Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1786 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 1786 conjugacy classes of subgroups — too many to draw. Only its 366 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C3×C8 | C2 | semidirect |
| 2. C2×C3×#466 | C2 | semidirect |
| 3. C3×#2847 | C2 | semidirect |
| 4. C3×#1893 | C2 | semidirect |
| 5. C3×#1808 | C2 | semidirect |
| 6. #32930 | C3 | direct (N × H) |
| 8. C22×C3×D8 | C22 | semidirect |
| 15. C2×C3×#107 | C22 | semidirect |
| 16. C2×C3×#102 | C22 | semidirect |
| 19. C3×#340 | C22 | semidirect |
| 20. C24×C8 | C2×C3 | semidirect |
| 21. C2×#466 | C2×C3 | semidirect |
| 22. #2847 | C2×C3 | semidirect |
| 23. #1893 | C2×C3 | semidirect |
| 24. #1808 | C2×C3 | semidirect |
| 37. C22×C3×C8 | D4 | semidirect |
| 50. C22×D8 | C22×C3 | semidirect |
| 57. C2×#107 | C22×C3 | semidirect |
| 58. C2×#102 | C22×C3 | semidirect |
| 61. #340 | C22×C3 | semidirect |
| 65. C24×C3 | D8 | semidirect |
| 105. C22×C8 | C3×D4 | semidirect |
| 144. C3×C8 | #120 | semidirect |
| 148. C24 | C3×D8 | semidirect |
| 184. C22×C3 | #466 | semidirect |
| 219. C8 | C3×#120 | semidirect |
| 233. C22 | C3×#466 | semidirect |
| 241. C3 | #32930 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C3×C8 | 2 |
| C2×C3×#466 | 2 |
| C3×#2847 | 2 |
| C3×#1893 | 2 |
| C3×#1808 | 2 |
| #32930 | 3 |