The augmented sixth chord we presented in the previous page and that appears below is often called Italian augmented sixth chord (iv degree triad in first inversion):
If we add the seventh to the Italian sixth chord, we get a German augmented sixth chord (IV degree seventh chord in first inversion). Note the parallel fifth between the German sixth and the dominant chord between the bass and tenor voices:
You can avoid the parallel fifth by using the tonic chord in second inversion before the dominant chord:
If we add an augmented fourth (from the bass note), we get the French augmented sixth chord (II degree seventh chord in second inversion):