= Subgroups of C4 times C2
{title2=$\#\{H:H\leq C_4\times C_2\}=8$}
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 $\langle c^2\rangle$, $\langle a\rangle$, $\langle c^2a\rangle$, and the three order-four groups $\langle c\rangle$, $\langle ca\rangle$, $\langle c^2,a\rangle$. 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>.
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