An hyperoctahedral group $G$ is the wreath product of $S_2$ and $S_n$, where $S_{n}$ is the symmetric group on $n$ letters, or in other words the semi-direct product $G=S_2^n\rtimes S_n$, w.r.t. the action on the indices. $G$ comes with a natural action on $[2]\times [n]$, so we consider it as a subgroup of $S_{2n}$.

$G$ has two normal subgroups of index two, the first $N_1$ is the preimage of $A_n$ under the quotient map $G\to S_n$ and the second $N_2$ is the intersection in $S_{2n}$ of $G$ and $A_{2n}$. This immediately gives a third normal subgroup of index two, coming from the diagonal in the Klein group $G/(N_1\cap N_2)$.

Are there any more maximal normal subgroups in $G$?

We checked this by computer for $n\leq 15$.