Hello, newbie - and welcome to English Forums. The rule is simple: for humans and human-like entities (like a pet dog or an institution or organization) , we use 's (the Anglo-Saxon genitive). For everything else we use 'of' genitive.
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eslnewbieIs there any kind of rule for 'of' possessives?No. There are only a few observations that can be called rules. For example, the car of you is ungrammatical; you have to use your car. The remaining observations are mostly just that - observations. These amount to guidelines rather than rules.