Generate some sort of "graph" of specific subgroups

Question

I need to look at all the conjugacy classes of subgroups of odd order in $\operatorname{GL}_6(2)$, and understand the respective inclusions. I was hoping to generate some sort of lattice of those conjugacy classes of subgroups.
For example, I have computed that there are three different conjugacy classes of subgroups isomorphic to $C_3$, and four conjugacy classes of subgroups of order $9$ (three elementary abelian and one $C_9$). I am interested in understanding what are the inclusion relations between those.

This seems like a task that GAP or MAGMA could easily do, as they are able to compute the subgroups and look if one is inside the other, but I have tried to find the right commands for such a task, and found nothing that could help me.

Answer 1

The Magma program below will do this for you quickly if I have got it right. I wrote this very quickly, so you will need to check it.

G:=GL(6,2);
S:=Subgroups(G: OrderDividing:=615195);
/* Sort in order of decreasing size */
Sort(~S,func<s,t|t`order - s`order>); 
/* Print pairs (i,j) such that S[i] contains a conjugate of S[j]
for i in [1..#S-1] do
  H := S[i]`subgroup;
  SH := [s`subgroup: s in Subgroups(H)];
  for j in [i+1..#S] do
    K := S[j]`subgroup;
    if #H mod #K eq 0 then
       for L in SH do
         if IsConjugate(G,K,L) then <i,j>; break; end if;
       end for;
    end if;
  end for;
end for;