Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
524 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 524 conjugacy classes of subgroups — too many to draw. Only its 124 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D4 | C2 | semidirect |
| 2. C22×D8 | C2 | semidirect |
| 3. C2×#121 | C2 | semidirect |
| 4. #447 | C2 | direct (N × H) |
| 5. C2×#102 | C2 | semidirect |
| 6. C2×C22⋊C8 | C2 | semidirect |
| 13. C22×D4 | C22 | semidirect |
| 14. C22×D4 | C22 | semidirect |
| 17. C2×D8 | C22 | semidirect |
| 18. C22×C8 | C22 | semidirect |
| 20. #121 | C22 | semidirect |
| 21. C2×C4⋊C4 | C22 | semidirect |
| 23. #102 | C22 | semidirect |
| 24. C22⋊C8 | C22 | semidirect |
| 42. C2×D4 | D4 | semidirect |
| 44. C2×D4 | C23 | semidirect |
| 53. C2×C8 | C23 | semidirect |
| 55. C4⋊C4 | C23 | semidirect |
| 71. C23 | D8 | semidirect |
| 80. D4 | C2×D4 | semidirect |
| 105. C22 | C2×D8 | semidirect |
| 115. C2 | #447 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4 | 2 |
| C22×D8 | 2 |
| C2×#121 | 2 |
| #447 | 2 |
| C2×#102 | 2 |
| C2×C22⋊C8 | 2 |