The prime graph or the Gruenberg-Kegel graph of a finite group G is the graph whose vertices are the prime divisors of the order of G and two distinct vertices p and q are adjacent if and only if G contains an element of order pq. This paper continues the study of the problem of describing the finite nonsolvable groups whose prime graphs do not contain triangles. We describe the groups in the case when a group has an element of order 6 and the order of its solvable radical is divisible by a prime greater than 3.