Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
531 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 531 conjugacy classes of subgroups — too many to draw. Only its 135 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 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3154 | C2 | semidirect |
| 2. C2×#492 | C2 | semidirect |
| 3. C2×#471 | C2 | semidirect |
| 4. #3057 | C2 | semidirect |
| 5. C2×#443 | C2 | semidirect |
| 6. #2005 | C2 | semidirect |
| 7. #2002 | C2 | semidirect |
| 8. #1977 | C2 | semidirect |
| 9. #1772 | C2 | semidirect |
| 51. C2×Q16 | D4 | semidirect |
| 62. C2×SD16 | D4 | semidirect |
| 64. C2×D8 | D4 | semidirect |
| 89. #108 | D4 | semidirect |
| 90. #107 | D4 | semidirect |
| 91. #106 | D4 | semidirect |
| 99. C4×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3154 | 2 |
| C2×#492 | 2 |
| C2×#471 | 2 |
| #3057 | 2 |
| C2×#443 | 2 |
| #2005 | 2 |
| #2002 | 2 |
| #1977 | 2 |
| #1772 | 2 |