Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
224 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C3×#493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#500 | C2 | semidirect |
| 3. C3×#492 | C2 | semidirect |
| 4. C3×#486 | C2 | semidirect |
| 5. C3×C82 | C2 | semidirect |
| 6. #1706 | C3 | direct (N × H) |
| 16. #500 | C2×C3 | semidirect |
| 18. #492 | C2×C3 | semidirect |
| 19. #486 | C2×C3 | semidirect |
| 20. C82 | C2×C3 | semidirect |
| 52. C3×C8 | D8 | semidirect |
| 53. C3×C8 | SD16 | semidirect |
| 84. C8 | C3×D8 | semidirect |
| 85. C8 | C3×SD16 | semidirect |
| 98. C3 | #1706 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#500 | 2 |
| C3×#493 | 2 |
| C3×#492 | 2 |
| C3×#486 | 2 |
| C3×C82 | 2 |
| #1706 | 3 |