Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
924 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 449 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2863 | C2 | semidirect |
| 2. #2843 | C2 | semidirect |
| 3. #2784 | C2 | semidirect |
| 4. #2748 | C2 | semidirect |
| 5. #2487 | C2 | semidirect |
| 6. #2417 | C2 | semidirect |
| 7. #2409 | C2 | semidirect |
| 47. #127 | Q8 | semidirect |
| 82. C2×D4 | C2×Q8 | semidirect |
| 110. C42 | C2×Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2863 | 2 |
| #2843 | 2 |
| #2784 | 2 |
| #2748 | 2 |
| #2487 | 2 |
| #2417 | 2 |
| #2409 | 2 |
| #2380 | 2 |
| C4×#127 | 2 |