Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
378 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 378 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22⋊C4 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C2×C22⋊C4 |
| prime radical | C22×#41 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×M16 | C2 | semidirect |
| 3. #413 | C2 | direct (N × H) |
| 4. C2×#101 | C2 | semidirect |
| 5. C2×#100 | C2 | semidirect |
| 19. C2×M16 | C22 | semidirect |
| 21. #101 | C22 | semidirect |
| 22. #100 | C22 | semidirect |
| 110. C2 | #413 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41 | 2 |
| C22×M16 | 2 |
| #413 | 2 |
| C2×#101 | 2 |
| C2×#100 | 2 |