Every fingering that fits a hand
x443
easy
988x
easy
9887
hard
988b
hard
9ccb
hard
What is true of it
Eaug is E, G♯ and B♯ — 1, 3 and #5 counted from E.
The easiest of the 5 fingerings below is x443: 3 fingers, held at the 3rd fret.
None of them uses an open string, so this chord moves up the neck unchanged — the same shape 1 fret higher is the chord a semitone above.
1 string stays silent.
Where its notes are on the neck
The brass dots are E, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
Eaug/G♯
100x
Eaug/B♯
5047
An inversion is the same chord with a different note lowest. The name changes to a slash name because the bass is what a listener hears as the foundation.