Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
143 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #3259 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #2238 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #2103 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #1341 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #1328 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #540 | — | 2 | 1 | no | — |
| 7 | #478 | — | 2 | 2 | no | — |
| 8 | #469 | C22 | 1 | 2 | yes | — |
| 9 | #466 | — | 2 | 1 | no | — |
| 10 | #459 | — | 2 | 2 | no | — |
| 11 | #430 | C22 | 1 | 1 | yes | characteristic |
| 12 | #424 | C22 | 1 | 1 | yes | characteristic |
| 13 | #365 | — | 2 | 2 | no | — |
| 14 | #361 | — | 2 | 2 | no | — |
| 15 | #349 | C22 | 1 | 2 | yes | — |
| 16 | #344 | — | 2 | 1 | no | — |
| 17 | #341 | — | 2 | 1 | no | — |
| 18 | #338 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×D8 | — | 2 | 1 | no | — |
| 20 | C2×D8 | D4 | 1 | 2 | yes | — |
| 21 | C2×M16 | — | 2 | 1 | no | — |
| 22 | C2×M16 | — | 2 | 1 | no | — |
| 23 | C2×M16 | C23 | 1 | 1 | yes | characteristic |
| 24 | C22×C8 | D4 | 1 | 1 | yes | characteristic |
| 25 | #127 | — | 2 | 1 | no | — |
| 26 | #125 | — | 2 | 1 | no | — |
| 27 | #123 | — | 4 | 2 | no | — |
| 28 | #121 | — | 2 | 2 | no | — |
| 29 | #121 | — | 4 | 2 | no | — |
| 30 | C4×D4 | — | 2 | 2 | no | — |
| 31 | #117 | D4 | 1 | 1 | yes | characteristic |
| 32 | M32 | — | 2 | 2 | no | — |
| 33 | #108 | — | 2 | 2 | no | — |
| 34 | #107 | — | 2 | 1 | no | — |
| 35 | #106 | D4 | 1 | 2 | yes | — |
| 36 | C4⋊C8 | — | 2 | 2 | no | — |
| 37 | #102 | — | 2 | 2 | no | — |
| 38 | #102 | — | 2 | 2 | no | — |
| 39 | C2×D4 | — | 2 | 2 | no | — |
| 40 | C2×D4 | — | 4 | 2 | no | — |
| 41 | C2×D4 | — | 4 | 2 | no | — |
| 42 | C22×C4 | — | 4 | 2 | no | — |
| 43 | C22×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 44 | D8 | — | 4 | 2 | no | — |
| 45 | D8 | — | 8 | 1 | no | — |
| 46 | M16 | — | 4 | 1 | no | — |
| 47 | M16 | — | 4 | 1 | no | — |
| 48 | M16 | — | 4 | 1 | no | — |
| 49 | M16 | — | 4 | 1 | no | — |
| 50 | M16 | — | 4 | 1 | no | — |
| 51 | C2×C8 | — | 2 | 1 | no | — |
| 52 | C2×C8 | — | 4 | 1 | no | — |
| 53 | C2×C8 | — | 4 | 1 | no | — |
| 54 | C2×C8 | — | 4 | 1 | no | — |
| 55 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 56 | C4⋊C4 | — | 2 | 1 | no | — |
| 57 | C4⋊C4 | — | 2 | 2 | no | — |
| 58 | C4⋊C4 | — | 4 | 2 | no | — |
| 59 | C22⋊C4 | — | 4 | 2 | no | — |
| 60 | C22⋊C4 | — | 4 | 2 | no | — |
| 61 | C22⋊C4 | SD16 | 1 | 2 | yes | — |
| 62 | C42 | — | 2 | 1 | no | — |
| 63 | C16 | — | 4 | 2 | no | — |
| 64 | C23 | — | 4 | 2 | no | — |
| 65 | C23 | C2×SD16 | 1 | 1 | yes | characteristic |
| 66 | D4 | — | 4 | 2 | no | — |
| 67 | D4 | — | 4 | 2 | no | — |
| 68 | D4 | — | 16 | 2 | no | — |
| 69 | D4 | — | 16 | 2 | no | — |
| 70 | C2×C4 | — | 2 | 1 | no | — |
| 71 | C2×C4 | — | 2 | 1 | no | — |
| 72 | C2×C4 | — | 2 | 2 | no | — |
| 73 | C2×C4 | — | 4 | 2 | no | — |
| 74 | C2×C4 | — | 16 | 2 | no | — |
| 75 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 76 | C2×C4 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 77 | C8 | — | 2 | 2 | no | — |
| 78 | C8 | — | 4 | 1 | no | — |
| 79 | C8 | — | 8 | 1 | no | — |
| 80 | C8 | — | 8 | 1 | no | — |
| 81 | C22 | — | 2 | 1 | no | — |
| 82 | C22 | — | 4 | 1 | no | — |
| 83 | C22 | — | 8 | 2 | no | — |
| 84 | C22 | — | 8 | 2 | no | — |
| 85 | C22 | — | 8 | 2 | no | — |
| 86 | C22 | #450 | 1 | 1 | yes | characteristic |
| 87 | C4 | — | 2 | 1 | no | — |
| 88 | C4 | — | 4 | 2 | no | — |
| 89 | C4 | — | 8 | 2 | no | — |
| 90 | C4 | #454 | 1 | 1 | yes | characteristic |
| 91 | C4 | #458 | 1 | 1 | yes | characteristic |
| 92 | C2 | — | 2 | 1 | no | — |
| 93 | C2 | — | 4 | 1 | no | — |
| 94 | C2 | — | 16 | 2 | no | — |
| 95 | C2 | #1616 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1616 |
| commutator, Frattini, cosocle | C2×M16 | C23 |
| prime radical | #3259 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2238 | C2 | semidirect |
| 3. #2103 | C2 | semidirect |
| 4. #1341 | C2 | semidirect |
| 5. #1328 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3259 | 2 |
| #2238 | 2 |
| #2103 | 2 |
| #1341 | 2 |
| #1328 | 2 |