Subgroups of $\mathbb{D}_6^n$

Question

Let $\mathbb{D}_6=\{1,x,x^2,y,xy,x^2y\}$ be the Dihedral group of order 6. I'm trying to find two subgroups $N\le \{1,x,x^2\}^n$ and $M\le \{1,y\}^n$ such that $MN=NM$ (so that the product is also a subgroup). Of course, I can take $N=\{1\}^n$ or $N=\{1,x,x^2\}^n$ with any $M$ but I'm looking for more interesting ones. Any hint/reference would be appreciated.

Answer 1

How about something like $N=\{(1,1),(x,x^2),(x^2,x)\}$ and $M=\{(1,1),(y,y)\}$ (for $n=2$)?

Or more generally (for general $n$), $N=\{(x_1,\ldots,x_n)|\prod_{i=1}^n x_i = 1\}$ and $M=<(y,\ldots,y)>$?