The prime graph or the Gruenberg–Kegel graph of a finite group $ G $ is the graphwhose vertices are the primedivisors of the order of $ G $ and two distinct vertices $ p $ and $ q $ areadjacent if and only if $ G $ contains an element oforder $ pq $. This paper continues the study of the problem of describingthe finite nonsolvable groups whoseprime graphs do not contain triangles. We describethe groups in the case when a grouphas an element of order 6 and the order of its solvable radical is divisibleby a prime greater than 3.