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We prove that if a finitely generated profinite group $G$ is not generated with positive probability by finitely many random elements, then every finite group $F$ is obtained as a quotient of an open subgroup of $G$ . 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 $G$ has at most $|G|^c$ maximal soluble subgroups, and show that this result is rather useful in various enumeration problems.