We describe a method for counting the number of extensions of
${\mathbb{Q}}_{p}$ with a given
Galois group
$G$,
founded upon the description of the absolute Galois group of
${\mathbb{Q}}_{p}$ due
to Jannsen and Wingberg. Because this description is only known for odd
$p$, our results do
not apply to
${\mathbb{Q}}_{2}$.
We report on the results of counting such extensions for
$G$ of order up to
$2000$ (except those
divisible by
$512$),
for
$p=3,5,7,11,13$.
In particular, we highlight a relatively short list of minimal
$G$
that do not arise as Galois groups. Motivated by this list,
we prove two theorems about the inverse Galois problem for
${\mathbb{Q}}_{p}$: one giving a necessary
condition for
$G$ to
be realizable over
${\mathbb{Q}}_{p}$
and the other giving a sufficient condition.
