Every fingering that fits a hand
x545
easy · no 5
a8a8
hard · no 5
acab
moderate · barre
a8ax
easy · no 5
a8ab
hard · no 5
What is true of it
Fm6 is F, A♭, C and D — 1, b3, 5 and 6 counted from F.
These are also the notes of Dm7♭5/F. 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 x545: 3 fingers, 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.
1 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 F, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
Fm6/A♭
1213
Fm6/C
5545
Fm6/D
7888
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.