My recollection is that for the moons of Uranus and Mars I have begun detecting them at around 250x or greater, usually greater as in ~300 or 350x and sometimes considerably more, particularly for the two inner moons of Uranus.
Magnification alone won't do the job, unless the seeing is adequate and one has the aperture to provide the needed resolution (smaller spurious disk). Using 350x on a 200 or 250x night isn't particularly helpful.
First you need to be able to collect enough light to actually "see" them, and they will be a dot. Unless you collect sufficent light no amount of magnification will help.
Need to get out of the idea that "magnification" is the answer to everything.
Then comes having reasonable quality optics.
It is less about sufficient light (aperture) than it is about contrast. For point sources, increasing magnification increases actual contrast, up to some limits. Seeing is an important part of that contrast, and tends to put a cap on effective magnification for these type of observations. When the seeing is stable, magnification dims the background surface brightness while the central intensity of dim objects is effectively unchanged. This increases contrast...until magnification is increased to the point that the seeing or diffraction pattern of the spurious disk begin to result in apparent size to the star/moon. When the object begins to have apparent size, further increases in magnification do not improve contrast vs. the background.
Mars' moons would be visible in nearly any telescope aperture in rural sky if this were merely about collecting sufficient light. Instead, the moons pose considerable difficulty even for large scopes because of the presence of Mars which makes its surroundings very bright. As the moons move further from the planet they are more readily seen, if conditions are sufficient, primarily seeing.
Uranus' moons are closer to the telescopic limiting magnitudes of small and medium scopes, so aperture is more of a factor. Fortunately, Uranus is not nearly as intense as Mars with respect to brightening the background sky. Even so the inner moons are close enough to be impacted and generally require more aperture and/or better seeing. Fortunately Titania and Oberon are far enough out to be observed with something like a 6" refractor or an 8" SCT.