Groups with single conjugacy class of subgroups. (modified)

Question

I am going to modify my previous question.

What are those finite non abelian groups in which non normal subgroups of same order are conjugate.

e.g. Dihedral groups of order $4n+2$.

Answer 1

There are lots of examples. Many Frobenius groups have this property (that includes the dihedral groups of twice odd order, and other examples such as nonabelian groups of order $pq$ for primes $p,q$).

$A_4$, $A_5$, ${\rm SL}(2,3)$ and ${\rm SL}(2,5)$ are examples. It seems likely that $A_5$ is the only nonabelian simple group with this property, and it should not be too difficult to check that. In fact I would guess that $A_5$ is the only possible nonabelian composition factor of such a group. The Frobenius group $11^2:{\rm SL}(2,5)$ is another interesting example.

But I think it would be difficult to come up with a complete description, so you might need to ask a more directed question.