Does every nontrivial finite group have a subgroup of prime index? I know that a finite group need not have a subgroup for any particular prime index (for example, $A_4$ has no subgroup of index $2$, but it does have one of index $3$). Is there a nontrivial finite group with no subgroups of prime index?
 A: The simple group $\operatorname{PSL}(2,8)$ has no subgroup of prime index. Its maximal subgroups have indices 9, 28, and 36.
In general $\operatorname{PSL}(2,q)$ has order $q(q-1)(q+1)/\gcd(q-1,2)$. Set $d=\gcd(q-1,2)$.
It has (usually) maximal subgroups that are dihedral of order $2(q+1)/d$ and $2(q-1)/d$. It has a maximal subgroup of order $q(q-1)/d$. If $q$ is a power of $r$ (with the power itself prime), then $\operatorname{PSL}(2,r)$ is a (usually) maximal subgroup. It also has (usually) maximal subgroups of orders $12$, $24$, or $60$ ($A_4$, $S_4$, or $A_5$ depending on some congruences on $q$).
For $q=8$, we get the order is 504. It has no subgroup of order 60 by Lagrange. So we check the (possibly) maximal subgroups of order $2(q+1)/d = 18$, $2(q-1)/d = 14$, $q(q-1)/d = 56$, $2(2-1)(2+1)/\gcd(2-1,2) = 6$ (not maximal in this exceptional case), and $24$ (which in this exceptional case is contained in the order 56 group).
These subgroups have index 28, 36, 9, 84, and 21.
The subgroup classification is in Huppert's Endliche Gruppen Kap II. §8, pages 191-214, culminating in Dickson's Hauptsatz 8.26 on page 213 (which is just the more precise version of what I said above).

Most simple groups have no subgroups of even prime power index. The list of exceptions is fairly short and is given in Guralnick (1983). In particular, the cyclic groups of prime order, a few bizarre PSL(n,q), alternating groups of prime power degree, the prime degree Mathieu groups, and PSp(4,3). Every other simple group is a counterexample to the claim that a non-identity group must have some proper subgroup of prime power index. For example the alternating group $A_{6} = \operatorname{PSL}(2,9)$ has maximal subgroup indices 6, 6, 10, 15, and 15.


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*Guralnick, Robert M.
“Subgroups of prime power index in a simple group.”
J. Algebra 81 (1983), no. 2, 304–311.
MR700286
