahedron whose sides, faces and volume, all rational numbers

Augmented truncated tetrahedron

augmented truncated tetrahedron, one of the Johnson solids

Parabigyrate rhombicosidodecahedron

parabigyrate rhombicosidodecahedron, one of the Johnson solids

Gyrate bidiminished rhombicosidodecahedron

gyrate bidiminished rhombicosidodecahedron, one of the Johnson solids

metabigyrate rhombicosidodecahedron

metabigyrate rhombicosidodecahedron, one of the Johnson solids

gyroelongated triangular cupola

gyroelongated triangular cupola, one of the Johnson solids

tridiminished rhombicosidodecahedron

Tridiminished rhombicosidodecahedron, one of the Johnson solids

Parabiaugmented hexagonal prism

parabiaugmented hexagonal prism, one of the Johnson solids

Metabiaugmented hexagonal prism

metabiaugmented hexagonal prism, one of the Johnson solids

Gyroelongated pentagonal rotunda

gyroelongated pentagonal rotunda, one of the Johnson solids

Bigyrate diminished rhombicosidodecahedron

bigyrate diminished rhombicosidodecahedron, one of the Johnson solids

gyroelongated pentagonal bicupola

gyroelongated pentagonal bicupola, one of the Johnson solids

elongated triangular orthobicupola

elongated triangular orthobicupola, one of the Johnson solids

elongated triangular gyrobicupola

elongated triangular gyrobicupola, one of the Johnson solids

Gyroelongated triangular bicupola

gyroelongated triangular bicupola, one of the Johnson solids

elongated pentagonal gyrocupolarotunda

elongated pentagonal gyrocupolarotunda, one of the Johnson solids

elongated pentagonal orthocupolarotunda

elongated pentagonal orthocupolarotunda, one of the Johnson solids

Trigyrate rhombicosidodecahedron

Trigyrate rhombicosidodecahedron, one of the Johnson solids

Parabidiminished rhombicosidodecahedron

Parabidiminished rhombicosidodecahedron, one of the Johnson solids

Augmented sphenocorona

Augmented sphenocorona, one of the Johnson solids, and is obtained by adding a square pyramid to one of the square faces of the sphenocorona

gyroelongated pentagonal birotunda

gyroelongated pentagonal birotunda, one of the Johnson solids

elongated pentagonal orthobirotunda

elongated pentagonal orthobirotunda, one of the Johnson solids

Elongated pentagonal gyrobirotunda

elongated pentagonal gyrobirotunda, one of the Johnson solids

Augmented tridiminished icosahedron

augmented tridiminished icosahedron, one of the Johnson solids

Gyroelongated pentagonal cupola

gyroelongated pentagonal cupola, one of the Johnson solids

Elongated pentagonal gyrobicupola

elongated pentagonal gyrobicupola, one of the Johnson solids

Elongated pentagonal dipyramid

elongated pentagonal dipyramid, one of the Johnson solids

Paragyrate diminished rhombicosidodecahedron

paragyrate diminished rhombicosidodecahedron, one of the Johnson solids

Elongated triangular dipyramid

elongated triangular dipyramid, one of the Johnson solids

elongated pentagonal orthobicupola

elongated pentagonal orthobicupola, one of the Johnson solids

Metabidiminished rhombicosidodecahedron

Metabidiminished rhombicosidodecahedron, one of the Johnson solids

gyroelongated pentagonal cupolarotunda

gyroelongated pentagonal cupolarotunda, one of the Johnson solids