Beelzebub and Astaroth were married at one point, but their marriage ended (for reasons unknown).
Astaroth is on bad terms with Beelzebub (on her side anyway). She treats him with great annoyance and irritation, going as far as dousing him with bug spray when he approaches her.
Beelzebub doesn't take offence to her actions, and has been implied to repeatedly ask her to marry him again, considering her beautiful and saying that her stubbornness makes her even lovelier. He was present at her "Funeral" with Lamia, seeing her off to Sleep.
Astaroth and Baalberith are civil to each other, but don't get along.
Baalberith calls Dantalion Astaroth's Nephilim protege. She was the one that put Dantalion forward as a candidate for the Substitute King, and supports him - often offering advice. The demons referred to her support as his "shield". 
They appear quite close, and Lamia holds high opinion of her mother - saying she would be the best one for the position of Substitute King. Lamio also shares her mother's disdain of her father, Beelzebub.
Astaroth does seem to let Lamia get a way with a lot.
Astaroth held high opinion of Solomon, and was one of his Pillars. She mentions feeling nostalgia when she met William Twining, Solomon's reincarnation.
She tells William that she felt like meeting Solomon because she was curious to what happened to one of her descendants, but never thought one of her descendants would be such a weak magician.
Astaroth holds some interest in William as the Elector and the descendant of Solomon. She has a bittersweet expression on her face when she mentions feeling nostalgia at their meeting, but being the only one that felt it.
She had no qualms in trying to force William into choosing Dantalion as the Substitute King. And has repeatedly expressed concern in protecting William from Heaven, which sought to make him ascend and turn in into their puppet.
Astaroth appears to be well loved by her subjects, as at her "Funeral" (departure to start Sleeping) her Warship was surrounded by a great entourage of Demons and Dragons seeing her off on her way to Limbo - the place where she will spend the next century Sleeping.