I wanted to know orders of all subgroups of the group $AGL(1,16)$ (of order 240). Indeed, regarding to the problem mentioned in https://mathoverflow.net/questions/143229/a-question-about-finite-groups-a-weak-version-of-the-converse-of-lagrange-theor, we are looking for the smallest group(s) which does not have the property.
***** Now, is it true that there exist subsets $A$, $B$ of $AGL(1,16)$ such that $|A|=6$, $|B|=40$ [resp. $|A|=10$, $|B|=24$] and $AGL(1,16)=AB$?
You can find the answer in GAP or Magma with a few seconds typing. It is $$\{1, 2, 3, 4, 5, 8, 12, 15, 16, 48, 80, 240 \}.$$
So there are no subgroups of orders $6, 10, 20, 24, 30, 40, 60, 120$. But you should be able to prove that yourself!