Let $G$ be a smooth group scheme over $S$, and let $H\subset G$ be a finite flat closed subgroup scheme (hence it's a Cartier divisor).
Let $nH$ be the closed subscheme corresponding to multiplying the Cartier divisor by the integer $n$. Is $nH$ also a subgroup scheme?
My intuition is that the answer is no. If that's the case, what breaks?