Finite groups whose prime graphs do not contain triangles. III

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 $3$.