Let $R$ be a commutative ring, suppose that $I$ and $J$ are ideals of $R$. Suppose that $R/I\cong S_1$ and $R/J\cong S_2$. It is true that if $S_1\subset S_2$ if then $J\subset I$?
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Consider the following. Let $k$ be a field. For all $a\in k$, we have $$k[X]/(X-a)\cong k.$$ However, each ideal $(X-a)$ is maximal, and so none is contained in the other.