There are three general types of augmented sixth chords—the Italian augmented sixth chord (“$\left.\text{It}^{+6}\right.$”), the French augmented sixth chord (“$\left.\text{Fr}^{+6}\right.$”), and the German augmented sixth chord (“$\left.\text{Ger}^{+6}\right.$”). These geographic labels have persisted throughout the years despite the fact that no reasoning has been found for these designations.  1 The 1964 Harvard Dictionary of Music states these chords are “rather pointlessly…distinguished as ‘Italian,’ ‘German,’ and ‘French’ sixth…”
All types of augmented sixth chords contain scale degrees ♭$\hat{6}$ and ♯$\hat{4}$ . To these two scale degrees, the $\left.\text{It}^{+6}\right.$ adds $\hat{1}$ . The three notes of the $\left.\text{It}^{+6}\right.$ (♭$\hat{6}$ , ♯$\hat{4}$ , and $\hat{1}$ ) form the foundation of the $\left.\text{Fr}^{+6}\right.$ and $\left.\text{Ger}^{+6}\right.$. The $\left.\text{Fr}^{+6}\right.$ adds $\hat{2}$ to the Italian augmented sixth chord's ♭$\hat{6}$ , ♯$\hat{4}$ , and $\hat{1}$ , and the $\left.\text{Ger}^{+6}\right.$ adds ♭$\hat{3}$ to the Italian's ♭$\hat{6}$ , ♯$\hat{4}$ , and $\hat{1}$ , as is shown in the example below.
The final chord on the first line—the Enharmonic German $\left.\text{}^{+6}\right.$ or $\left.\text{EnGer}^{+6}\right.$—respells the ♭$\hat{3}$ as a ♯$\hat{2}$ because the $\left.\text{EnGer}^{+6}\right.$ resolves only to major $\left.\text{I}^{6}_{4}\right.$. The $\left.\text{EnGer}^{+6}\right.$ does not occur in minor.