Finite Group DatabasebetaFinite Group DatabasebetaOrder 576.
639 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 639 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C22×D9 |
| commutator, cosocle | C22×C9 | C24 |
| Frattini | C22×C3 | C23×D3 |
| Fitting | C2×C9×C22⋊C4 | C2 |
| socle | C23×C3 | C22×D3 |
| hypercenter | C2×C22⋊C4 | D9 |
| hypocenter | C9 | C2×#121 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#611 | C2 | semidirect |
| 2. C22×#611 | C2 | semidirect |
| 3. C22×D36 | C2 | semidirect |
| 4. C22×C4×D9 | C2 | semidirect |
| 5. C2×C9×C22⋊C4 | C2 | semidirect |
| 6. C2×(C2×D9)⋊C4 | C2 | semidirect |
| 7. C2×Dic9⋊C4 | C2 | semidirect |
| 8. #63446 | C2 | direct (N × H) |
| 17. C2×#611 | C22 | semidirect |
| 18. C2×#611 | C22 | semidirect |
| 22. C2×D36 | C22 | semidirect |
| 24. C2×C4×D9 | C22 | semidirect |
| 25. C9×C22⋊C4 | C22 | semidirect |
| 26. (C2×D9)⋊C4 | C22 | semidirect |
| 27. Dic9⋊C4 | C22 | semidirect |
| 46. C22×D9 | D4 | semidirect |
| 56. C2×Dic9 | D4 | semidirect |
| 95. C2×D9 | C2×D4 | semidirect |
| 99. Dic9 | C2×D4 | semidirect |
| 107. C2×C22⋊C4 | D9 | semidirect |
| 134. C2×C9 | #121 | semidirect |
| 157. C22⋊C4 | C2×D9 | semidirect |
| 178. C9 | C2×#121 | semidirect |
| 235. C2 | #63446 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#611 | 2 |
| C22×D36 | 2 |
| C22×C4×D9 | 2 |
| C2×C9×C22⋊C4 | 2 |
| C2×(C2×D9)⋊C4 | 2 |
| C2×Dic9⋊C4 | 2 |
| #63446 | 2 |
| C2×#884 | 3 |