Big Fancy House: both the Kujous' and Karasumas', to the point where both Karin and Ami have gotten lost in them. This is lampshaded when Karin wonders why they all have such big houses. Jin's mansion is much bigger, and sparkly. It turns out to be a subversion, as Jin doesn't live in the house himself, but lives in a shack, and has never been inside the actual house.
Elevator School: Sakuragaoka Academy has elementary, middle, and high school branches.
Everyone Hates Hades: Averted. Jin, who draws his power from Hades, is a "bad boy" type but actually a pretty cool guy, a Chivalrous Pervert at worst. Further played with once it's revealed he's actually a vessel created by Kirihikofor his eventual revival. Jin is still a completely separate person from Kirihiko, though, and 'Hiko takes his power from a different god (Zeus) anyway.
Kazune and Himeka are initially introduced as cousins with Kazuto being Kazune's father, but actually Kazune is a clone of Kazuto and Himeka is Kazuto's daughter. Karin is either a clone of or a deaged (depending on canon) Kazuto's wife/Himeka's mother. Kazusa is said to be Kazune's twin sister but is implied to be a clone of Himeka.
Kazune finds out Jin is (supposedly) the son of Kirihiko's relative partway through Chu, but he's actually a clone(?) of Kirihiko.
The two Himekas are the odd name out in their respective families, but still tied in by the fact that they sharetheir names.
And finally, Suzune's name is written with kanji from Kazune and Karin/Suzuka's names.
Fangirl: in-universe example: Kazune, Micchi, and Jin all have their own respective fan clubs: the Kazune-Z, Michirians, and Jeens. Jin's fangirls can get downright violent.
Fountain of Youth: In the backstory, Kazuto Kujou used this plus a Memory Gambit on his wife, Suzuka (better known as Karin Hanazono), to help them hide the research of the Gods' rings from Kirihiko Karasuma.
Meido: Karin wears a maid outfit while kissing Kazune, leading Ami to call her "the scandalous maid".
Mind Screw: Well, maybe not that much, but the story gets pretty complicated. The anime introduces a bunch of new plot threads, some of them seemingly related to Chu, but then never got continued, falling further into this unintentionally.
Neuro-Vault: Himeka is where her father, Kazuto Kujou, had hidden his research on the Gods' Rings. Then, he separated her into twohalves.
Not What It Looks Like: In a side chapter, Karin and Kazune get their hands stuck together and have to change into their gym clothes in the storage shed. Kirio walks in on them in a... rather compromising position.
Parental Abandonment: Karin, Kazune, Himeka, Micchi, Kirio and Kirika all have dead parents. Kazune and Karin, however, are clones of Kazune's supposed parents, and the Kujous are actually Himeka's parents, though they're still quite dead.
In the anime, only Kazune is a clone. Karin was turned into a young child/infant with no memories of her previous life.