$|G|=8$, then $G$ has proper nontrivial subgroups of order $4$
126113
Prove the existence of order 4 subgroups of order 8 groups
28181
Sylow subgroups of $S_3$ and $S_4$
133298
Assume $|G|=40$. Show that the subgroup of order $8$ is normal and unique.
7973
Groups of order $pq$ have a proper normal subgroup
21673
If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$
63798
How can you show there are only 2 nonabelian groups of order 8?
154398
Show that H is a subgroup of G
149030
Nonisomorphic groups of order 12.
14611
$A_4$ has no subgroup of order $6$?
58033
$G\times H\cong G$ with $H$ non-trivial
41688
An elementary problem in Group Theory: the unique noncyclic group of order 4
8066
Sort-of-simple non-Hopfian groups
139451
Prove that any two nontrivial subgroups of $\mathbb{Q}$ have nontrivial intersection
7640
Exactly one nontrivial proper subgroup
15706
"Every Group of order $216$ contains a normal subgroup of order $27$ or $9$
46904
Let $G$ be a non-trivial group with no non-trivial proper subgroup. Prove that $G$ cannot be infinite group.
157853
$Q_8$ is isomorphic to a subgroup of $S_8$, but not isomorphic to a subgroup of $S_n$ for $n\leq 7$.