Suppose that $\varphi$ is a homomorphism from $\mathbb{Z}_{36}$ to a group of order $24$. Determine the possible homomorphic images
I'm trying to solve this exercise from Gallian's book, and I'm stuck here.
I know by the First Isomorphisms Theorem that $|\varphi(\mathbb{Z}_{36})|\Bigl ||\mathbb{Z}_{36}|$
And by Lagrange's Theorem that $|\varphi(\mathbb{Z}_{36})|\Bigl | 24$
So the possible orders of $\varphi(\mathbb{Z}_{36})$ are $1,2,3,4,6$ or $12$.
Please give me some tips of what should I do next to determine the possibilities of $\varphi(\mathbb{Z}_{36}).$