Finite Group DatabasebetaFinite Group DatabasebetaOrder 34432.
1492 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 557 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 | C24×D269 |
| commutator, socle, cosocle | C22×C269 | C25 |
| Fitting | C23×Q8×C269 | C2 |
| hypercenter | C23×Q8 | D269 |
| hypocenter | C269 | #3486 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#22575501 | C2 | semidirect |
| 2. Q8×C22⋊D269 | C2 | semidirect |
| 3. #31771266 | C2 | semidirect |
| 4. #31771179 | C2 | semidirect |
| 5. #31771116 | C2 | semidirect |
| 6. #31771083 | C2 | semidirect |
| 7. C23×Q8×C269 | C2 | semidirect |
| 9. C2×#16864403 | C22 | semidirect |
| 11. C2×Q8×D269 | C22 | semidirect |
| 12. #22575451 | C22 | semidirect |
| 13. #22575442 | C22 | semidirect |
| 16. #22575429 | C22 | semidirect |
| 19. Q8×Dic269 | C22 | semidirect |
| 44. C2×Q8×C269 | D4 | semidirect |
| 75. C22×C269 | 2−1+4 | semidirect |
| 88. C269 | #3486 | semidirect |
| 91. C23×Q8 | D269 | semidirect |
| 123. C2×Q8 | C22⋊D269 | semidirect |
| 166. C22 | #22575501 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#22575501 | 2 |
| Q8×C22⋊D269 | 2 |
| #31771266 | 2 |
| #31771179 | 2 |
| #31771116 | 2 |
| #31771083 | 2 |
| C23×Q8×C269 | 2 |
| #3486 | 269 |