Prime divisors and the number of conjugacy classes of finite groups

We prove that there exists a universal constant $D$ such that if $p$ is a prime divisor of the index of the Fitting subgroup of a finite group $G$, then the number of conjugacy classes of G is at least $Dp\mathrm{/}lo{g}_{2}p$. We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1\mathrm{/}3$.