  In this paper, we consider infinite Galois extensions of number fields and
study the relation between their local degrees and the structure of their
Galois groups. It is known that, if $K$ is a number field and $L/K$ is an
infinite Galois extension of group $G$, then the local degrees of $L$ are
uniformly bounded at all rational primes if and only if $G$ has finite
exponent. In this note we show that the non uniform boundedness of the local
degrees is not equivalent to any group theoretical property. More precisely, we
exhibit several groups that admit two different realisations over a given
number field, one with bounded local degrees at a given set of primes and one
with infinite local degrees at the same primes.
