Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
444 conjugacy classes of subgroups; only the 240 normal subgroups are shown.
This group has 444 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, prime radical | C22×C32 | C2×D8 |
| commutator | C8 | C22×C32×C4 |
| Frattini, cosocle | C2×C8 | C23×C32 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ C32×C42 ◁ C32×C4×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C3×#2176 | C3 | direct (N × H) |
| 9. C32×Q32 | C4 | semidirect |
| 31. #2176 | C32 | direct (N × H) |
| 34. C3×Q32 | C3×C4 | semidirect |
| 73. Q32 | C32×C4 | semidirect |
| 107. C32 | #2176 | direct (N × H) |
| 125. C3 | C3×#2176 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×Q32 | 2 |
| C32×C4×Q16 | 2 |
| C32×#367 | 2 |
| C32×#358 | 2 |
| C32×#346 | 2 |
| C3×#2176 | 3 |