What are the three subgroups of $\mathbb{Z}_4\times\mathbb{Z}_6$ of 12 elements? My formatting didn't work in the title, here is the question again:
What are the three subgroups of $\mathbb{Z}_4\times\mathbb{Z}_6$ of 12 elements?
I know that this group does not have order 24 since $\gcd(4, 6) \ne 1$, but I am at a loss as to where to start.

Think of homomorphisms $\mathbb Z_4 \times \mathbb Z_6 \to \mathbb Z_2$. If you choose them surjective, the kernel will have order $12$. Conversely, a subgroup of index $2$ is always normal, so corresponds to such an homomorphism. 
A morphism $G \times H \to K$ corresponds to a pair of homomorphisms $G \to K$ and $H \to K$. In your case, the groups are all cyclic, so you only have to check where to map the generator. 
Hope that helps,
