Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
515 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2867 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2864 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #2041 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #1961 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #1745 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#127 | — | 2 | 4 | no | — |
| 7 | C2×#127 | C22 | 1 | 2 | yes | — |
| 8 | C2×#127 | C22 | 1 | 4 | yes | — |
| 9 | C2×#102 | C22 | 1 | 16 | yes | — |
| 10 | C2×#97 | C22 | 1 | 2 | yes | — |
| 11 | #389 | C22 | 1 | 4 | yes | — |
| 12 | #387 | — | 2 | 4 | no | — |
| 13 | #382 | C22 | 1 | 2 | yes | — |
| 14 | C43 | C22 | 1 | 1 | yes | characteristic |
| 15 | #336 | C22 | 1 | 4 | yes | — |
| 16 | C22×D4 | — | 2 | 4 | no | — |
| 17 | C22×D4 | — | 2 | 8 | no | — |
| 18 | C22×D4 | C23 | 1 | 4 | yes | — |
| 19 | C22×C8 | C23 | 1 | 4 | yes | — |
| 20 | #127 | — | 2 | 8 | no | — |
| 21 | #127 | — | 4 | 8 | no | — |
| 22 | #127 | — | 4 | 16 | no | — |
| 23 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 24 | C2×C4⋊C4 | C23 | 1 | 4 | yes | — |
| 25 | C2×C42 | D4 | 1 | 4 | yes | — |
| 26 | C2×C42 | C23 | 1 | 2 | yes | — |
| 27 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 28 | #102 | — | 4 | 16 | no | — |
| 29 | #97 | — | 2 | 4 | no | — |
| 30 | #95 | — | 2 | 4 | no | — |
| 31 | #95 | — | 2 | 4 | no | — |
| 32 | C24 | — | 2 | 4 | no | — |
| 33 | C2×D4 | — | 2 | 8 | no | — |
| 34 | C2×D4 | — | 2 | 16 | no | — |
| 35 | C2×D4 | — | 4 | 16 | no | — |
| 36 | C2×D4 | — | 4 | 16 | no | — |
| 37 | C2×D4 | — | 4 | 16 | no | — |
| 38 | C2×D4 | — | 4 | 8 | no | — |
| 39 | C2×D4 | — | 4 | 16 | no | — |
| 40 | C22×C4 | — | 2 | 4 | no | — |
| 41 | C22×C4 | C2×D4 | 1 | 2 | yes | — |
| 42 | C22×C4 | C2×D4 | 1 | 4 | yes | — |
| 43 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 44 | C2×C8 | — | 2 | 4 | no | — |
| 45 | C2×C8 | — | 2 | 8 | no | — |
| 46 | C4⋊C4 | — | 2 | 8 | no | — |
| 47 | C4⋊C4 | — | 4 | 8 | no | — |
| 48 | C42 | — | 2 | 8 | no | — |
| 49 | C42 | — | 2 | 4 | no | — |
| 50 | C42 | #41 | 1 | 4 | yes | — |
| 51 | C23 | — | 4 | 4 | no | — |
| 52 | C23 | — | 4 | 8 | no | — |
| 53 | C23 | — | 4 | 16 | no | — |
| 54 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 55 | D4 | — | 4 | 16 | no | — |
| 56 | D4 | — | 8 | 16 | no | — |
| 57 | D4 | — | 8 | 32 | no | — |
| 58 | C2×C4 | — | 2 | 4 | no | — |
| 59 | C2×C4 | — | 2 | 4 | no | — |
| 60 | C2×C4 | — | 2 | 4 | no | — |
| 61 | C2×C4 | — | 4 | 4 | no | — |
| 62 | C2×C4 | — | 4 | 8 | no | — |
| 63 | C2×C4 | #124 | 1 | 4 | yes | — |
| 64 | C2×C4 | AGL(1,ℤ/8) | 1 | 8 | yes | — |
| 65 | C2×C4 | C2×#41 | 1 | 2 | yes | — |
| 66 | C2×C4 | H(5,𝔽2) | 1 | 4 | yes | — |
| 67 | C8 | — | 4 | 4 | no | — |
| 68 | C22 | — | 8 | 8 | no | — |
| 69 | C22 | — | 8 | 4 | no | — |
| 70 | C22 | — | 8 | 16 | no | — |
| 71 | C22 | C2×#124 | 1 | 1 | yes | characteristic |
| 72 | C22 | #535 | 1 | 2 | yes | — |
| 73 | C22 | C2×AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 74 | C4 | — | 2 | 4 | no | — |
| 75 | C4 | — | 2 | 8 | no | — |
| 76 | C4 | — | 8 | 4 | no | — |
| 77 | C4 | #540 | 1 | 4 | yes | — |
| 78 | C2 | — | 16 | 4 | no | — |
| 79 | C2 | #2641 | 1 | 1 | yes | characteristic |
| 80 | C2 | #3138 | 1 | 2 | yes | — |
| 81 | C2 | #3217 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2867 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2864 | C2 | semidirect |
| 3. #2041 | C2 | semidirect |
| 4. #1961 | C2 | semidirect |
| 5. #1745 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2867 | 2 |
| #2864 | 2 |
| #2041 | 2 |
| #1961 | 2 |
| #1745 | 2 |