Wow man there are so many things you can do to get a loud clean sound but it seems to me the issue may start and end with your bands volume!
Sounds like your guitar player(s) are going overkill on the volume and or your drummer is perhaps an outright basher! First thing is to try to get the group to play at a musical level, so the band can be heard and not each individual trying to be the loudest one!
Also EQ and amp placement are essential! There is only so much available space in the sonic spectrum. If your guitarist(s) is going for a big fat heavy low-mid sound, all of the instrument are starting to fight for the same sonic space, and none of them will be heard cleanly. Beginning musicians often make the mistake of sitting at home and dialing in their amps until it sounds a certain way and leave the knobs there because "That's my sound", ignoring the thought of the bands sound! Likewise you need to adjust your EQ settings so they compliment the guitar sound as well.
Amp placement: Been in hundreds of bands (so it seems) and 99.99% set up their amps for rehearsal and stand right in front of them. The sound waves are being projected at their kneecaps, past the player and across the room, they cant hear themselves so they turn up! In turn the other amps are doing the same thing screaming right past the player who can hardly hear it but they are blasting out the other guys standing across the room! No one can hear themselves so they turn up and it's an out of controll volume war with no hope. This is ESPECIALLY TRUE FOR BASS! Bass waves take longer to develop, try standing farther away from your amp, you may be surprised that the farther away from it you are, the more you can hear it. Also try using your amps like monitors and set them a few feet in front of you and aim them back at you. I've done this in bands before where we stand just like we would on stage, with the amps in a line in front of us and the results were an amazing even blend. Tiltback amp stands also work very well for guitarists to hear themselves without making others eardrums bleed!
Room acoustics are every bit as important if not more so! If you are playing in a live room with lots of hard surfaces, the sound will reflect off the walls/floor/celing and will make the upper registers much louder. Playing in a square room, can couse the sound waves to "bounce" off the wall back and forth eventually cancelling each other out (it's called comb filtering) and can actually make a signal virtually dissapear. What happens? The amp gets turned up louder! Imagine playing in a square room with lots of hard reflective surfaces, it's an audio nightmare!!!!!!!!!!
Playing a bass with a maple fingerboard will also give your bass a cutting presence in the mix, try it you'll like it! Also speaker cabs with 10 inch speakers seem to have more presence than say 15's.
I would give sereous consideration to all of the above as well as all the great advice from the previous posters!
Last but not least of course you want a bass rig that will deliver the sound you are looking for. I would reccomend a solid state or tube-hybrid amp with tons of headroom, tube amps by nature tend to break up when you crank them!
Good luck, hope this helps!