It depends.
IMTU a mainworld may or may not have established proprietary claims over the outworlds (that is, planets other than the mainworld) in the star system. A mainworld of Pop 6+ and TL 8+ will most likely establish a formal claim to the system outworlds, though there are a fair number of exceptions to this general guideline IMTU.
Colonies (gov 6) established on outworlds by the mainworld are the simplest and most direct example of a proprietary claim - mainworld colonies share a unified government system with the mainworld and are accorded Imperial membership along with the mainworld if applicable. Some corporation-run outworlds (gov 1) would also share this status as well, most often when the outworld is a wholly-owned subsidiary of a corporate entity headquartered and legally operated on the mainworld.
Other mainworld proprietary claims to outworlds may exist as well. Self-governing outworlds (gov 1-4) may be established by charters, compacts, treaties, or similar agreements under the proprietary authority of the main-world. For example, in one system IMTU a balkanized mainworld established a multinational development authority over several outworlds and satellites for purposes of mining and other resource extraction. These outworlds may or may not share the Imperial status of the mainworld, depending on the specifics of the enabling agreement, though IMTU most are not Imperial members.
Proprietary claims to a system's outworlds may also be made by a mainworld in another star system, most often in cases where the local mainworld does not have the population or technological resources to exert its own influence over the outworlds in its own system.
Both the local mainworld and a distant mainworld may exert influence over a far companion star system's outworlds.
Independent out worlds are also present in some star systems - these may be former colonies or other proprietary worlds that have achieved independence from the mainworld, amicably or otherwise, or they may have been colonized separately from the mainworld, making them independent from the time they were established. Independent outworlds IMTU are usually Imperial members under seperate charters from the mainworld, although there are exceptions. Independent outworlds may have their own colonies or other proprietary outworlds in the system as well! Independent corporate outworlds usually belong to megacorporations IMTU.
Finally there are self-governing inhabited outworlds (gov 0-4) that fall into a gray area - they may occupy an outworld or planetoid belt for which no proprietary claim exists, or where a claim exists that is effectively unenforceable. Belters often create "independent mining districts" or "belters benevolent associations" that provide a semblence of government, primarily to promote less lethal means of resolving disagreements over claims and so forth. These self-governing outworlds hold no legal claim to their planets or belts and cannot be recognized by the Imperium until such a claim is formally recognized.
