Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
328 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 172 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 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, socle, cosocle | C23 | C24 |
| prime radical | C24 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×Q8 | C2 | semidirect |
| 4. #391 | C2 | semidirect |
| 5. #389 | C2 | semidirect |
| 6. #387 | C2 | semidirect |
| 9. C4×C4⋊C4 | C2 | semidirect |
| 10. C4×C4⋊C4 | C2 | semidirect |
| 16. C22⋊Q8 | C4 | direct (N × H) |
| 41. C2×Q8 | C2×C4 | semidirect |
| 50. C22×C4 | Q8 | semidirect |
| 65. C4⋊C4 | C2×C4 | semidirect |
| 66. C4⋊C4 | C2×C4 | semidirect |
| 123. C22 | C4×Q8 | semidirect |
| 140. C4 | C22⋊Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C22⋊Q8 | 2 |
| C2×C4×Q8 | 2 |
| C22×C42 | 2 |
| #391 | 2 |
| #389 | 2 |
| #387 | 2 |
| #385 | 2 |
| #380 | 2 |
| C4×C4⋊C4 | 2 |
| C4×C22⋊C4 | 2 |