Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
528 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 204 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×C32 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C32 |
| prime radical | C32×#459 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C32×#486 | C2 | semidirect |
| 2. C32×#476 | C2 | semidirect |
| 4. C32×C8×D4 | C2 | semidirect |
| 5. C32×#334 | C2 | semidirect |
| 6. C32×#333 | C2 | semidirect |
| 7. C32×#331 | C2 | semidirect |
| 8. C3×#1677 | C3 | direct (N × H) |
| 24. C3×#486 | C2×C3 | semidirect |
| 25. C3×#476 | C2×C3 | semidirect |
| 27. C3×C8×D4 | C2×C3 | semidirect |
| 28. C3×#334 | C2×C3 | semidirect |
| 29. C3×#333 | C2×C3 | semidirect |
| 30. C3×#331 | C2×C3 | semidirect |
| 49. #1677 | C32 | direct (N × H) |
| 87. #486 | C2×C32 | semidirect |
| 88. #476 | C2×C32 | semidirect |
| 90. C8×D4 | C2×C32 | semidirect |
| 91. #334 | C2×C32 | semidirect |
| 92. #333 | C2×C32 | semidirect |
| 93. #331 | C2×C32 | semidirect |
| 200. C32 | #1677 | direct (N × H) |
| 242. C3 | C3×#1677 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#486 | 2 |
| C32×#476 | 2 |
| C32×#459 | 2 |
| C32×C8×D4 | 2 |
| C32×#334 | 2 |
| C32×#333 | 2 |
| C32×#331 | 2 |
| C3×#1677 | 3 |