The following is an important paper about FFT algorithms:
Tran-Thong, “Algebraic Formulation of the Fast Fourier Transform,” IEEE Circuits and
Systems magazine, vol. 3, no. 2, June 1981, pp. 9-19.
In the above, the author distinguishes 16 basic types of FFT algorithm, characterized by: 1) decimation type (DIT - type T below, or DIF - type F below), 2) input order, 3) output order, and 4) geometry. The author also shows graphs, but they will not be shown here.
Alg. Input Order Output Order Geometry
T1 bit reverse sequential in-place
T2 sequential bit reverse in-place
T3 bit reverse bit reverse same output
T4 sequential sequential same output
T5 bit reverse bit reverse same input
T6 sequential sequential same input
T7 bit reverse sequential isogeometric
T8 sequential bit reverse isogeometric
F1 bit reverse sequential in-place
F2 sequential bit reverse in-place
F3 bit reverse bit reverse same output
F4 sequential sequential same output
F5 bit reverse bit reverse same input
F6 sequential sequential same input
F7 bit reverse sequential isogeometric
F8 sequential bit reverse isogeometric
Although many of the algorithms can be attributed to different authors, as stated in the above paper: "The Cooley-Tukey algorithm and its modifications are DIT algorithms. The Gentleman-Sande algorithms and their modifications are DIF algorithms."
There are many more FFT algorithms other than the ones described above, including higher and mixed radix ones. But they also exhibit similar characteristics. For instance, take a look at the 6 point DIT radix-3/radix-2 graph shown on page 13 in the following reference:
It is two radix-3 butterflies followed by three radix-2 butterflies. It is bit-reversed input and sequential output.
Of course, given that the FFT can be viewed as a matrix transpose, one can also decompose a 6 point FFT into three radix-2 butterflies, followed by two radix-3 butterflies. You can also have DIT or DIT (or mixed decimation). So you have several options to choose from.