Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
236 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 96 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #548 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#137 | C2 | semidirect |
| 2. C22×Q16 | C2 | semidirect |
| 3. D4×Q8 | C2 | semidirect |
| 5. #497 | C2 | semidirect |
| 6. #470 | C2 | semidirect |
| 7. #465 | C2 | semidirect |
| 8. #464 | C2 | semidirect |
| 9. #462 | C2 | semidirect |
| 10. #452 | C2 | semidirect |
| 11. #451 | C2 | semidirect |
| 12. #451 | C2 | semidirect |
| 13. #448 | C2 | semidirect |
| 14. #441 | C2 | semidirect |
| 15. #435 | C2 | semidirect |
| 92. Q16 | D4 | semidirect |
| 176. C22 | #137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#137 | 2 |
| C22×Q16 | 2 |
| D4×Q8 | 2 |
| #548 | 2 |
| #497 | 2 |
| #470 | 2 |
| #465 | 2 |
| #464 | 2 |
| #462 | 2 |
| #452 | 2 |
| #451 | 2 |
| #448 | 2 |
| #441 | 2 |
| #435 | 2 |