Upper bound on the number of ramified primes for odd order solvable groups

Let $G$ be a finite group and let $ra{m}^t(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\le ra{m}^t(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly $d(G)$ ramified primes. We will show that $3\cdot log(\mathrm{\mid }G\mathrm{\mid })$ is an upper bound for $ra{m}^t(G)$ for all odd order solvable group $G$.