Working out cyclic subgroups of $D_{8}$ [duplicate]

First of all, I think your way of finding the subgroups of $D_4$ by considering Lagrange's Theorem and orders of the subgroups is systematic enough (at least for groups of smaller orders, this way can handle the problem well enough).

HINT: Now, as for the subgroups isomorphic to $V_4$, note that we can write generators and relation of $V_4$ as well, as you did for $D_4$:

$$V_4 = \langle a,b\mid a^2 = b^2 = e, \ ab = ba \rangle$$

This tells us that $V_4$ is generated by two elements of order $2$, which commute. Then, we can seek for such elements in $D_4$ among the elements of order $2$, namely $r^2, s, sr, sr^2, sr^3$ in order to find the subgroups isomorphic to $V_4$.