Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
804 conjugacy classes of subgroups; only the 224 normal subgroups are shown.
This group has 804 conjugacy classes of subgroups — too many to draw. Only its 224 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, Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×#121 | C2 | semidirect |
| 2. C3×#396 | C2 | semidirect |
| 3. C3×#392 | C2 | semidirect |
| 4. C3×#392 | C2 | semidirect |
| 5. C3×#385 | C2 | semidirect |
| 6. #2675 | C3 | direct (N × H) |
| 18. C2×#121 | C2×C3 | semidirect |
| 19. #396 | C2×C3 | semidirect |
| 20. #392 | C2×C3 | semidirect |
| 21. #392 | C2×C3 | semidirect |
| 22. #385 | C2×C3 | semidirect |
| 39. C3×C22⋊C4 | D4 | semidirect |
| 86. C22⋊C4 | C3×D4 | semidirect |
| 139. C3 | #2675 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#121 | 2 |
| C3×#396 | 2 |
| C3×#392 | 2 |
| C3×#385 | 2 |
| #2675 | 3 |