Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
549 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 549 conjugacy classes of subgroups — too many to draw. Only its 127 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#578 | C2 | semidirect |
| 2. C2×#510 | C2 | semidirect |
| 3. C2×#508 | C2 | semidirect |
| 4. #3061 | C2 | semidirect |
| 5. #2189 | C2 | semidirect |
| 6. #2188 | C2 | semidirect |
| 7. #2186 | C2 | semidirect |
| 8. #2185 | C2 | semidirect |
| 9. #2139 | C2 | semidirect |
| 10. C2×#358 | C2 | semidirect |
| 11. #2113 | C2 | semidirect |
| 63. #135 | D4 | semidirect |
| 69. C2×Q16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#578 | 2 |
| C2×#510 | 2 |
| C2×#508 | 2 |
| #3061 | 2 |
| #2189 | 2 |
| #2188 | 2 |
| #2186 | 2 |
| #2185 | 2 |
| #2139 | 2 |
| C2×#358 | 2 |
| #2113 | 2 |