Let $K$ be a $p$-adic field, or more generally a local field. Let $D$ be a $d^2$-dimensional division algebra over $K$. Then $D$ is necessarily of the form $D = \bigoplus_{i=0}^{d-1} Le^i$, where $L$ is the unique unramified extension of $K$ of degree $d$, and $e^n =a$ for some $a \in L$. Then the Hasse invariant $\operatorname{inv}_K(D)$ is equal to $\frac{1}{d}\operatorname{ord}_L(a) \in \frac{1}{d} \mathbb Z/\mathbb Z$.
Now let $D'$ be a subalgebra of $D$. It is necessarily a division algebra over $K$. I believe one can use the primary decomposition of $D$ to show that $\dim_K D = e^2$ for some $e \mid d$.
I'm wondering: how is the Hasse invariant of $D'$ related to the Hasse invariant of $D$?