Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
780 conjugacy classes of subgroups; only the 520 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 520 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 | C24 |
| commutator | C2 | C22×C3×C42 |
| Frattini, cosocle | C23 | C24×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C4×C4⋊C4 | C2 | semidirect |
| 3. C3×C4×C22⋊C4 | C2 | semidirect |
| 4. C3×#376 | C2 | semidirect |
| 5. C2×C3×#95 | C2 | semidirect |
| 6. #2275 | C3 | direct (N × H) |
| 9. C3×C4×D4 | C4 | semidirect |
| 18. C4×C4⋊C4 | C2×C3 | semidirect |
| 19. C4×C22⋊C4 | C2×C3 | semidirect |
| 20. #376 | C2×C3 | semidirect |
| 21. C2×#95 | C2×C3 | semidirect |
| 25. C22×C3×C4 | C2×C4 | semidirect |
| 30. C3×C4⋊C4 | C2×C4 | semidirect |
| 31. C3×C22⋊C4 | C2×C4 | semidirect |
| 33. C3×C42 | C2×C4 | semidirect |
| 36. C4×D4 | C3×C4 | semidirect |
| 48. C3×D4 | C42 | semidirect |
| 61. C22×C4 | C2×C3×C4 | semidirect |
| 66. C4⋊C4 | C2×C3×C4 | semidirect |
| 67. C22⋊C4 | C2×C3×C4 | semidirect |
| 69. C42 | C2×C3×C4 | semidirect |
| 84. D4 | C3×C42 | semidirect |
| 108. C3 | #2275 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×D4 | 2 |
| C3×C4×C4⋊C4 | 2 |
| C3×C4×C22⋊C4 | 2 |
| C3×#376 | 2 |
| C2×C3×#95 | 2 |
| #2275 | 3 |