Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 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×D191 |
| commutator, socle, cosocle | C22×C191 | C24 |
| Fitting | C4⋊Q8×C191 | C2 |
| prime radical | C2×D764 | C22 |
| hypercenter | C4⋊Q8 | D191 |
| hypocenter | C191 | D4×Q8 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×D191 | C2 | semidirect |
| 2. #20259321 | C2 | semidirect |
| 3. #20259300 | C2 | semidirect |
| 4. #20259298 | C2 | semidirect |
| 5. C4×Dic382 | C2 | semidirect |
| 8. C4⋊Q8×C191 | C2 | semidirect |
| 28. Dic382 | D4 | semidirect |
| 29. D764 | Q8 | semidirect |
| 43. C2×D191 | C2×Q8 | semidirect |
| 47. C4×C191 | C2×Q8 | semidirect |
| 55. C191 | D4×Q8 | semidirect |
| 59. C4⋊Q8 | D191 | semidirect |
| 106. C4 | Q8×D191 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D191 | 2 |
| #20259321 | 2 |
| #20259300 | 2 |
| #20259298 | 2 |
| C4×Dic382 | 2 |
| C4×D764 | 2 |
| #20259284 | 2 |
| C4⋊Q8×C191 | 2 |
| D4×Q8 | 191 |