Is it possible for a group (nonsimple and nonabelian) that solvability of all of its proper subgroups leads the whole group to be solvable?

$\begingroup$ Minor edits. In any case, the question may be too elementary for MO. $\endgroup$ – Jim Humphreys Mar 25 '11 at 12:27

1$\begingroup$ Sorry Jim for my elementary questions. $\endgroup$ – mrs Mar 25 '11 at 12:32
No. $SL(2,5)$ is a nonsimple nonsolvable group with the property that all its proper subgroups are solvable.

3$\begingroup$ And that's the smallest example. Also called the binary icosahedral group. Maps onto $A_5$ with kernel of order $2$. $\endgroup$ – Tom Goodwillie Mar 25 '11 at 11:06
An even simpler counter example is $A_5$.
I believe that finite simple groups in which every proper subgroup is solvable are called minimal finite simple groups and as I recall they were classified by J. Thompson before the calssofication of all finite simple groups. This classification is useful, I think J. Wilson used them to study identities of solvable groups.

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$\begingroup$ J. Thompson, in his (famous) series of papers, dealt also with not necessarily simple groups, as far as I remember. $\endgroup$ – Pasha Zusmanovich May 8 '11 at 10:38
The minimal nonsolvable group surely has the property that all proper subgroups are solvable.

2$\begingroup$ The only (very slight) subtlety is that the OP asked for a nonsimple example. $\endgroup$ – HJRW Mar 26 '11 at 22:16