1. Subgrounds of Symmetric Groups

I'm trying to find all of the subgroups of the symmetric $S_{3}$. But am not sure how to generate these groups as all of the properties of groups must be satisfied.

I am thinking it is something along the lines of:
{(123)->(123)} (no change)
{(123)->(123), (312), (231)}
{(123)->(123),(132), (213), (321)}

Is this the right track? Or am out out in left field?

2. Originally Posted by apsis
I'm trying to find all of the subgroups of the symmetric $S_{3}$. But am not sure how to generate these groups as all of the properties of groups must be satisfied.

I am thinking it is something along the lines of:
{(123)->(123)} (no change)
{(123)->(123), (312), (231)}
{(123)->(123),(132), (213), (321)}

Is this the right track? Or am out out in left field?
The subgroups are $\{ e \}, \left< (12) \right>, \left< (13)\right>, \left< (23) \right>, \left< (123) \right>,S_3$.