Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
760 conjugacy classes of subgroups; only the 407 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 407 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | #3633 |
| commutator, hypocenter | C17 | C22×C4×C8 |
| Frattini | C2 | C23×#3633 |
| Fitting | C22×C4×C17 | C8 |
| socle | C23×C17 | C2×C8 |
| prime radical | C23×D17 | C2×C4 |
| cosocle | C2×#595 | C24 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C22×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×#3633 | C2 | direct (N × H) |
| 6. C22×#3633 | C4 | direct (N × H) |
| 11. C4×#3633 | C22 | direct (N × H) |
| 13. C22×#595 | C8 | semidirect |
| 18. C2×#3633 | C2×C4 | direct (N × H) |
| 20. C22×C4×C17 | C8 | semidirect |
| 21. C22×Dic17 | C8 | semidirect |
| 22. C2×C4×D17 | C8 | semidirect |
| 32. C2×#595 | C2×C8 | semidirect |
| 37. #3633 | C22×C4 | direct (N × H) |
| 38. C2×C4×C17 | C2×C8 | semidirect |
| 39. C2×Dic17 | C2×C8 | semidirect |
| 40. C4×D17 | C2×C8 | semidirect |
| 43. C22×C17 | C4×C8 | semidirect |
| 45. C2×D17 | C4×C8 | semidirect |
| 51. #595 | C22×C8 | semidirect |
| 53. C4×C17 | C22×C8 | semidirect |
| 54. Dic17 | C22×C8 | semidirect |
| 58. C2×C17 | C2×C4×C8 | semidirect |
| 60. D17 | C2×C4×C8 | semidirect |
| 71. C17 | C22×C4×C8 | semidirect |
| 81. C22×C4 | #3633 | direct (N × H) |
| 100. C2×C4 | C2×#3633 | direct (N × H) |
| 107. C22 | C4×#3633 | direct (N × H) |
| 113. C4 | C22×#3633 | direct (N × H) |
| 117. C2 | C2×C4×#3633 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#3633 | 2 |
| C2×C4×#3633 | 2 |
| C22×C4×#595 | 2 |
| C22×C4×C8 | 17 |