Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
4732 conjugacy classes of subgroups; only the 1114 normal subgroups are shown.
This group has 4732 conjugacy classes of subgroups — too many to draw. Only its 1114 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25×C3 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C22×C3×Q8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×2−1+6 | C2 | semidirect |
| 2. C3×#3486 | C2 | semidirect |
| 3. C3×#3028 | C2 | semidirect |
| 4. C3×#3025 | C2 | semidirect |
| 5. C3×#2884 | C2 | semidirect |
| 6. #60400 | C3 | direct (N × H) |
| 11. C23×C3×Q8 | C22 | semidirect |
| 14. C3×#536 | C22 | semidirect |
| 15. C3×#525 | C22 | semidirect |
| 16. C3×#519 | C22 | semidirect |
| 17. C3×#458 | C22 | semidirect |
| 18. C3×B(4,𝔽2) | C22 | semidirect |
| 19. C3×#410 | C22 | semidirect |
| 20. 2−1+6 | C2×C3 | semidirect |
| 21. #3486 | C2×C3 | semidirect |
| 22. #3028 | C2×C3 | semidirect |
| 23. #3025 | C2×C3 | semidirect |
| 24. #2884 | C2×C3 | semidirect |
| 33. C2×C3×#41 | D4 | semidirect |
| 37. C22×C3×Q8 | D4 | semidirect |
| 56. C23×Q8 | C22×C3 | semidirect |
| 59. #536 | C22×C3 | semidirect |
| 60. #525 | C22×C3 | semidirect |
| 61. #519 | C22×C3 | semidirect |
| 62. #458 | C22×C3 | semidirect |
| 63. B(4,𝔽2) | C22×C3 | semidirect |
| 64. #410 | C22×C3 | semidirect |
| 101. C2×#41 | C3×D4 | semidirect |
| 105. C22×Q8 | C3×D4 | semidirect |
| 216. C3 | #60400 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×2−1+6 | 2 |
| C3×#3486 | 2 |
| C3×#3028 | 2 |
| C3×#3025 | 2 |
| C3×#2884 | 2 |
| #60400 | 3 |