Example of not so simple group ?? Help me with an example of a group having subgroups but it doesn't admit a normal subgroup.. ??

The alternate definition of simple groups using non trivial homomorphic image. Searching for a map that doesn't have a non trivial kernal ??
One classic example is $A_5$, which has plenty of non-trivial subgroups, none of which are normal.  In fact, this is the smallest suitable example.
Hint: Any subgroup of a commutative group is normal. What noncommutative groups do you know.

Bigger Hint: Think about subgroups of $S_5$.

Even Bigger Hint: What about the subgroup of the simple group $A_5$ generated by a product transpositions?

SUPER OMEGA HINT: What is $(12)(45)(12)(34)(12)(45)$?