How to find all conjugacy classes of $A_4$?

Question

I want to find all conjugacy classes of $A_4$. So basically what I did, I took all elements of $A_4$ and calculated their conjugates. I had no problems with $$\{e\}, \{(123),(134),(142),(243)\}, \{(132),(143),(124),(234)\}$$but I don't understand why the rest of $A_4$ elements $$(12)(34),(13)(24),(23)(14) $$is in one conjugacy class? Because, for example, computing the cojugates of (12)(34) gives me: $$(12)(34)(12)(34)(12)(34)=(12)(34)$$ $$(13)(24)(12)(34)(13)(24)=(12)(34)$$ $$(14)(23)(12)(34)(14)(23)=(12)(34)$$ (here $(ab)(cd)^{-1}=(ab)(cd)$). My result is that $(12)(34)$ is the only member of the conjugacy class generated by it. Same for $(13)(24)$ and $(14)(23)$. I get three different conjugacy classes. Where am I wrong?