For a finite commutative ring, any quotient ring is isomorphic to some subring?

The ring $\mathbb{Z}/4\mathbb{Z}$ has no proper subrings, but has quotient $\mathbb{Z}/2\mathbb{Z}$.

Note the ideal $(2)\subseteq \mathbb{Z}/4\mathbb{Z}$ is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ as an additive group. However its multiplicative structure is different, as $2\times 2 = 0$ in $\mathbb{Z}/4\mathbb{Z}$, but $1\times 1=1$ in $\mathbb{Z}/2\mathbb{Z}$.

So in one case the additive generator squares to the additive identity, but in the other case it squares to the additive generator.

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