Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
594 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 594 conjugacy classes of subgroups — too many to draw. Only its 240 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 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C32 |
| prime radical | C32×#127 | C22 |
| 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×#500 | C2 | semidirect |
| 2. C32×#494 | C2 | semidirect |
| 5. C32×#322 | C2 | semidirect |
| 6. C3×#1722 | C3 | direct (N × H) |
| 16. C3×#500 | C2×C3 | semidirect |
| 17. C3×#494 | C2×C3 | semidirect |
| 20. C3×#322 | C2×C3 | semidirect |
| 33. #1722 | C32 | direct (N × H) |
| 54. #500 | C2×C32 | semidirect |
| 55. #494 | C2×C32 | semidirect |
| 58. #322 | C2×C32 | semidirect |
| 120. C32 | #1722 | direct (N × H) |
| 142. C3 | C3×#1722 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×#500 | 2 |
| C32×#494 | 2 |
| C32×#492 | 2 |
| C32×#486 | 2 |
| C32×#322 | 2 |
| C3×#1722 | 3 |