Basically exactly what Papa Gates said. They are all inversions on one another because they are made up of stacked Major 3rds. And the Diminished one’s with Minor 3rds.
There’s some seriously cool things that you can also do with that info in a lead setting too,for example (Hopefully this isn’t going too off topic):
If I’m playing a D Augmented chord, you are free to play a D Augmented arpeggio, or an F# Augmented Arpeggio, or an A# Augmented Arpeggio over it.
You can do this because the D Augmented chord shares the same notes as every one of those arpeggios.
These are both the notes in D, F#, & A# Augmented and their corresponding arpeggios:
D Aug: D F# A#
F# Aug: F# A# D
A# Aug: A# D F#
See how it’s just the same notes in a different order? Inversions basically
I’m not sure how savvy you are with your Modes as of yet (If not just ignore this bit) but if you’re familiar with the Lydian mode (Major Scale with a Sharpened 4th Degree), you can adjust that mode to fit over these Augmented Chords too. The Mode you’d be looking for is the ‘Lydian **Augmented** Mode’, which is just the regular Lydian Mode with a Sharpened 5th degree (Augmented 5th).
You can use the same, moving in Major thirds technique with this whole scale over each Augmented chord. For example:
Over a D Augmented chord, you could play:
D Lyd Aug, F# Lyd Aug, A# Lyd Aug
It gives off a really interesting sound to mess around with!
Hopefully that wasn’t too off topic, just happy to help out with some options related to what you are learning
Hope you’re well,
Chris from Sunny Scotland