Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
687 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2865 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #2759 | C2 | 1 | 21 | yes | maximal, complemented |
| 3 | #2736 | C2 | 1 | 7 | yes | maximal |
| 4 | #401 | C22 | 1 | 7 | yes | complemented |
| 5 | #397 | C22 | 1 | 21 | yes | — |
| 6 | #395 | C22 | 1 | 7 | yes | complemented |
| 7 | #393 | C22 | 1 | 21 | yes | — |
| 8 | #391 | C22 | 1 | 21 | yes | complemented |
| 9 | #388 | C22 | 1 | 21 | yes | — |
| 10 | #388 | C22 | 1 | 21 | yes | — |
| 11 | #387 | C22 | 1 | 21 | yes | complemented |
| 12 | #383 | C22 | 1 | 7 | yes | complemented |
| 13 | #379 | C22 | 1 | 7 | yes | — |
| 14 | C43 | C22 | 1 | 1 | yes | characteristic, complemented |
| 15 | C25 | C23 | 1 | 1 | yes | characteristic |
| 16 | C22×Q8 | C23 | 1 | 7 | yes | — |
| 17 | C2×C4⋊C4 | C23 | 1 | 21 | yes | — |
| 18 | C2×C22⋊C4 | C23 | 1 | 21 | yes | — |
| 19 | C2×C22⋊C4 | C23 | 1 | 21 | yes | — |
| 20 | C2×C42 | C23 | 1 | 7 | yes | — |
| 21 | C2×C42 | C23 | 1 | 7 | yes | — |
| 22 | #95 | C23 | 1 | 21 | yes | — |
| 23 | #95 | C23 | 1 | 21 | yes | — |
| 24 | #95 | C23 | 1 | 21 | yes | — |
| 25 | #95 | C23 | 1 | 7 | yes | — |
| 26 | C24 | — | 4 | 7 | no | — |
| 27 | C24 | C24 | 1 | 3 | yes | — |
| 28 | C2×Q8 | — | 4 | 21 | no | — |
| 29 | C22×C4 | C24 | 1 | 21 | yes | — |
| 30 | C22×C4 | C24 | 1 | 7 | yes | — |
| 31 | C4⋊C4 | — | 4 | 21 | no | — |
| 32 | C22⋊C4 | — | 4 | 21 | no | — |
| 33 | C22⋊C4 | — | 4 | 21 | no | — |
| 34 | C42 | — | 4 | 7 | no | — |
| 35 | C42 | — | 4 | 7 | no | — |
| 36 | C23 | — | 2 | 21 | no | — |
| 37 | C23 | — | 8 | 14 | no | — |
| 38 | C23 | C25 | 1 | 1 | yes | characteristic |
| 39 | Q8 | — | 8 | 14 | no | — |
| 40 | C2×C4 | — | 2 | 21 | no | — |
| 41 | C2×C4 | — | 2 | 21 | no | — |
| 42 | C2×C4 | — | 2 | 42 | no | — |
| 43 | C22 | — | 4 | 21 | no | — |
| 44 | C22 | — | 8 | 8 | no | — |
| 45 | C22 | #585 | 1 | 7 | yes | — |
| 46 | C4 | — | 4 | 7 | no | — |
| 47 | C4 | — | 4 | 21 | no | — |
| 48 | C2 | — | 8 | 3 | no | — |
| 49 | C2 | #3568 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| 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. #2865 | C2 | semidirect |
| 2. #2759 | C2 | semidirect |
| 4. #401 | C22 | semidirect |
| 6. #395 | C22 | semidirect |
| 8. #391 | C22 | semidirect |
| 11. #387 | C22 | semidirect |
| 12. #383 | C22 | semidirect |
| 14. C43 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2865 | 2 |
| #2759 | 2 |
| #2736 | 2 |