Let $k$ be a field, $D$ be a central division algebra of degree $n$ over $k$. We call $k'$ a splitting field of $D$ if $D\otimes_kk'\cong M_n(k')$. Splitting fields may not be isomorphic, can we say more about them?
Let $L_1,L_2$ be splitting fields of $D/k$, do we know if $\mathrm{Gal}(L_i/k)$ are isomorphic?
Let me suppose $k$ is a number field, and $D/k$ is a central division algebra. If $K/k$ is a number field embedding in $D$, then $K$ splits $D$ if and only if $K$ is maximal. Any field containing such a $K$ will also split $D$.
Say $n$ is the degree of $D$, i.e., $n^2 = dim_k D$. Then the maximal subfields of are the extensions of $k$ of degree $n$ which have suitable splitting behavior at any prime where $D$ is ramified. E.g., suppose $D_v$ is division at any place it is ramified. Then a degree $n$ field extension $K$ of $k$ embeds in $D$ if and only it it is not split ($K_v$ is a field) at all $v$ ramifying in $D$.
If $K_1$ and $K_2$ are maximal subfields of $D$, then you can only guarantee their Galois groups over $k$ are isomorphic when $n \le 2$.