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If you think that such subsets exist, perhaps you could find them with some random search. But if you think that the group is a candidate for the counterexample, it won't help.
Here is a one-liner in GAP that shows no subgroups of index 21 in $PSL_2(8)$: G:=PSL(2,8);ForAny(ConjugacyClassesSubgroups(G),c->Index(G,Representative(c))=21);. Same for index 24.
