Augmented Sixth Chords Theory Things   William Wieland
2 pitches (the A6) resolve to the dominant by step.
   raised 4 up arrow 5
lowered 6 up arrow 5
fi up arrow sol
leup arrow sol
All augmented sixth chords also include tonic. Real Music Examples
Italian Sixth Chord
 only 3 notes (A6 + tonic)
 since it does not have 4 notes
      it has a hole, like macaroni
c:   It6 V 
Beethoven Symphony 5
  (near the opening)
German Sixth Chord
 4 notes (A6 + major triad)
reliable, like a dominant 7th
      (enharmonically equivalent)
the cadential 6/4 avoids
      parallel fifths.
c:    Gr6  i
Joplin The Chrysanthemum
  (before the first repeat)
French Sixth Chord
 4 notes (A6 + two M3)
mysterious, like tritones
      and the whole tone scale
c:    Fr6 V 
Joplin The Cascades
  (top of p. 4)
Every part on this page either moves by half step or remains on the same tone.
Note Augmented sixth chords may trick you. We identify chords by stacking pitches in thirds, line-line-line or space-space-space, i.e. every other letter. However, F#-Ab-C-Eb is not a common seventh chord. How can you tell? F sharp up to A flat is a diminished 3rd. (Diminished 3rds are enharmonically equivalent to major 2nds.)
DryErasedTheory - Augmented Sixth Chord Review (3:30) awesome video by richard johnson