Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1620 conjugacy classes of subgroups; only the 573 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 573 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D97 |
| commutator, cosocle | C22×C97 | C25 |
| Frattini | C22 | C24×D97 |
| Fitting | C2×C4×Q8×C97 | C2 |
| socle | C23×C97 | C24 |
| hypercenter | C2×C4×Q8 | D97 |
| hypocenter | C97 | C2×#548 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#13095512 | C2 | semidirect |
| 2. C22×Q8×D97 | C2 | semidirect |
| 3. #20261397 | C2 | direct (N × H) |
| 4. C2×#15690798 | C2 | semidirect |
| 5. C2×#15690790 | C2 | semidirect |
| 6. C2×#15690785 | C2 | semidirect |
| 7. C2×C4×D388 | C2 | semidirect |
| 8. C2×C4×Q8×C97 | C2 | semidirect |
| 10. C2×#13095512 | C22 | semidirect |
| 13. C2×Q8×D97 | C22 | semidirect |
| 18. #15690798 | C22 | semidirect |
| 19. #15690790 | C22 | semidirect |
| 20. #15690785 | C22 | semidirect |
| 21. C4×D388 | C22 | semidirect |
| 27. C4×Q8×C97 | C22 | semidirect |
| 50. C2×Q8×C97 | D4 | semidirect |
| 63. C22×D97 | #41 | semidirect |
| 66. Q8×C97 | C2×D4 | semidirect |
| 93. C2×D97 | C2×#41 | semidirect |
| 99. C2×C97 | #548 | semidirect |
| 106. C97 | C2×#548 | semidirect |
| 113. C2×C4×Q8 | D97 | semidirect |
| 128. C4×Q8 | C2×D97 | semidirect |
| 142. C2×Q8 | D388 | semidirect |
| 173. Q8 | C2×D388 | semidirect |
| 207. C2 | #20261397 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13095512 | 2 |
| C22×Q8×D97 | 2 |
| #20261397 | 2 |
| C2×#15690798 | 2 |
| C2×#15690790 | 2 |
| C2×#15690785 | 2 |
| C2×C4×D388 | 2 |
| C2×C4×Q8×C97 | 2 |
| C2×#548 | 97 |