If every quotient group of a group G by non-trivial normal subgroups is abelian, G is abelian

Question

If every quotient group of a group G by non-trivial normal subgroups is abelian, G is abelian.

I'm asked about the veracity of this statement. I thought about the group $\mathrm{D_3}$ and the only normal subgroup that I came across was $\mathrm{\{\mathbb{id},\rho,\rho^2}\}$, that has index 2, so it's normal, but I'm not sure about how I can efficiently show that its quotient is abelian.

Answer 1

Try a non-abelian simple group. Does this satisfy the condition?

Answer 2

HINT: What is the order of the quotient group $G/H$ if $G = D_3$, $H=\mathrm{\{\mathbb{id},\rho,\rho^2}\}$? What does this tell about it?