Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
294 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 294 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #379 |
| commutator | C2×C4 | C22×C4 |
| Frattini, cosocle | C22×C4 | C23 |
| prime radical | C2×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #421 | C2 | semidirect |
| 3. #418 | C2 | semidirect |
| 4. #413 | C2 | semidirect |
| 5. #410 | C2 | semidirect |
| 9. C2×#41 | C4 | semidirect |
| 14. C22×D4 | C4 | semidirect |
| 16. C2×M16 | C22 | semidirect |
| 17. #117 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2) | 2 |
| #421 | 2 |
| #418 | 2 |
| #413 | 2 |
| #410 | 2 |