A regular dodecahedron has 120 symmetries. How to see this? Take each pentagonal face and chop into 10 triangles as shown here. Since there are 12 pentagonal faces, that's a total of 10 x 12 = 120 triangles. If you pick one triangle as your favorite, there's exactly one symmetry carrying it to any other triangle (or itself) - so there are 120 symmetries!

If your favorite triangle was black, any rotation of the dodecahedron will carry it to a black triangle, while any reflection will carry it to a blue one. So, there are just 60 rotational symmetries of the dodecahedron.

This trick of subdividing the faces into triangles also works to count the symmetries of the other Platonic solids. It also works for higher-dimensional Platonic solids, but instead of triangles we must use 'simplexes': the higher-dimensional relatives of triangles, for example tetrahedra. The subdivided shape we get this way is called a Coxeter complex:

http://en.wikipedia.org/wiki/Coxeter_complex

and the symmetry groups we can study this way are called Coxeter groups:

http://en.wikipedia.org/wiki/Coxeter_group

The Coxeter complex below is drawn on the surface of a round sphere, instead of literally taking a dodecahedon and chopping the pentagons into triangles. That doesn't matter too much, but it's sort of nice.

Puzzle: what does the Coxeter complex of the icosahedron look like?