AMC 12 2005a Problem 25

Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\{0, 1, 2\}$. How many equilateral triangles have all their vertices in $S$?

$\textbf{(A) } 72 \textbf{(B) } 76 \textbf{(C) } 80 \textbf{(D) } 84 \textbf{(E) } 88$