Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
199 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C24 |
| commutator, socle | C22 | C23×C4 |
| prime radical | C2×#41 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #446 | C2 | semidirect |
| 3. #435 | C2 | semidirect |
| 4. C8×D4 | C2 | semidirect |
| 5. #432 | C2 | semidirect |
| 6. #431 | C2 | semidirect |
| 7. #423 | C2 | semidirect |
| 8. C2×C4⋊C8 | C2 | semidirect |
| 64. D4 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#41 | 2 |
| #446 | 2 |
| #435 | 2 |
| C8×D4 | 2 |
| #432 | 2 |
| #431 | 2 |
| #423 | 2 |
| C2×C4⋊C8 | 2 |