existence of a subgroup of a solvable group G of order d for any divisor d of |G|, with d<|G|^(1/2)

Question

Let $G$ be a solvable group of order $n$ and $d<\sqrt{n}$ be any divisor of $n$. Is there any subgroup of $G$ of order $d$?

Answer 1

The group $A_4$ is solvable and has no subgroup of size $6$, so $A_4\times \mathbf{Z}/101\mathbf{Z}$ is a counterexample for you.