2006 AIME II Problem 8

There is an unlimited supply of congruent equilateral triangles made of colored paper.
Each triangle is a solid color with the same color on both sides of the paper.
A large equilateral triangle is constructed from four of these paper triangles as shown.
Two large triangles are considered distinguishable if it is not possible to place one on the other,
using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color.
Given that there are six different colors of triangles from which to choose, how many distinguishable
large equilateral triangles can be constructed?