Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
944 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 944 conjugacy classes of subgroups — too many to draw. Only its 163 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 | #393×C97 | C2 |
| hypercenter | #393 | D97 |
| hypocenter | C97 | #2388 |
| LCS | G ▷ C23×C97 ▷ C97 | |
| UCS | C1 ◁ C23 ◁ #393 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#13095512 | C2 | semidirect |
| 2. C2×#15690770 | C2 | semidirect |
| 3. #20260654 | C2 | semidirect |
| 4. #20260653 | C2 | semidirect |
| 5. #20260644 | C2 | semidirect |
| 6. #20260582 | C2 | semidirect |
| 7. #393×C97 | C2 | semidirect |
| 34. C2×C4×D97 | D4 | semidirect |
| 46. C23×C97 | #41 | semidirect |
| 64. C2×Dic97 | #41 | semidirect |
| 89. C97 | #2388 | semidirect |
| 94. #393 | D97 | semidirect |
| 140. C23 | #13095512 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#13095512 | 2 |
| C2×#15690770 | 2 |
| #20260654 | 2 |
| #20260653 | 2 |
| #20260644 | 2 |
| #20260582 | 2 |
| #393×C97 | 2 |
| #2388 | 97 |