Trinculo,
Nope. The red LEDs are actually about 1.8V, not 1.5V. If you put 4 in series then their total voltage is 7.2V. Add a current-limiting resistor in series that is calculated for 20 mA through it with the difference voltage of 1.8V across it makes the resistor's value 90 ohms, but use a standard 100 ohms. Let us call that a string of 4 LEDs (and a resistor).
Make 6 of those strings and connect them to the battery. The total current is 120mA.
Use 2 white 3.3V LEDs in series, and also in series with a current-limiting resistor. Call that 1 string. If you want 20mA through each LED then 12 strings will draw 240mA.
A rectangular 9V battery contains 6 tiny cells that don't have much capacity. AA cells are much bigger so have much more capacity. 3 AA cells in series produce 4.5V, which is enough for strings of 2 red LEDs but only 1 white LED. Of course, all LED strings need a current-limiting resistor. 3 AA cells in series will power 12 strings of 2 red LEDs for about 3.2 hours. The same amount of time for 12 white LEDs.
Watch out for the battery spec's. The AA cell is rated with a "capacity" of 2850mA/hr, but to an end voltage of only 0.8V (extremely low) and with a low operating current of only 25mA. See the detailed graphs for your own end voltage (1.2V) and operating current (120mA or 240mA) and you will see that the actual capacity will be much less.