4T4 $A_4$ $12$ $1$ yes 6T4, 12T4
6T4 $A_4$ $12$ $1$ yes $C_3$ 4T4, 12T4
12T4 $A_4$ $12$ $1$ yes $C_3$, $A_4$, $A_4$ 4T4, 6T4
Results are complete for degrees $\leq 23$.