Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
444 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 162 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 | C23×#87604 |
| commutator, socle | C22×C97 | C23×C4 |
| Fitting | C4×Q8×C97 | C4 |
| prime radical, cosocle | C22×D97 | C24 |
| hypercenter | C4×Q8 | #87604 |
| hypocenter | C97 | #2320 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 21. Q8×Dic97 | C4 | semidirect |
| 22. C4×Dic194 | C4 | semidirect |
| 23. C4×Dic194 | C4 | semidirect |
| 24. #15690775 | C4 | semidirect |
| 25. #15690775 | C4 | semidirect |
| 30. C4×Q8×C97 | C4 | semidirect |
| 116. C97 | #2320 | semidirect |
| 131. C4×Q8 | #87604 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| (C2×#87604)⋊Q8 | 2 |
| #20261686 | 2 |
| #20261684 | 2 |
| #20261680 | 2 |
| #20261679 | 2 |
| C4×C4⋊#87604 | 2 |
| #20261671 | 2 |
| C4×Q8×D97 | 2 |
| #2320 | 97 |