I am trying to apply Artin-Wedderburn to finite semisimple rings, but I am getting confused on determining the isomorphism classes. For an example, suppose I am trying to find all semisimple rings of order $720=2^4\cdot3^2\cdot5$. So far my argument runs like this:
We know all finite rings are Artinian so we know that the ring $ R$ is the product of matrix rings over a division ring(which is finite and therefore a field).
Now we determine the isomorphism classes. $ 1\times 1$ matrices over a field $\mathbb{F}_q $ are isomorphic to $F_q $, so for $5^1$, we will have a copy of $\mathbb{F}_5 $. For $3^2$ we have either $\mathbb{F}_9 $ or $\mathbb{F}_3 \times \mathbb{F}_3 $. For $2^4 $ I would like to take $\mathbb{F}_{16},\mathbb{F}_8\times \mathbb{F}_2,\mathbb{F}_4\times \mathbb{F}_4,\mathbb{F}_4\times \mathbb{F}_2 \times \mathbb{F}_2, \mathbb{F}_2\times \mathbb{F}_2 \times \mathbb{F}_2\times \mathbb{F}_2$ and then $ M_{2\times 2} (\mathbb{F}_2)$ and then we take products accordingly and getting twelve isomorphism classes.
$ M_{2\times 2} (\mathbb{F}_2) \times \mathbb{F}_9 \times \mathbb{F}_5 $
$M_{2\times 2} (\mathbb{F}_2) \times \mathbb{F}_3 \times \mathbb{F}_3\times \mathbb{F}_5 $
$\mathbb{F}_{16} \times \mathbb{F}_9\times \mathbb{F}_5 $
$ \mathbb{F}_{16} \times \mathbb{F}_3 \times \mathbb{F}_3\times \mathbb{F}_5 $
$\mathbb{F}_{8}\times\mathbb{F}_{2} \times \mathbb{F}_9\times \mathbb{F}_5 $
$\mathbb{F}_{8}\times\mathbb{F}_{2} \times \mathbb{F}_3 \times \mathbb{F}_3\times \mathbb{F}_5 $
$\mathbb{F}_{4}\times\mathbb{F}_{4} \times \mathbb{F}_9\times \mathbb{F}_5 $
$\mathbb{F}_{4}\times\mathbb{F}_{4} \times \mathbb{F}_3 \times \mathbb{F}_3\times \mathbb{F}_5 $
$\mathbb{F}_{4}\times\mathbb{F}_{2} \times \mathbb{F}_{2} \times \mathbb{F}_{9} \times \mathbb{F}_5 $
$\mathbb{F}_{4}\times\mathbb{F}_{2} \times \mathbb{F}_{2} \times \mathbb{F}_3 \times \mathbb{F}_3 \times \mathbb{F}_5 $
$\mathbb{F}_{2}\times\mathbb{F}_{2}\times\mathbb{F}_{2} \times \mathbb{F}_{2} \times \mathbb{F}_9 \times \mathbb{F}_5 $
$\mathbb{F}_{2}\times\mathbb{F}_{2}\times\mathbb{F}_{2} \times \mathbb{F}_{2} \times \mathbb{F}_3 \times \mathbb{F}_3 \times \mathbb{F}_5 $
I have been told, however, that this is incorrect. In particular this list shouldn't include those with $\mathbb{F}_8$ and $\mathbb{F}_{16}$.
Am I missing a simple isomorphism here?