Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
760 conjugacy classes of subgroups; only the 498 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 498 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | #595 |
| commutator, hypocenter | C17 | C22×C4×C8 |
| Frattini | C22 | C23×#595 |
| Fitting | C23×C4×C17 | C4 |
| socle, prime radical | C24×C17 | C2×C4 |
| cosocle | C2×Dic17 | C24 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×#3624 | C2 | direct (N × H) |
| 7. C22×#3624 | C4 | direct (N × H) |
| 11. C4×#3624 | C22 | direct (N × H) |
| 13. C22×C4×C17 | C8 | semidirect |
| 15. C22×Dic17 | C8 | semidirect |
| 20. C2×#3624 | C2×C4 | direct (N × H) |
| 28. C2×C4×C17 | C2×C8 | semidirect |
| 31. C2×Dic17 | C2×C8 | semidirect |
| 36. #3624 | C22×C4 | direct (N × H) |
| 38. C22×C17 | C4×C8 | semidirect |
| 43. C4×C17 | C22×C8 | semidirect |
| 45. Dic17 | C22×C8 | semidirect |
| 50. C2×C17 | C2×C4×C8 | semidirect |
| 62. C17 | C22×C4×C8 | semidirect |
| 69. C22×C4 | #3624 | direct (N × H) |
| 85. C2×C4 | C2×#3624 | direct (N × H) |
| 88. C22 | C4×#3624 | direct (N × H) |
| 96. C4 | C22×#3624 | direct (N × H) |
| 97. C2 | C2×C4×#3624 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#3624 | 2 |
| C2×C4×#3624 | 2 |
| C22×C4×Dic17 | 2 |
| C22×C4×C8 | 17 |