Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
924 conjugacy classes of subgroups; only the 201 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 201 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D97 |
| commutator, socle, cosocle | C23×C97 | C24 |
| Fitting | C2×#121×C97 | C2 |
| hypercenter | C2×#121 | D97 |
| hypocenter | C97 | #2856 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ C2×#121 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690806 | C2 | semidirect |
| 2. C2×#15690784 | C2 | semidirect |
| 3. C2×#15690782 | C2 | semidirect |
| 4. C22×C4×Dic97 | C2 | semidirect |
| 5. #20260638 | C2 | semidirect |
| 6. #20260636 | C2 | semidirect |
| 7. #20260635 | C2 | semidirect |
| 8. #20260604 | C2 | semidirect |
| 9. C2×#121×C97 | C2 | semidirect |
| 43. C22×Dic97 | D4 | semidirect |
| 54. C4×Dic97 | D4 | semidirect |
| 64. C22×C4×C97 | D4 | semidirect |
| 118. C22×C97 | #127 | semidirect |
| 131. C4×C97 | #121 | semidirect |
| 134. Dic97 | #121 | semidirect |
| 144. C97 | #2856 | semidirect |
| 148. C2×#121 | D97 | semidirect |
| 200. C22×C4 | C22⋊D97 | semidirect |
| 267. C22 | #15690806 | semidirect |
| 278. C4 | #15690782 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690806 | 2 |
| C2×#15690784 | 2 |
| C2×#15690782 | 2 |
| C22×C4×Dic97 | 2 |
| #20260638 | 2 |
| #20260636 | 2 |
| #20260635 | 2 |
| #20260604 | 2 |
| C2×#121×C97 | 2 |
| #2856 | 97 |