We prove that if a finitely generated profinite group
G
G
is not generated with positive probability by finitely many random elements, then every finite group
F
F
is obtained as a quotient of an open subgroup of
G
G
. The proof involves the study of maximal subgroups of profinite groups, as well as techniques from finite permutation groups and finite Chevalley groups. Confirming a conjecture from Ann. of Math. 137 (1993), 203–220, we then prove that a finite group
G
G
has at most
|
G
|
c
|G|^c
maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.