Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
242 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C24 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #391 | C2 | semidirect |
| 5. #387 | C2 | semidirect |
| 6. #387 | C2 | semidirect |
| 8. C4×C4⋊C4 | C2 | semidirect |
| 12. C22⋊Q8 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊Q8 | 2 |
| #391 | 2 |
| #388 | 2 |
| #387 | 2 |
| #381 | 2 |
| C4×C4⋊C4 | 2 |
| C4×C22⋊C4 | 2 |