Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
281 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #2645 | C2 | 1 | 2 | yes | maximal |
| 2 | #2086 | C2 | 1 | 3 | yes | maximal |
| 3 | #2068 | C2 | 1 | 6 | yes | maximal |
| 4 | #2056 | C2 | 1 | 3 | yes | maximal |
| 5 | #1843 | C2 | 1 | 1 | yes | maximal, characteristic |
| 6 | C22×Q16 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#107 | C22 | 1 | 3 | yes | — |
| 8 | C2×C4⋊C8 | C22 | 1 | 3 | yes | — |
| 9 | C2×C4⋊C8 | C22 | 1 | 3 | yes | — |
| 10 | C2×#103 | C22 | 1 | 6 | yes | — |
| 11 | #401 | — | 2 | 2 | no | — |
| 12 | #400 | — | 2 | 6 | no | — |
| 13 | #391 | C22 | 1 | 6 | yes | — |
| 14 | #389 | C22 | 1 | 6 | yes | — |
| 15 | #376 | C22 | 1 | 1 | yes | characteristic |
| 16 | #340 | C22 | 1 | 3 | yes | — |
| 17 | #340 | C22 | 1 | 3 | yes | — |
| 18 | C22×Q8 | C23 | 1 | 2 | yes | — |
| 19 | C2×Q16 | — | 4 | 3 | no | — |
| 20 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 21 | C22×C8 | C23 | 1 | 3 | yes | — |
| 22 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 23 | C2×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 24 | C2×C42 | C23 | 1 | 3 | yes | — |
| 25 | #107 | — | 4 | 3 | no | — |
| 26 | C4⋊C8 | — | 4 | 3 | no | — |
| 27 | C4⋊C8 | — | 4 | 3 | no | — |
| 28 | #103 | — | 4 | 6 | no | — |
| 29 | #95 | — | 2 | 6 | no | — |
| 30 | #95 | — | 2 | 6 | no | — |
| 31 | #95 | — | 2 | 6 | no | — |
| 32 | #95 | D4 | 1 | 3 | yes | — |
| 33 | #95 | D4 | 1 | 1 | yes | characteristic |
| 34 | C2×Q8 | — | 2 | 6 | no | — |
| 35 | C2×Q8 | — | 2 | 6 | no | — |
| 36 | C22×C4 | — | 2 | 2 | no | — |
| 37 | C22×C4 | — | 2 | 6 | no | — |
| 38 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 39 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 40 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 41 | Q16 | — | 4 | 4 | no | — |
| 42 | C2×C8 | — | 2 | 6 | no | — |
| 43 | C2×C8 | — | 2 | 3 | no | — |
| 44 | C2×C8 | — | 2 | 3 | no | — |
| 45 | C4⋊C4 | — | 2 | 6 | no | — |
| 46 | C4⋊C4 | — | 2 | 6 | no | — |
| 47 | C4⋊C4 | — | 8 | 6 | no | — |
| 48 | C42 | — | 2 | 3 | no | — |
| 49 | C42 | — | 2 | 3 | no | — |
| 50 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 51 | Q8 | — | 4 | 2 | no | — |
| 52 | Q8 | — | 4 | 6 | no | — |
| 53 | C2×C4 | — | 2 | 3 | no | — |
| 54 | C2×C4 | — | 2 | 3 | no | — |
| 55 | C2×C4 | — | 2 | 3 | no | — |
| 56 | C2×C4 | — | 2 | 6 | no | — |
| 57 | C2×C4 | — | 2 | 3 | no | — |
| 58 | C2×C4 | — | 4 | 6 | no | — |
| 59 | C2×C4 | — | 4 | 6 | no | — |
| 60 | C2×C4 | — | 4 | 6 | no | — |
| 61 | C2×C4 | — | 4 | 6 | no | — |
| 62 | C2×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 63 | C2×C4 | 2−1+4 | 1 | 3 | yes | — |
| 64 | C8 | — | 4 | 3 | no | — |
| 65 | C8 | — | 4 | 1 | no | — |
| 66 | C22 | #537 | 1 | 3 | yes | — |
| 67 | C22 | #576 | 1 | 1 | yes | characteristic |
| 68 | C22 | #578 | 1 | 3 | yes | — |
| 69 | C4 | — | 4 | 3 | no | — |
| 70 | C4 | — | 4 | 3 | no | — |
| 71 | C4 | — | 8 | 2 | no | — |
| 72 | C4 | — | 8 | 6 | no | — |
| 73 | C4 | #560 | 1 | 1 | yes | characteristic |
| 74 | C4 | #563 | 1 | 3 | yes | — |
| 75 | C2 | #2823 | 1 | 1 | yes | characteristic |
| 76 | C2 | #3261 | 1 | 3 | yes | — |
| 77 | C2 | #3262 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2645 | 2 |
| #2086 | 2 |
| #2068 | 2 |
| #2056 | 2 |
| #1843 | 2 |