For $n\in \mathbb{Z}^+$ and $k < n,$ let $R_k = \begin{bmatrix}\cos (\theta_k) & -\sin(\theta_k)\\ \sin(\theta_k) & \cos(\theta_k)\end{bmatrix}$ and $F_k = \begin{bmatrix}\cos (\theta_k) & \sin(\theta_k)\\ \sin(\theta_k) & -\cos(\theta_k)\end{bmatrix},$ where $\theta_k = \frac{2\pi k}n.$ Find all subgroups of the dihedral group $D_n = \{R_0,\cdots, R_{n-1}, F_0,\cdots, F_{n-1}\}.$
I know that by Lagrange's Theorem, the order of a subgroup must divide the order of a group. Since $|D_n| = 2n,$ any subgroup must have an order that's a divisor of $2n.$ But I don't think this helps much. I also know the following properties: for $a,b \in \mathbb{Z}_n$,
$$\begin{align} R_a R_b& = R_{a+b},\\ F_aF_b &= R_{a-b}, \\ R_a F_b &= F_{b+a},\\ F_a R_b &= F_{a-b}. \end{align}$$
Every subgroup contains the identity element of the group and for any elements $a,b$ in a subgroup $S, ab^{-1} \in S.$ I also know the subgroups of a cyclic group $G = \langle a\rangle , |G| = n$ are the subgroups $\langle a^d\rangle, d \mid n,$ but I'm not sure if this is useful.