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Are there examples of finite groups which are not simple, but its normal subgroups (except $1$) are non-abelian?

(One example I was thinking was about take two non-isomorphic simple groups and take their direct product; but then the order of resultant group was quite big; I was looking for smallest order non-trivial example).

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    $\begingroup$ Is $S_5$ too large? $\endgroup$
    – user169852
    Dec 6, 2016 at 3:18
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    $\begingroup$ In any case why insist that the simple groups are non-isomorphic? $A_5 \times A_5$ is an example. $\endgroup$
    – Derek Holt
    Dec 6, 2016 at 9:03

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This is equivalent to having trivial soluble radical. Such groups are sometimes called semisimple. They have quite a nice structure. In particular, such a group acts faithfully on its socle by conjugation, so it can be embedded in the automorphism group of its socle, which is a direct product of simple groups. (And, indeed $S_5$ is the smallest example.)

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