Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
584 conjugacy classes of subgroups; only the 208 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 208 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×AGL(1,ℤ/8) | C2 | semidirect |
| 2. C2×C3×#127 | C2 | semidirect |
| 3. C3×C4×#41 | C2 | semidirect |
| 4. C3×#460 | C2 | semidirect |
| 5. C3×#459 | C2 | semidirect |
| 6. C3×#423 | C2 | semidirect |
| 7. C3×#418 | C2 | semidirect |
| 8. #3038 | C3 | direct (N × H) |
| 13. C2×C3×SD16 | C22 | semidirect |
| 14. C2×C3×D8 | C22 | semidirect |
| 19. C3×C4×Q8 | C22 | semidirect |
| 21. C3×C4×D4 | C22 | semidirect |
| 26. C3×C4⋊C8 | C22 | semidirect |
| 27. C3×#102 | C22 | semidirect |
| 28. C2×AGL(1,ℤ/8) | C2×C3 | semidirect |
| 29. C2×#127 | C2×C3 | semidirect |
| 30. C4×#41 | C2×C3 | semidirect |
| 31. #460 | C2×C3 | semidirect |
| 32. #459 | C2×C3 | semidirect |
| 33. #423 | C2×C3 | semidirect |
| 34. #418 | C2×C3 | semidirect |
| 37. C3×#41 | D4 | semidirect |
| 69. C2×SD16 | C22×C3 | semidirect |
| 70. C2×D8 | C22×C3 | semidirect |
| 75. C4×Q8 | C22×C3 | semidirect |
| 77. C4×D4 | C22×C3 | semidirect |
| 82. C4⋊C8 | C22×C3 | semidirect |
| 83. #102 | C22×C3 | semidirect |
| 90. C3×Q8 | C2×D4 | semidirect |
| 98. C3×D4 | C2×D4 | semidirect |
| 113. #41 | C3×D4 | semidirect |
| 153. C3×C4 | AGL(1,ℤ/8) | semidirect |
| 162. Q8 | C2×C3×D4 | semidirect |
| 170. D4 | C2×C3×D4 | semidirect |
| 200. C4 | C3×AGL(1,ℤ/8) | semidirect |
| 203. C3 | #3038 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×AGL(1,ℤ/8) | 2 |
| C2×C3×#127 | 2 |
| C3×C4×#41 | 2 |
| C3×#460 | 2 |
| C3×#459 | 2 |
| C3×#423 | 2 |
| C3×#418 | 2 |
| #3038 | 3 |