Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
626 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 196 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 | C23×D4 | C2 |
| 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. C2×C4×D4 | C2 | semidirect |
| 3. C22×C22⋊C4 | C2 | semidirect |
| 4. #390 | C2 | semidirect |
| 5. #386 | C2 | semidirect |
| 6. #380 | C2 | semidirect |
| 7. #379 | C2 | semidirect |
| 8. C4×C22⋊C4 | C2 | semidirect |
| 11. C22×D4 | C4 | semidirect |
| 16. C23×C4 | C22 | semidirect |
| 18. C2×C4⋊C4 | C22 | semidirect |
| 23. C2×C22⋊C4 | C22 | semidirect |
| 24. C2×C22⋊C4 | C22 | semidirect |
| 25. C2×C22⋊C4 | C22 | semidirect |
| 26. C2×C42 | C22 | semidirect |
| 27. #95 | C22 | semidirect |
| 28. #95 | C22 | semidirect |
| 33. C24 | C2×C4 | semidirect |
| 41. C2×D4 | C2×C4 | semidirect |
| 42. C2×D4 | C2×C4 | semidirect |
| 48. C22×C4 | C2×C4 | semidirect |
| 60. C22⋊C4 | D4 | direct (N × H) |
| 85. D4 | C22⋊C4 | direct (N × H) |
| 100. C2×C4 | C2×D4 | semidirect |
| 115. C22 | C2×C22⋊C4 | semidirect |
| 116. C22 | C4×D4 | semidirect |
| 126. C4 | C2×C22⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4 | 2 |
| C2×C4×D4 | 2 |
| C22×C22⋊C4 | 2 |
| #390 | 2 |
| #386 | 2 |
| #380 | 2 |
| #379 | 2 |
| C4×C22⋊C4 | 2 |