So I know that the primitve roots of unity are given by $$\xi_n = \left[1, \frac{2\pi}{n} \right] = \cos\left(\frac{2\pi}{n}\right)+i \sin\left(\frac{2\pi}{n}\right).$$
Let $\xi_4=\left[1,\frac{\pi}{2} \right]=i$ be the primitive fourth root of unity and $\xi_8=\left[1, \frac{\pi}{4} \right]$ the primitive eighth root of unity.
I need to find the subgroups of $(\mathbb{C} \setminus\{0\},\cdot)$ generated by $\xi_4$ resp $\xi_8$ and the cosets of $C_4$ in $C_8$.
For the elements I had the idea that they would be given by the $n$-th unit roots
$${\xi_n}^{j} = \left[1, \frac{2\pi j}{n} \right], j \in \{0,1,...,n-1\}$$
So if I am right then the elements in $C_4$ would be $\{\xi^{0}_{4},\xi^{1}_{4}, \xi^{2}_{4}, \xi^{3}_{4} \}$ and for $C_8=\{\xi^{0}_{8},\xi^{1}_{8}, \xi^{2}_{8}, \xi^{3}_{8}, \xi^{4}_{8}, \xi^{5}_{8}, \xi^{6}_{8}, \xi^{7}_{8} \}$ (still not 100% sure about this).
However writing down the elements makes it clear that $C_4 \subseteq C_8$. What I still struggle with are the cosets, are the left and right cosets equal here? and how can I determine them?