The dogs are the group of friends that Alpha and Matthew hang out with. They include Callum, Dean, and all the other boys that kick it with them. The dogs are bound to each other. They're a group no matter what they're doing:
They were going through life together—whether the pranks they pulled were dumb or brilliant. She was going through life with no one. (33.23-24)
Frankie admires that they all have each other's back and that they have this bond that she doesn't see anywhere else. She still longs for the pack at the end, when she sees Matthew goes by with them, but she knows that she can't be a part of it. She's better off alone.
You might think of Star as Frankie's counterpart in that she is the other sophomore girlfriend of a senior (in this case Dean's). But that's pretty much where the similarity ends. She's a vapid, kind of stupid girl who tries to act really cool but simply falls apart when Dean breaks up with her. That's a nice wakeup call for Frankie, who realizes that Star's fate could be hers if she doesn't stand up for herself.
Elizabeth Heywood is the she-wolf, otherwise known as Alpha's girlfriend. She is an interesting character because she doesn't have the same pedigree as the other students:
[…] she differed from her fellow students in the way that new money differed from old. She was extremely well known around campus—without being exactly popular. (15.2-3)
She's made her own money as an actress and should therefore be very independent. But even though she feigns disinterest in everything, every time Alpha tells her to do something, she does it. She's still bound to her relationship.
Frankie's pop (whom she's named after) is a former member of the Loyal Order of the Basset Hounds and is the reason that Frankie knows anything about them at all. He always goes on and on about how Alabaster is where you meet the connections you'll keep for the rest of your life.
Senior Banks was a doctor specializing in lung problems. Mentally, however, he was an Old Boy—more concerned with his network of Ivy League cronies than he was with the diseases of his patients. He had attended Alabaster (back when it was all male), followed by Harvard, just as his father had attended Alabaster followed by Harvard. (4.4)
Senior definitely represents the Old Boys' Club, which Frankie originally wants to be a part of because of all his yammering on about it. But eventually, she just wants to overthrow all that her father stands for.
Frankie's mother, Ruth, is extremely protective of her daughter and continually calls her Bunny Rabbit. She is constantly fretting over Frankie, not allowing her to go into town alone (even though her younger cousins can) and telling Frankie that she's just glad that she has a boyfriend now because she has someone to "take care of her." The whole thing is rather infuriating for Frankie, who wants her mom to see that she's independent and can take care of herself just fine, thank you very much.
Headmaster Richmond is the ultimate symbol for control at the school. He makes all the rules and enforces them, and he's more concerned with rule following than considering the actual intent behind them. He is ultimately very annoyed by all the pranks and is one of the figureheads of the patriarchy at Alabaster that Frankie is rebelling against the whole time.