Cosets and Lagrange's Theorem · Hard problem

$D_{12}$, the symmetries of a regular $12$-gon, has order $24$: $12$ rotations $r^0,\dots,r^{11}$ (where $r^k$ has order $12/\gcd(12,k)$) and $12$ reflections (each of order $2$). How many elements of $D_{12}$ have order exactly $4$?