Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
244 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 244 conjugacy classes of subgroups — too many to draw. Only its 116 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 | C2×C42 |
| Frattini, cosocle | C22×C4 | C23 |
| socle | C23 | C22⋊C4 |
| prime radical | C24 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #339 | C2 | direct (N × H) |
| 9. C2×M16 | C4 | semidirect |
| 14. #117 | C4 | semidirect |
| 16. C2×C4⋊C4 | C4 | semidirect |
| 21. C2×C42 | C4 | semidirect |
| 40. M16 | C2×C4 | semidirect |
| 44. C4⋊C4 | C2×C4 | semidirect |
| 51. C42 | C2×C4 | semidirect |
| 103. C2 | #339 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16 | 2 |
| C2×#117 | 2 |
| C22×C42 | 2 |
| #339 | 2 |