Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
380 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 380 conjugacy classes of subgroups — too many to draw. Only its 108 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 | #120 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#137 | C2 | semidirect |
| 3. C2×#124 | C2 | semidirect |
| 4. #455 | C2 | direct (N × H) |
| 5. C2×#104 | C2 | semidirect |
| 6. C2×#100 | C2 | semidirect |
| 12. C22×Q8 | C22 | semidirect |
| 17. #137 | C22 | semidirect |
| 20. C2×M16 | C22 | semidirect |
| 22. #124 | C22 | semidirect |
| 24. C2×C42 | C22 | semidirect |
| 25. #104 | C22 | semidirect |
| 26. #100 | C22 | semidirect |
| 39. C2×Q8 | C23 | semidirect |
| 59. M16 | C23 | semidirect |
| 63. C42 | C23 | semidirect |
| 117. C2 | #455 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(5,𝔽2) | 2 |
| C2×#137 | 2 |
| C2×#124 | 2 |
| #455 | 2 |
| C2×#104 | 2 |
| C2×#100 | 2 |