Emulate many-pole, dual throw switch with transistors
You can implement this with semiconductors provided the power supply to the circuit stays on all the while you want to have the transistors conducting current. This problem is also much easier to solve if you design for the current flow through the switches to always be the same direction.

Let's say that you design for 12V coming into the C (common) terminal of each "switch" and the NC (normally closed) and NO (normally open) "connections" will always be used to gate power to a ground referenced load from the incoming 12V.

For each pole you will need two FETs driven from a pair of control signals that are logically inverted from each other. You drive one FET ON while the other is OFF and flip the states when the "switch" changes state. In this 12V design you may find it easiest to use a P-FET since turning on the FET works nicely by pulling the FET gate to GND with the control signal. (If you use N-FETs it is necessary to provide sufficient gate drive above the 12V level signals that you are trying to switch). The two FET drive signals can derive from a single control signal by adding an inverter within the path to one of the FET gate drives. 

Example Schematic - Note that you have to select the appropriate P-FET per your requirements.