Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
2525 conjugacy classes of subgroups; only the 357 normal subgroups are shown.
This group has 2525 conjugacy classes of subgroups — too many to draw. Only its 357 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C22×A4 |
| commutator, hypocenter | Q8 | C24×C9 |
| Frattini | C2×C3 | C24×A4 |
| Fitting, prime radical | C22×C3×H(5,𝔽2) | C3 |
| cosocle | C3×Q8 | C24×C3 |
| hypercenter | C22×C3×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23×C3 ◁ C22×C3×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#63700 | C2 | direct (N × H) |
| 2. C22×#41⋊C9 | C2 | semidirect |
| 5. #63700 | C22 | direct (N × H) |
| 6. C2×#41⋊C9 | C22 | semidirect |
| 7. C22×Q8⋊C9 | C22 | semidirect |
| 15. #41⋊C9 | C23 | semidirect |
| 16. C2×Q8⋊C9 | C23 | semidirect |
| 17. C22×H(5,𝔽2) | C9 | semidirect |
| 36. Q8⋊C9 | C24 | semidirect |
| 38. C2×H(5,𝔽2) | C2×C9 | semidirect |
| 68. H(5,𝔽2) | C22×C9 | semidirect |
| 151. C22 | #63700 | direct (N × H) |
| 158. C2 | C2×#63700 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#63700 | 2 |
| C22×#41⋊C9 | 2 |
| C22×C3×H(5,𝔽2) | 3 |
| C22×Q8×C9 | 4 |