Find a subgroup of index 3 of dihedral group $D_{12}$

Question

Find a subgroup of index 3 in the dihedral group $D_{12}$. I know the number of elements in $D_{12}$ is 24 and also that is we have this subgroup of index 3, then we obtain that $|D_{12}:H|=8$, where $H$ is our wanted subgroup, but I don`t know how to go further. I am new to this type of problems and I do not have many examples, could you provide a full proof, or at least in the form of an answer, such that it would serve as a model for similar problems I encounter? Thank you very much!!!

Answer 1

$D_{12}$ is generated by $a,b$ with $a^2=b^{12}=1, aba=b^{-1}$. Then $b^3$ generates a subgroup $A$ of order $12/3=4$. That subgroup is normal in $D_{12}$ because $ab^3a=b^{-3}=b^9=(b^3)^3$. Then $a$ and $A$ generate a subgroup $H$ of order 8. The index of that subgroup is then 3.