p-div gps over a finite field (e.g., $\mathbb F_p$) are classified by Dieudonne modules.
There are also results for p-div gps over integers of local fields (e.g., over $\mathbb Z_p$).
However, are there results for p-div gps over local fields (e.g.,$\mathbb Q_p$)? I couldn't find any references.. Or more generally, p-div gps over other fields?