Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
241 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 241 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 |
| 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×#135 | C2 | semidirect |
| 2. C22×Q16 | C2 | semidirect |
| 3. #548 | C2 | semidirect |
| 4. #495 | C2 | semidirect |
| 5. #467 | C2 | semidirect |
| 6. #466 | C2 | semidirect |
| 7. #464 | C2 | semidirect |
| 8. #452 | C2 | semidirect |
| 9. #448 | C2 | semidirect |
| 10. C4×Q16 | C2 | semidirect |
| 11. C8×D4 | C2 | semidirect |
| 58. Q16 | D4 | semidirect |
| 119. C22 | #135 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#135 | 2 |
| C22×Q16 | 2 |
| #548 | 2 |
| #495 | 2 |
| #467 | 2 |
| #466 | 2 |
| #464 | 2 |
| #452 | 2 |
| #448 | 2 |
| C4×Q16 | 2 |
| C8×D4 | 2 |