By M. Kisin, let $k$ be an algebraically closed field of characteristic $p$, and $K$ be a totally ramified extension of $B(k)$, the fraction field of the Witt vector ring $W(k)$, the category of finite flat group schemes over $\mathcal{O}_K$ which are killed by $p$ is equivalent to the category of Kisin modules over $\mathfrak{S}_1=k[[u]]$, whose objects are finite free $k[[u]]$ modules endowed with a Frobenius $\phi$ such that the cokernel is killed by the Eisenstein polynomial of a uniformizer in $K$.

My question is, there are a lot of submodules of a Kisin module whose quotient is not a free $k[[u]]$ module. However on the finite group scheme side, the quotient of a subgroup scheme is always again a finite flat group scheme. Where am I wrong in this inconsistency argument?

Thanks a lot!