Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
282 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 282 conjugacy classes of subgroups — too many to draw. Only its 116 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 |
| prime radical | C2×#121 | C2 |
| 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. C22×SD16 | C2 | semidirect |
| 2. C23×C8 | C2 | semidirect |
| 3. C2×C22⋊Q8 | C2 | semidirect |
| 5. #465 | C2 | direct (N × H) |
| 6. C2×#106 | C2 | semidirect |
| 7. C2×#103 | C2 | semidirect |
| 8. C2×#102 | C2 | semidirect |
| 14. C2×SD16 | C22 | semidirect |
| 18. C22×C8 | C22 | semidirect |
| 20. C22⋊Q8 | C22 | semidirect |
| 28. #106 | C22 | semidirect |
| 29. #103 | C22 | semidirect |
| 30. #102 | C22 | semidirect |
| 55. C2×C8 | D4 | semidirect |
| 71. C23 | SD16 | semidirect |
| 97. C8 | C2×D4 | semidirect |
| 107. C22 | C2×SD16 | semidirect |
| 119. C2 | #465 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×SD16 | 2 |
| C23×C8 | 2 |
| C2×C22⋊Q8 | 2 |
| C2×#121 | 2 |
| #465 | 2 |
| C2×#106 | 2 |
| C2×#103 | 2 |
| C2×#102 | 2 |