Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
396 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C23 | C2×D4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C8 | C2 | semidirect |
| 3. C2×#117 | C2 | semidirect |
| 4. #416 | C2 | direct (N × H) |
| 5. C2×#103 | C2 | semidirect |
| 6. C2×#102 | C2 | semidirect |
| 9. C2×#41 | C4 | semidirect |
| 18. C22×C8 | C22 | semidirect |
| 20. #117 | C22 | semidirect |
| 24. #103 | C22 | semidirect |
| 25. #102 | C22 | semidirect |
| 30. #41 | C2×C4 | semidirect |
| 104. C2 | #416 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41 | 2 |
| C23×C8 | 2 |
| C2×#117 | 2 |
| #416 | 2 |
| C2×#103 | 2 |
| C2×#102 | 2 |