Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
460 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 184 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 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#577 | C2 | semidirect |
| 2. C3×#492 | C2 | semidirect |
| 3. C3×#471 | C2 | semidirect |
| 4. C3×#456 | C2 | semidirect |
| 5. C3×#453 | C2 | semidirect |
| 6. C3×#443 | C2 | semidirect |
| 7. #3291 | C3 | direct (N × H) |
| 9. C3×H(5,𝔽2) | C22 | semidirect |
| 11. C3×#137 | C22 | semidirect |
| 17. C2×C3×SD16 | C22 | semidirect |
| 18. C3×#131 | C22 | semidirect |
| 19. C3×C4⋊Q8 | C22 | semidirect |
| 21. C3×#108 | C22 | semidirect |
| 22. C3×#104 | C22 | semidirect |
| 23. C3×#101 | C22 | semidirect |
| 24. C3×#100 | C22 | semidirect |
| 25. C3×C4×C8 | C22 | semidirect |
| 26. #577 | C2×C3 | semidirect |
| 27. #492 | C2×C3 | semidirect |
| 28. #471 | C2×C3 | semidirect |
| 29. #456 | C2×C3 | semidirect |
| 30. #453 | C2×C3 | semidirect |
| 31. #443 | C2×C3 | semidirect |
| 43. C3×Q16 | D4 | semidirect |
| 48. C3×SD16 | D4 | semidirect |
| 50. C3×D8 | D4 | semidirect |
| 58. H(5,𝔽2) | C22×C3 | semidirect |
| 60. #137 | C22×C3 | semidirect |
| 66. C2×SD16 | C22×C3 | semidirect |
| 67. #131 | C22×C3 | semidirect |
| 68. C4⋊Q8 | C22×C3 | semidirect |
| 70. #108 | C22×C3 | semidirect |
| 71. #104 | C22×C3 | semidirect |
| 72. #101 | C22×C3 | semidirect |
| 73. #100 | C22×C3 | semidirect |
| 74. C4×C8 | C22×C3 | semidirect |
| 106. Q16 | C3×D4 | semidirect |
| 111. SD16 | C3×D4 | semidirect |
| 113. D8 | C3×D4 | semidirect |
| 162. C3 | #3291 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#577 | 2 |
| C3×#492 | 2 |
| C3×#471 | 2 |
| C3×#456 | 2 |
| C3×#453 | 2 |
| C3×#443 | 2 |
| #3291 | 3 |