AMC 12 2017a Problem 17

There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number?

\textbf{(A)}\ 0 \textbf{(B)}\ 4 \textbf{(C)}\ 6 \textbf{(D)}\ 12 \textbf{(E)}\ 24