Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
320 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | #953 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#538 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #538 | D3 |
| hypocenter | C3 | #1600 |
| LCS | G ▷ C22×C3×C4 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #538 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#538 | C2 | semidirect |
| 2. #5046 | C2 | semidirect |
| 3. #5036 | C2 | semidirect |
| 4. #4960 | C2 | semidirect |
| 5. #4538 | C2 | semidirect |
| 6. #4537 | C2 | semidirect |
| 7. #4525 | C2 | semidirect |
| 28. #538 | D3 | semidirect |
| 37. C2×C3×Q8 | D4 | semidirect |
| 41. C2×C3×D4 | D4 | semidirect |
| 54. C3×C4⋊C4 | D4 | semidirect |
| 55. C3×C4⋊C4 | D4 | semidirect |
| 129. C2×Q8 | #67 | semidirect |
| 139. C2×D4 | #67 | semidirect |
| 150. C4⋊C4 | #67 | semidirect |
| 151. C4⋊C4 | #67 | semidirect |
| 262. C3 | #1600 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#538 | 2 |
| #5046 | 2 |
| #5036 | 2 |
| #4960 | 2 |
| #4538 | 2 |
| #4537 | 2 |
| #4525 | 2 |
| #1600 | 3 |