Question: How many subgroups of order 6 does $D_6$ have? How many does $D_{12}$ have? Generalize to $D_n$ where n is a positive integer divisible by 6. Here, order of $D_n = 2n$. Please don't use normal subgroup & cosets because those two topics is from the next chapters. It's from "Contemporary Abstract Algebra TENTH EDITION Joseph A. Gallian", problem 40 of chapter 3. 'Finite Groups; Subgroups'.
My attempt: For $D_6$ I can find two subgroups of order 6, $S_1$ and $S_2$.
$ S_1 $ = {$ {R_0, R_{60}, R_{120}, R_{180}, R_{240}, R_{300}} $} and
$S_2$ = {${R_0, R_{120}, R_{240}, F, FR_{120}, FR_{240}}$}.
Is there any more ? Note: F stands for flip/reflection.
For $D_{12}$, I can find the same two subgroups $S_1$ and $S_2$, as in for $D_6$. Is there any more subgroup ?
What about the generalization?