Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
420 conjugacy classes of subgroups; only the 300 normal subgroups are shown.
This group has 420 conjugacy classes of subgroups — too many to draw. Only its 300 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C32×C4 | C2×D8 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C2×C32 | C2×#107 |
| prime radical | C23×C32 | Q16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ 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×#368 | C2 | semidirect |
| 4. C3×#2161 | C3 | direct (N × H) |
| 13. C3×#368 | C2×C3 | semidirect |
| 23. #2161 | C32 | direct (N × H) |
| 41. #368 | C2×C32 | semidirect |
| 90. C32 | #2161 | direct (N × H) |
| 109. C3 | C3×#2161 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×M32 | 2 |
| C2×C32×#108 | 2 |
| C32×#368 | 2 |
| C3×#2161 | 3 |