This seems largely country dependent with some exceptions.

In the US, once upon a time, elementary/middle/highschools might be attached to something like schoolname.district.state.gov. But now, even my local area school now has a .com. It seems that older hierarchy style is falling out of fashion for smaller/shorter domains across public services, schools, government agencies, etc.

Now here it seems to be either a .com, .gov, .org, or a totally different and newer tld. Even .net has fallen out of fashion.