Mistake in my Second Isomorphism Theorem example

Question

Suppose that $G = \mathbb{Z}$, $N = 6\mathbb{Z}$, and $H = 4\mathbb{Z}$.

1. Is $HN = 24\mathbb{Z}$?
2. Is $H \cap N = 12\mathbb{Z}$?
3. Is $HN/N \cong H/(H \cap N)$?

I am having trouble with why this isomorphism doesn't work in this case. What am I doing wrong here? Any hints would be ok.

I know that $N$ should be a normal subgroup of $HN$, but what's wrong here?