I will try to help you out with this. The reason the game did not allow you to win is because you also had exposed the

. In order to have the triple mixed chow open, you can only open the parts of that hand of those chows. Therefore in your case, you can only open the 4-5-6 sequences of each suit and nothing else, the rest of the hand must remain concealed, unless you can get a pon of dragon tiles, your seat wind, or the round wind.
Let me give you an example of another hand that is like this, which is the straight-through. In a straight-through you have 1-9 of one suit and the rest of your hand is whatever. If you were to open your hand and still want the straight through you can only open these parts, the 123 sequence, 456, and 789 sequences, nothing else in your hand can be opened. Unless once again you get a pon of dragon tiles, your seat wind, or the round wind.
Hope that helps clear it up for you. If you have anymore questions please ask.