The arabic, Bayer, and Flamsteed designations exist only for the very brightest, visible. Most stars are known by catalog names. Stars in this catalog are designated as HR xxxx or BS xxxx. There are 9110 entries in this catalog, which includes most stars brighter than visual magnitude 6.5. This catalog contains much supplementary information about these stars.
Most bright stars have many names.
There are other names too, including entries in double star and variable star catalogs. Variable Stars Variable stars are named by constellation, and in the order that they are identified. That works for 9 stars. The next set uses two letter designations, beginning with RR , followed by RS through RZ , then SS through SZ and so on to ZZ . This can account for 334 variable stars in a constellation.