There are three main types of augmented sixth chords, commonly known as the Italian sixth, the French sixth, and the German sixth.
The Italian sixth (It+6 or It6 or ♯iv6) is derived from iv6 with an altered fourth scale degree. This is the only augmented sixth chord comprising just three distinct notes; in four part writing the tonic pitch is doubled.
The French sixth (Fr+6 or Fr4 3) is similar to the Italian, but with an additional tone. The notes of the French sixth chord are all contained within the same whole tone scale lending a sonority common to French music in the 19th century (especially associated with impressionistic music)
The German sixth (Ger+6 or Ger6 5) is also like the Italian, but with an added tone.
In Classical music, however, it appears in much the same places as the other variants, though perhaps less often because of the contrapuntal difficulties outlined below. It appears frequently in the works of Beethoven, and in ragtime music. The German sixth chord is enharmonically equivalent to a dominant seventh chord though it functions differently.