Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
370 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 61 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 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #2541 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3208 | C2 | semidirect |
| 2. #3077 | C2 | semidirect |
| 4. #1545 | C2 | semidirect |
| 5. #1503 | C2 | semidirect |
| 6. #1478 | C2 | semidirect |
| 7. #1303 | C2 | semidirect |
| 39. C4⋊Q8 | D4 | semidirect |
| 40. C4⋊Q8 | D4 | semidirect |
| 68. C4⋊C8 | D4 | semidirect |
| 69. C4⋊C8 | D4 | semidirect |
| 102. C2×D4 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3208 | 2 |
| #3077 | 2 |
| #2541 | 2 |
| #1545 | 2 |
| #1503 | 2 |
| #1478 | 2 |
| #1303 | 2 |