Every fingering that fits a hand
x444
moderate · barre · no 5
989x
easy · no 5
9897
hard · no 5
9b9b
moderate · barre
989b
hard · no 5
What is true of it
E6 is E, G♯, B and C♯ — 1, 3, 5 and 6 counted from E.
These are also the notes of C♯m7/E and D♭m7/E. Which name is right depends on what is underneath and what came before — the notes alone do not decide.
On an instrument that cannot reach all 4, the 5 goes first, in that order.
The easiest of the 5 fingerings below is x444: one finger, one of them barring the 4th fret, held at the 4th 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.
2 of the 5 need a barre.
It leaves out the 5. 4 strings cannot carry 4 notes, and the 5 is the one this chord can lose without changing what it is.
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
E6/G♯
1102
E6/B
4444
E6/C♯
6877
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.