Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1112 conjugacy classes of subgroups; only the 575 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 575 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 | Q82×C97 | C2 |
| prime radical | C22×D97 | C24 |
| hypercenter | Q82 | D97 |
| hypocenter | C97 | C2×Q82 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×Dic194 | C2 | semidirect |
| 2. #20261465 | C2 | semidirect |
| 5. Q82×C97 | C2 | semidirect |
| 21. Q8×D97 | Q8 | direct (N × H) |
| 34. Q8×C97 | C2×Q8 | semidirect |
| 42. Dic194 | C2×Q8 | semidirect |
| 52. D97 | Q82 | direct (N × H) |
| 54. C97 | C2×Q82 | semidirect |
| 57. Q82 | D97 | direct (N × H) |
| 90. Q8 | Q8×D97 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×Dic194 | 2 |
| #20261465 | 2 |
| C4⋊Q8×D97 | 2 |
| C4×Q8×D97 | 2 |
| Q82×C97 | 2 |
| C2×Q82 | 97 |