A GAP code for a class of small groups

Question

I need a GAP code for checking the following question

Let $n$ be a given positive integer. Is it true that every group $G$ of order $n$ is either solvable or satisfies the condition: "For any positive divisor $d$ of $n$ there exists a subgroup of $G$ with order $d$ or $n/d$"?

We know that all $n<240$ satisfy the above property. So, the first interested number is $n=240$. Indeed, we are looking for a natural number $n$ which does not have the above property.

Any advice?

Thanks in advance

Answer 1

I hope the following brute force code answers your question:

nmax:=240;   
for n in [240..nmax] do
     if IsPrime(n) then continue; fi;    
     div:=DivisorsInt(n);    
     gl:=AllSmallGroups(n);    
     for g in gl do    
          if IsSolvableGroup(g) then continue; fi;         
          ll:=LatticeSubgroups(g);    
          cc:=ConjugacyClassesSubgroups(ll);    
          szs:=List(cc,c->Size(Representative(c)));    
          if not ForAll(div, d->d in szs or n/d in szs) then 
                  Display(IdSmallGroup(g)); 
          fi;    
      od;    
od;

It is possible to add a break statement after the Display if only one example is wanted.

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UPDATE:

There seems to be an exception : The group PSL(2,8) is not solvable and lacks subgroups of order $12, 21, 24, 28, 36, 42, 63, 72, 84, 126, 168, 252$. The question that rises now is :"Could one have predicted this result on theoretical grounds?".