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Is there a way to classify all finite groups with trivial centre?

As far as I know, all finite symmetric groups, all non-abelian finite simple groups, $A_4$, the trivial group, and all direct products of aforementioned groups satisfy that condition.

But are there any other ones?

Any help will be appreciated.

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    $\begingroup$ direct products of any of these. $\endgroup$ – Lord Shark the Unknown Dec 27 '17 at 12:44
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    $\begingroup$ $A_4$ is not in your list. $\endgroup$ – Mark Bennet Dec 27 '17 at 12:53
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    $\begingroup$ All (nontrivial) semidirect products with faithful actions. $\endgroup$ – tomasz Dec 27 '17 at 13:08
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Such a classification would include all solvable non-nilpotent groups with trivial center. It is well known, that this is "in some sense impossible", e.g., see this MO-question.

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