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I'm looking for examples of (families of) finite groups that are not semidirect products.

When first learning group theory, the first such group that one encounters is $Q_8$. In my search for other groups that are not semidirect products, the only examples I could find were simple groups, which clearly cannot be semidirect products since they don't have normal subgroups.

Does anyone have example of non-simple finite groups that are not semidirect products?

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    $\begingroup$ All the generalized quaternion groups satisfy this, as they have a unique element of order $2$. $\endgroup$ – Tobias Kildetoft Jul 24 '13 at 18:04
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    $\begingroup$ Cyclic groups of prime-power order also work (or, as it were, don't work - they are not semidirect products). Note that these are not always simple, 'cause I said prime-power. $\endgroup$ – user1729 Jul 24 '13 at 18:11
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    $\begingroup$ quasi-simple groups (nonsplit central extensions of simple groups) also work. For instance SL(n,q) or the valentiner group. $\endgroup$ – Jack Schmidt Jul 24 '13 at 18:49
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The easiest examples of non-simple groups, that are not a semidirect product of two non-trivial subgroups, are cyclic groups of order $p^n$, where $p$ is prime and $n \geq 2$, and generalized quaternion groups

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