Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
384 conjugacy classes of subgroups; only the 228 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 228 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 | #117 |
| commutator | C2×C4 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| socle | C2×C32 | #432 |
| prime radical | C23×C32 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C4 ◁ C32×#97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C32×#364 | C2 | semidirect |
| 4. C32×#347 | C2 | semidirect |
| 5. C3×#2165 | C3 | direct (N × H) |
| 15. C3×#364 | C2×C3 | semidirect |
| 17. C3×#347 | C2×C3 | semidirect |
| 30. #2165 | C32 | direct (N × H) |
| 52. #364 | C2×C32 | semidirect |
| 54. #347 | C2×C32 | semidirect |
| 118. C32 | #2165 | direct (N × H) |
| 142. C3 | C3×#2165 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C32×C4×M16 | 2 |
| C32×#364 | 2 |
| C32×#349 | 2 |
| C32×#347 | 2 |
| C3×#2165 | 3 |