Finite Group DatabasebetaFinite Group DatabasebetaOrder 3968.
1620 conjugacy classes of subgroups; only the 573 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 573 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 | C22×C31 | C25 |
| Frattini | C22 | C24×D31 |
| Fitting | C2×C4×Q8×C31 | C2 |
| socle | C23×C31 | C24 |
| hypercenter | C2×C4×Q8 | D31 |
| hypocenter | C31 | C2×#548 |
| LCS | G ▷ C22×C31 ▷ C31 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#92499 | C2 | semidirect |
| 2. C22×Q8×D31 | C2 | semidirect |
| 3. #14521711 | C2 | direct (N × H) |
| 4. C2×#11758452 | C2 | semidirect |
| 5. C2×#11758474 | C2 | semidirect |
| 6. C2×#11758472 | C2 | semidirect |
| 7. C2×C4×D124 | C2 | semidirect |
| 8. C2×C4×Q8×C31 | C2 | semidirect |
| 11. C2×#92499 | C22 | semidirect |
| 13. C2×Q8×D31 | C22 | semidirect |
| 19. C4×Q8×C31 | C22 | semidirect |
| 26. #11758474 | C22 | semidirect |
| 27. #11758472 | C22 | semidirect |
| 28. #11758452 | C22 | semidirect |
| 29. C4×D124 | C22 | semidirect |
| 32. C2×Q8×C31 | D4 | semidirect |
| 63. C22×D31 | #41 | semidirect |
| 66. Q8×C31 | C2×D4 | semidirect |
| 94. C2×D31 | C2×#41 | semidirect |
| 106. C2×C4×Q8 | D31 | semidirect |
| 108. C2×C31 | #548 | semidirect |
| 127. C4×Q8 | C2×D31 | semidirect |
| 135. C31 | C2×#548 | semidirect |
| 142. C2×Q8 | D124 | semidirect |
| 173. Q8 | C2×D124 | semidirect |
| 207. C2 | #14521711 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#92499 | 2 |
| C22×Q8×D31 | 2 |
| #14521711 | 2 |
| C2×#11758452 | 2 |
| C2×#11758474 | 2 |
| C2×#11758472 | 2 |
| C2×C4×D124 | 2 |
| C2×C4×Q8×C31 | 2 |
| C2×#548 | 31 |