Finite Group DatabasebetaFinite Group DatabasebetaOrder 12352.
260 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 91 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×D193 |
| commutator, socle, cosocle | C22×C193 | C24 |
| Fitting | #124×C193 | C2 |
| hypercenter | #124 | D193 |
| hypocenter | C193 | H(3,𝔽4) |
| LCS | G ▷ C22×C193 ▷ C193 | |
| UCS | C1 ◁ C22 ◁ #124 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20259888 | C2 | semidirect |
| 2. #20259887 | C2 | semidirect |
| 3. #20259885 | C2 | semidirect |
| 4. #20259869 | C2 | semidirect |
| 5. #20259867 | C2 | semidirect |
| 6. #124×C193 | C2 | semidirect |
| 7. C23×D193 | C22 | semidirect |
| 8. C2×Dic386 | C22 | semidirect |
| 12. C22⋊Dic193 | C22 | semidirect |
| 13. #15690532 | C22 | semidirect |
| 14. #15690532 | C22 | semidirect |
| 15. #15690532 | C22 | semidirect |
| 16. C4×Dic193 | C22 | semidirect |
| 17. C2×Q8×C193 | C22 | semidirect |
| 19. C22⋊C4×C193 | C22 | semidirect |
| 20. C42×C193 | C22 | semidirect |
| 50. C193 | H(3,𝔽4) | semidirect |
| 55. #124 | D193 | semidirect |
| 60. C2×Q8 | C2×D193 | semidirect |
| 69. C22⋊C4 | C2×D193 | semidirect |
| 71. C42 | C2×D193 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20259888 | 2 |
| #20259887 | 2 |
| #20259885 | 2 |
| #20259869 | 2 |
| #20259867 | 2 |
| #124×C193 | 2 |
| H(3,𝔽4) | 193 |