Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
680 conjugacy classes of subgroups; only the 236 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 236 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| prime radical | C23×D4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C4⋊C4 | C2 | semidirect |
| 3. C22×C22⋊C4 | C2 | semidirect |
| 4. C22×C42 | C2 | semidirect |
| 5. #390 | C2 | direct (N × H) |
| 9. C22×D4 | C4 | semidirect |
| 15. C2×C4⋊C4 | C22 | semidirect |
| 17. C2×C22⋊C4 | C22 | semidirect |
| 19. C2×C42 | C22 | semidirect |
| 30. C2×D4 | C2×C4 | semidirect |
| 38. C22×C4 | D4 | semidirect |
| 68. C2×C4 | C22⋊C4 | semidirect |
| 72. C2×C4 | C2×D4 | semidirect |
| 91. C4 | C2×C22⋊C4 | semidirect |
| 93. C2 | #390 | 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×C4⋊C4 | 2 |
| C22×C22⋊C4 | 2 |
| C22×C42 | 2 |
| #390 | 2 |