Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
266 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 144 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. #387 | C2 | semidirect |
| 3. #387 | C2 | semidirect |
| 5. #383 | C2 | semidirect |
| 6. #382 | C2 | semidirect |
| 8. C4×C4⋊C4 | C2 | semidirect |
| 10. #376 | C2 | semidirect |
| 14. #117 | C4 | semidirect |
| 54. C4⋊C4 | C2×C4 | semidirect |
| 55. C4⋊C4 | C2×C4 | semidirect |
| 61. C42 | C2×C4 | semidirect |
| 62. C42 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#117 | 2 |
| #387 | 2 |
| #385 | 2 |
| #383 | 2 |
| #382 | 2 |
| #381 | 2 |
| C4×C4⋊C4 | 2 |
| C4×C22⋊C4 | 2 |
| #376 | 2 |
| C2×#95 | 2 |