Detecting the prime divisors of the character degrees and the class sizes by a subgroup generated with few elements

We prove that every finite group G contains a three-generated subgroup H with the following property‎: ‎a prime p divides the degree of an irreducible character of G if and only if it divides the degree of an irreducible character of H. There is no analogous result for the prime divisors of the sizes of the conjugacy classes‎.