Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
117 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 117 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #99 |
| commutator | C23 | C42 |
| Frattini, prime radical, cosocle | C22×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C23×C4 | C4 | semidirect |
| 9. C2×C22⋊C4 | C4 | semidirect |
| 11. #100 | C4 | semidirect |
| 13. #99 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#100 | 2 |
| C2×#99 | 2 |
| #386 | 2 |