Finite Group DatabasebetaFinite Group DatabasebetaOrder 3968.
5872 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.
This group has 5872 conjugacy classes of subgroups — too many to draw. Only its 3221 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D31 |
| commutator, cosocle | C2×C31 | C26 |
| Frattini | C2 | C25×D31 |
| Fitting | C23×Q8×C31 | C2 |
| socle | C23×C31 | C24 |
| hypercenter | C23×Q8 | D31 |
| hypocenter | C31 | C22×2−1+4 |
| LCS | G ▷ C2×C31 ▷ C31 | |
| UCS | C1 ◁ C23 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#11758564 | C2 | direct (N × H) |
| 2. C22×#92499 | C2 | semidirect |
| 3. C22×#92495 | C2 | semidirect |
| 4. C22×Q8×D31 | C2 | semidirect |
| 5. C23×Q8×C31 | C2 | semidirect |
| 7. C22×Q8×C31 | C22 | semidirect |
| 10. #11758564 | C22 | direct (N × H) |
| 11. C2×#92499 | C22 | semidirect |
| 12. C2×Q8×D31 | C22 | semidirect |
| 13. C2×#92495 | C22 | semidirect |
| 20. C2×Q8×C31 | C23 | semidirect |
| 25. #92499 | C23 | semidirect |
| 26. Q8×D31 | C23 | semidirect |
| 27. #92495 | C23 | semidirect |
| 46. C22×C31 | 2−1+4 | semidirect |
| 57. C23×Q8 | D31 | semidirect |
| 59. C2×C31 | C2×2−1+4 | semidirect |
| 68. C22×Q8 | C2×D31 | semidirect |
| 73. C31 | C22×2−1+4 | semidirect |
| 81. C2×Q8 | C22×D31 | semidirect |
| 104. C22 | #11758564 | direct (N × H) |
| 111. C2 | C2×#11758564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#11758564 | 2 |
| C22×#92499 | 2 |
| C22×#92495 | 2 |
| C22×Q8×D31 | 2 |
| C23×Q8×C31 | 2 |
| C22×2−1+4 | 31 |