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The quaternion group $Q_8$ is a common counterexample to many statements. For example, even though every subgroup is normal, it is not abelian, a direct product, or even a semidirect product! In addition, $Q_8/Z(Q_8) \cong C_2 \times C_2$ but every subgroup of order 4 is cyclic.

What other "conjectures" does the quaternion group disprove, and are there other groups which are common counterexamples?

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closed as too broad by Grigory M, Nick Peterson, Brian Rushton, Olivier Bégassat, Dan Rust Jan 5 '14 at 18:08

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  • $\begingroup$ groupprops.subwiki.org/wiki/Quaternion_group $\endgroup$ – JPLF Jan 5 '14 at 16:54
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    $\begingroup$ even though every subgroup is normal, it is not abelian, a direct product, or even a semidirect product! - The shock! The horror! :D $\endgroup$ – Lucian Jan 5 '14 at 18:05
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The quaternion group $Q_{8}$ is also used to show that two non-isomorphic groups may have the same character table. The character table of $Q_{8}$ is the same as the character table of $D_{8}$, the dihedral group of order 8.

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The (generalized) quaternion groups are the only non-cyclic finite $p$-groups with a unique subgroup of order $p$.

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