Subgroups of C4 times C2 ($\#\{H:H\le C_4\times C_2\}=8$)

With the generators $c,a$ of the direct product of cyclic groups of orders four and two, the subgroups are the trivial group, the whole group, the three order-two groups $⟨c^2⟩$, $⟨a⟩$, $⟨c^{2}a⟩$, and the three order-four groups $⟨c⟩$, $⟨ca⟩$, $⟨c^2,a⟩$. These are exhaustive by Lagrange's theorem: a proper nontrivial subgroup has order two or four; an order-four subgroup either contains an element of order four and is one of the two cyclic examples, or is precisely the four-element set killed by squaring. The latter is a Klein four-group.