Let be a finite group. The set of all prime divisors of the order of is called the prime spectrum of and is denoted by . A group is called prime spectrum minimal if for any proper subgroup of. We prove that every prime spectrum minimal group all whose non-abelian composition factors are isomorphic to the groups from the set is generated by two conjugate elements. Thus, we expand the correspondent result for finite groups with Hall maximal subgroups. Moreover, we study the normal structure of a finite prime spectrum minimal group which has a simple non-abelian composition factor whose order is divisible by different primes only.