Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
460 conjugacy classes of subgroups; only the 170 normal subgroups are shown.
This group has 460 conjugacy classes of subgroups — too many to draw. Only its 170 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 | #2331 |
| 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 |
|---|---|---|
| 20. #15690810 | C4 | semidirect |
| 21. #15690809 | C4 | semidirect |
| 22. #15690809 | C4 | semidirect |
| 23. #15690808 | C4 | semidirect |
| 27. C4⋊Q8×C97 | C4 | semidirect |
| 35. C4×#87604 | Q8 | semidirect |
| 103. C97 | #2331 | semidirect |
| 116. 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 |
| #20261679 | 2 |
| C4⋊C4×#87604 | 2 |
| #20261671 | 2 |
| C4⋊Q8×D97 | 2 |
| #2331 | 97 |