Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
456 conjugacy classes of subgroups; only the 312 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 312 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C32×C4 | D8 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C22×C32 | C2×D8 |
| prime radical | C23×C32 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C32×C4 ◁ C22×C32×C4 ◁ C22×C32×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C32×#367 | C2 | semidirect |
| 4. C32×#366 | C2 | semidirect |
| 5. C3×#2158 | C3 | direct (N × H) |
| 8. C2×C32×C16 | C4 | semidirect |
| 15. C3×#367 | C2×C3 | semidirect |
| 16. C3×#366 | C2×C3 | semidirect |
| 27. #2158 | C32 | direct (N × H) |
| 30. C2×C3×C16 | C3×C4 | semidirect |
| 46. #367 | C2×C32 | semidirect |
| 47. #366 | C2×C32 | semidirect |
| 67. C2×C16 | C32×C4 | semidirect |
| 101. C32 | #2158 | direct (N × H) |
| 121. C3 | C3×#2158 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C32×C16 | 2 |
| C32×#427 | 2 |
| C32×#367 | 2 |
| C32×#366 | 2 |
| C3×#2158 | 3 |