This is Exercise 8.28 of Gallian's "Contemporary Abstract Algebra".
Answers that use only methods from the textbook prior to the exercise are preferred.
Here $G_1\oplus G_2$ is the external direct product of $G_1$ by $G_2$.
Here $G_1\le G_2$ means $G_1$ is a subgroup or equal to the group $G_2$.
Find a subgroup of $\Bbb Z_4\oplus\Bbb Z_2$ not of the form $H\oplus K$ for some $H\le \Bbb Z_4, K\le \Bbb Z_2$.
I must confess: I cheated here a little bit by looking up the subgroups of $\Bbb Z_4\times \Bbb Z_2$. But notice the difference in notation! I think in terms of just plain old direct products (because aren't internal and external direct products equivalent? Yes! But this is not established in the textbook yet; indeed, the former is not even mentioned at this point).
It appears to me to be a trick question. Here are the subgroups of $\Bbb Z_4\times \Bbb Z_2$. Where is the subgroup of the desired form?
My guess is that there's some technical aspect of external direct products that is being emphasised here.
Please help :)