Homomorphic image if smaller fails to exist

Question

Suppose a finite group $G$ has no homomorphic image of order $n$. Is it possible for $G$ to have a homomorphic image of order a multiple of $n$?

My gut says "no", as the larger homomorphic image could be made into the smaller by "throwing out" information.

Answer 1

Hint: There is no surjective homomorphism from $S_3$ to a group of order $3$ (do you know why?), but there is an obvious surjective homomorphism from $S_3$ to a certain group of order $6$.