Isomorphisms between Groups (Integers mod n)

Question

1. How many isomorphisms are there from $\mathbb{Z}_{18}$ to $\mathbb{Z}_{2} \times \mathbb{Z}_{9}$?

2. How many isomorphisms are there from $\mathbb{Z}_{18}$ to $\mathbb{Z}_{2} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3}$?

My "solution":

1. There are $6$ generators of $\mathbb{Z}_{18}$, $6$ generators of $\mathbb{Z}_{9}$, and $1$ generators of $\mathbb{Z}_{2}$. Thus $\mathbb{Z}_{2} \times \mathbb{Z}_{9}$ has $6$ generators. An isomorphism will take generators to generators - there will be $6*6=36$ isomorphism.

2. Similar strategy.

The solution:

1. $6$ isomorphisms
2. $\mathbb{Z}_{2} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3}$ is not cyclic thus no isomorphism exists.

I think I see #2 -- $\mathbb{Z}_{2} \times \mathbb{Z}_{3} \times \mathbb{Z}_{3}$, the orders of components are not relatively prime, so they won't combine to a cyclic group. Is that correct?

I however, fail to see #1. Thoughts?