Finite Group DatabasebetaFinite Group DatabasebetaOrder 12224.
290 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 290 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 | C22×Q8×C191 | C2 |
| hypercenter | C22×Q8 | D191 |
| hypocenter | C191 | #548 |
| LCS | G ▷ C22×C191 ▷ C191 | |
| UCS | C1 ◁ C22 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#15690288 | C2 | semidirect |
| 2. #20259311 | C2 | semidirect |
| 3. #20259298 | C2 | semidirect |
| 4. #20259296 | C2 | semidirect |
| 5. Q8×Dic191 | C2 | semidirect |
| 6. C4×C22⋊D191 | C2 | semidirect |
| 7. C22×Q8×C191 | C2 | semidirect |
| 31. Q8×C191 | D4 | semidirect |
| 37. C22×C191 | #41 | semidirect |
| 52. C191 | #548 | semidirect |
| 55. C22×Q8 | D191 | semidirect |
| 78. Q8 | C22⋊D191 | semidirect |
| 93. C22 | #15690288 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#15690288 | 2 |
| #20259311 | 2 |
| #20259298 | 2 |
| #20259296 | 2 |
| Q8×Dic191 | 2 |
| C4×C22⋊D191 | 2 |
| C22×Q8×C191 | 2 |
| #548 | 191 |