A question in a section on external direct products in Gallian’s 9th edition Contemporary Abstract Algebra asks for an order $12$ cyclic subgroup and an order $12$ non-cyclic subgroup from $\mathrm{Z}_{40}\oplus\mathrm{Z}_{30}$. Finding the cyclic subgroup was easy, but I don’t know where to start with finding the non-cyclic one. I tried including elements of the form $(a,0)$ and $(0,b)$ in order to generate other potential elements for such a subgroup, but I didn’t succeed.
As a matter of fact, I always end up getting stuck in these questions asking for non-cyclic subgroups of groups. Sometimes computing a subgroup of the form $\langle x,y \rangle = \{x^sy^t \mid s,t \in \mathrm{Z}\}$ works, but what is a good strategy? Filtering out elements with tools like Lagrange’s Theorem?
Any help would be appreciated.