Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
600 conjugacy classes of subgroups; only the 444 normal subgroups are shown.
This group has 600 conjugacy classes of subgroups — too many to draw. Only its 444 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C24 |
| commutator, Frattini, cosocle | C22 | C25 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #554 | C2 | direct (N × H) |
| 2. C2×C4⋊Q8 | C2 | semidirect |
| 3. C2×#125 | C2 | semidirect |
| 4. C2×C22⋊Q8 | C2 | semidirect |
| 5. C2×C22⋊Q8 | C2 | semidirect |
| 6. C2×C4×Q8 | C2 | semidirect |
| 9. C22×C4⋊C4 | C2 | semidirect |
| 15. C4⋊Q8 | C22 | semidirect |
| 16. #125 | C22 | semidirect |
| 17. C22⋊Q8 | C22 | semidirect |
| 18. C22⋊Q8 | C22 | semidirect |
| 19. C4×Q8 | C22 | semidirect |
| 29. C2×C4⋊C4 | C22 | semidirect |
| 39. C2×D4 | Q8 | semidirect |
| 70. D4 | C2×Q8 | semidirect |
| 103. C2 | #554 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #554 | 2 |
| C2×C4⋊Q8 | 2 |
| C2×#125 | 2 |
| C2×C22⋊Q8 | 2 |
| C2×C4×Q8 | 2 |
| C2×C4×D4 | 2 |
| C22×C4⋊C4 | 2 |