$L|_k$ be a finite Galois extension and $M,N$ are subfields containing $k$ such that $[M:k], [N:k]$ are powers of $2$. Is $[MN:k]$ power of $2$?

Question

Let $L/k$ be a finite Galois extension, and let $M$ and $N$ be subfields containing $k$ such that $[M:k]$ and $[N:k]$ are powers of $2$. Is $[MN:k]$ a power of $2$?

I think this is false, but I have tried with different fields and extensions but didn't get a counterexample. Can anyone help me with this problem?

Thanks for help in advance.