Minor chords
There are three widely used ways to play minor chords: with the root fretted on the lower E string, with the root fretted on the A string or in an inversed shape usually known as m7/5b (half-diminished). These three positions depend upon the chord being played in a similar fashion as for major chords.
minor 6 chords
Root on the lower E string (Gm6 here) mostly played with the root between F# and C:
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Root on the A string (Cm6 here) mostly played with the root between B and F with the middle finger muting the lower E string.
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Half-diminished chord type (Em6 here) with the index muting the lower E string, 4 frets to the right of the root played on the lower E string or 3 frets to the left of the root note played on the A string, mostly played with the root beetwen Db and G:
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minor 69 chords
Root fretted with the thumb on the lower E string (Gm6/9 here) mostly played with the root between F and C:
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