Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1240 conjugacy classes of subgroups; only the 501 normal subgroups are shown.
This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 501 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×#555 |
| 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. #20261451 | C2 | direct (N × H) |
| 2. C2×#15690799 | C2 | semidirect |
| 3. C2×#15690785 | C2 | semidirect |
| 4. C2×C4×D388 | C2 | semidirect |
| 5. C2×#15690768 | C2 | semidirect |
| 6. C2×C4⋊Q8×C97 | C2 | semidirect |
| 10. #15690799 | C22 | semidirect |
| 11. #15690785 | C22 | semidirect |
| 12. C4×D388 | C22 | semidirect |
| 13. #15690768 | C22 | semidirect |
| 19. C4⋊Q8×C97 | C22 | semidirect |
| 69. C2×C97 | #555 | semidirect |
| 74. C97 | C2×#555 | semidirect |
| 76. C2×C4⋊Q8 | D97 | semidirect |
| 85. C4⋊Q8 | C2×D97 | semidirect |
| 144. C2 | #20261451 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261451 | 2 |
| C2×#15690799 | 2 |
| C2×#15690785 | 2 |
| C2×C4×D388 | 2 |
| C2×#15690768 | 2 |
| C2×C4⋊Q8×C97 | 2 |
| C2×#555 | 97 |