Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1492 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D97 |
| commutator, socle, cosocle | C22×C97 | C25 |
| Fitting | C23×Q8×C97 | C2 |
| hypercenter | C23×Q8 | D97 |
| hypocenter | C97 | #3486 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690858 | C2 | semidirect |
| 2. Q8×C22⋊D97 | C2 | semidirect |
| 3. #20261396 | C2 | semidirect |
| 4. #20261309 | C2 | semidirect |
| 5. #20261246 | C2 | semidirect |
| 6. #20261213 | C2 | semidirect |
| 7. C23×Q8×C97 | C2 | semidirect |
| 9. C2×#13095512 | C22 | semidirect |
| 11. C2×Q8×D97 | C22 | semidirect |
| 12. #15690808 | C22 | semidirect |
| 13. #15690799 | C22 | semidirect |
| 16. #15690786 | C22 | semidirect |
| 19. Q8×Dic97 | C22 | semidirect |
| 44. C2×Q8×C97 | D4 | semidirect |
| 75. C22×C97 | 2−1+4 | semidirect |
| 89. C97 | #3486 | semidirect |
| 91. C23×Q8 | D97 | semidirect |
| 123. C2×Q8 | C22⋊D97 | semidirect |
| 166. C22 | #15690858 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690858 | 2 |
| Q8×C22⋊D97 | 2 |
| #20261396 | 2 |
| #20261309 | 2 |
| #20261246 | 2 |
| #20261213 | 2 |
| C23×Q8×C97 | 2 |
| #3486 | 97 |