Is it possible for a group (non-simple and non-abelian) that solvability of all of its proper subgroups leads the whole group to be solvable?

-
Minor edits. In any case, the question may be too elementary for MO. – Jim Humphreys Mar 25 2011 at 12:27
Sorry Jim for my elementary questions. – Babak Sorouh Mar 25 2011 at 12:32

No. $SL(2,5)$ is a non-simple non-solvable group with the property that all its proper subgroups are solvable.

-
And that's the smallest example. Also called the binary icosahedral group. Maps onto $A_5$ with kernel of order $2$. – Tom Goodwillie Mar 25 2011 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.

-
The OP asked for a non-simple group. – Zev Chonoles Mar 25 2011 at 10:14
Oops you are right. – Yiftach Barnea Mar 25 2011 at 11:03
J. Thompson, in his (famous) series of papers, dealt also with not necessarily simple groups, as far as I remember. – Pasha Zusmanovich May 8 2011 at 10:38

The minimal non-solvable group surely has the property that all proper subgroups are solvable.

-
The only (very slight) subtlety is that the OP asked for a non-simple example. – HW Mar 26 2011 at 22:16