List of spherical symmetry groups

Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.
This article lists the groups by Schoenflies notation, Coxeter notation, orbifold notation, and order. John Conway used a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion.
Hermann–Mauguin notation is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.

Involutional symmetry

There are four infinite cyclic symmetry families, with n = 2 or higher.

There are three infinite dihedral symmetry families, with n = 2 or higher.

All of the discrete point symmetries are subgroups of certain continuous symmetries. They can be classified as products of orthogonal groups O or special orthogonal groups SO. O is a single orthogonal reflection, dihedral symmetry order 2, Dih₁. SO is just the identity. Half turns, C₂, are needed to complete.