Find all characteristic subgroups of the Dihedral group $D_{12}$.

Question

In my notation $$D_{12}=\langle \rho,\tau : \rho^6=\tau^2=1,\ \rho \tau \rho=\tau\rangle$$ So firstly, I know that all characteristic subgroups are normal. Thus, the possible candidates of $D_{2n}$ for the characteristic subgroups are $$\{1\},D_{12},<\rho>,<\rho^2>,<\rho^3>,<\rho^2,\tau>,<\rho^2,\rho\tau>$$ Is there now a property or tool which I can use to pick out the characteristic subgroups. I know for sure that $D_{12}$ and $\{1\}$ as well as the commutator subgroup $D_{12}'=<\rho>$ are characteristics, but what about the others.

Answer 1

For starters, consider the orders of the subgroups. If there is only one subgroup of a certain order, surely it must be characteristic. Of the three remaining subgroups, only one is cyclic, and so it must also be characteristic. For the remaining two groups, it is not hard to check whether there is an automorphism mapping one to the other.

Answer 2

As for the remaining $2$ subgroups, $\langle \rho^2,\tau\rangle$ and $\langle \rho^2,\rho\tau\rangle$ there is an automorphism of $D_{12}$ sending them to each other... Namely, the one determined by $\rho\to\rho$ and $\tau\to\rho\tau$. (By the relation $\rho\tau\rho=\tau$, the elements $\tau$ and $\rho\tau$ both have order $2$; or, they are both reflections.)

So, these two subgroups aren't characteristic...