Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
69 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #1662 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1413 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #1356 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #562 | C4 | 1 | 1 | yes | characteristic |
| 5 | #491 | C22 | 1 | 1 | yes | characteristic |
| 6 | #489 | — | 2 | 1 | no | — |
| 7 | #356 | — | 2 | 1 | no | — |
| 8 | #347 | — | 2 | 1 | no | — |
| 9 | #330 | C4 | 1 | 1 | yes | characteristic |
| 10 | #329 | — | 2 | 1 | no | — |
| 11 | C4⋊Q8 | D4 | 1 | 1 | yes | characteristic |
| 12 | #125 | C2×C4 | 1 | 1 | yes | characteristic |
| 13 | #124 | — | 2 | 1 | no | — |
| 14 | #123 | — | 4 | 1 | no | — |
| 15 | C22⋊Q8 | — | 4 | 1 | no | — |
| 16 | #121 | — | 4 | 1 | no | — |
| 17 | M32 | — | 4 | 1 | no | — |
| 18 | #103 | — | 4 | 1 | no | — |
| 19 | #103 | — | 4 | 1 | no | — |
| 20 | #102 | — | 4 | 1 | no | — |
| 21 | #101 | — | 4 | 1 | no | — |
| 22 | #97 | — | 2 | 1 | no | — |
| 23 | #97 | D4 | 1 | 1 | yes | characteristic |
| 24 | C2×Q8 | — | 2 | 1 | no | — |
| 25 | C2×Q8 | — | 2 | 1 | no | — |
| 26 | C2×D4 | — | 2 | 1 | no | — |
| 27 | C2×D4 | — | 4 | 1 | no | — |
| 28 | C22×C4 | — | 4 | 1 | no | — |
| 29 | M16 | — | 8 | 1 | no | — |
| 30 | C2×C8 | — | 2 | 1 | no | — |
| 31 | C2×C8 | — | 4 | 1 | no | — |
| 32 | C4⋊C4 | — | 2 | 1 | no | — |
| 33 | C4⋊C4 | — | 4 | 1 | no | — |
| 34 | C4⋊C4 | — | 4 | 1 | no | — |
| 35 | C4⋊C4 | — | 4 | 1 | no | — |
| 36 | C22⋊C4 | — | 4 | 1 | no | — |
| 37 | C22⋊C4 | — | 4 | 1 | no | — |
| 38 | C22⋊C4 | — | 4 | 1 | no | — |
| 39 | C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 40 | C16 | — | 4 | 2 | no | — |
| 41 | C23 | — | 4 | 1 | no | — |
| 42 | Q8 | — | 8 | 1 | no | — |
| 43 | Q8 | — | 8 | 1 | no | — |
| 44 | D4 | — | 8 | 1 | no | — |
| 45 | D4 | — | 16 | 1 | no | — |
| 46 | C2×C4 | — | 2 | 1 | no | — |
| 47 | C2×C4 | — | 4 | 1 | no | — |
| 48 | C2×C4 | — | 4 | 1 | no | — |
| 49 | C2×C4 | — | 4 | 1 | no | — |
| 50 | C2×C4 | — | 16 | 1 | no | — |
| 51 | C2×C4 | #99 | 1 | 1 | yes | characteristic |
| 52 | C8 | — | 4 | 1 | no | — |
| 53 | C8 | — | 8 | 1 | no | — |
| 54 | C8 | — | 16 | 1 | no | — |
| 55 | C22 | — | 8 | 1 | no | — |
| 56 | C22 | — | 8 | 1 | no | — |
| 57 | C22 | — | 8 | 1 | no | — |
| 58 | C22 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 59 | C4 | — | 2 | 1 | no | — |
| 60 | C4 | — | 4 | 1 | no | — |
| 61 | C4 | — | 8 | 1 | no | — |
| 62 | C4 | — | 8 | 1 | no | — |
| 63 | C4 | — | 8 | 1 | no | — |
| 64 | C2 | — | 2 | 1 | no | — |
| 65 | C2 | — | 16 | 1 | no | — |
| 66 | C2 | #1404 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1404 |
| commutator | #125 | C2×C4 |
| Frattini, cosocle | #491 | C22 |
| prime radical | #562 | C4 |
| LCS | G ▷ #125 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #491 ◁ G | |
| LECS | G ▷ #491 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #1413 | C2 | semidirect |
| 3. #1356 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #1662 | 2 |
| #1413 | 2 |
| #1356 | 2 |