Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
181 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 181 conjugacy classes of subgroups — too many to draw. Only its 88 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 | #127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #557 | C2 | semidirect |
| 3. #497 | C2 | semidirect |
| 5. #464 | C2 | semidirect |
| 6. #462 | C2 | semidirect |
| 7. #461 | C2 | semidirect |
| 9. #446 | C2 | semidirect |
| 10. #440 | C2 | semidirect |
| 11. C4×Q16 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #559 | 2 |
| #557 | 2 |
| #497 | 2 |
| #492 | 2 |
| #464 | 2 |
| #462 | 2 |
| #461 | 2 |
| #460 | 2 |
| #446 | 2 |
| #440 | 2 |
| C4×Q16 | 2 |