Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
400 conjugacy classes of subgroups; only the 236 normal subgroups are shown.
This group has 400 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, prime radical | C24 | C23 |
| commutator | C22 | C23×C4 |
| Frattini, cosocle | C23 | C24 |
| 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 |
|---|---|---|
| 5. #389 | C2 | direct (N × H) |
| 14. C2×C4⋊C4 | C4 | semidirect |
| 44. C4⋊C4 | C2×C4 | semidirect |
| 67. C2×C4 | C4⋊C4 | semidirect |
| 92. C4 | C2×C4⋊C4 | semidirect |
| 93. C2 | #389 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4⋊C4 | 2 |
| C22×C42 | 2 |
| #389 | 2 |
| C2×#95 | 2 |