You might get that to work if you change the inverter to 74HC04 as suggested, and add a 100nF (0.1uF) capacitor between VCC and GND, with leads as short as possible. You could try it across your power rails (in the center of the picture), but right across the IC , pins 7-14, would be better.
Those breadboards have notoriously high stray capacitance. If you could shorten the wires, you would have a better chance. Stand the resistor on end, squeeze the leads of the crystal close together (you may have to extend them by soldering on some short bare wires), and move the caps so that all the parts go into the pin 12-13 buses (except the other ends of the caps, which you can stretch and bridge over to your GND rail).
A better option is to build the oscillator on a little piece of perf board that has a GND plane (or at least a good GND bus), and make it really tight, with the power supply decpupling cap added as mentioned. You only need 3 wires from the perf board to your breadboard.