Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
559 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#376 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #1443 | C2 | 1 | 8 | yes | maximal |
| 3 | C4⋊#95 | C2 | 1 | 6 | yes | maximal |
| 4 | C22×C4⋊C4 | C22 | 1 | 4 | yes | — |
| 5 | C22×C42 | C22 | 1 | 3 | yes | — |
| 6 | #376 | C4 | 1 | 8 | yes | complemented |
| 7 | C2×#95 | C22 | 1 | 24 | yes | — |
| 8 | C2×#95 | C22 | 1 | 4 | yes | — |
| 9 | C23×C4 | C23 | 1 | 8 | yes | — |
| 10 | C23×C4 | C23 | 1 | 6 | yes | — |
| 11 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×C4⋊C4 | — | 2 | 24 | no | — |
| 13 | C2×C42 | — | 2 | 12 | no | — |
| 14 | C2×C42 | C2×C4 | 1 | 12 | yes | — |
| 15 | #95 | — | 4 | 24 | no | — |
| 16 | #95 | C2×C4 | 1 | 16 | yes | — |
| 17 | C24 | C24 | 1 | 1 | yes | characteristic |
| 18 | C22×C4 | — | 2 | 6 | no | — |
| 19 | C22×C4 | — | 2 | 6 | no | — |
| 20 | C22×C4 | — | 2 | 12 | no | — |
| 21 | C22×C4 | — | 2 | 12 | no | — |
| 22 | C22×C4 | — | 2 | 24 | no | — |
| 23 | C22×C4 | — | 2 | 24 | no | — |
| 24 | C22×C4 | — | 2 | 8 | no | — |
| 25 | C22×C4 | C22×C4 | 1 | 12 | yes | — |
| 26 | C22×C4 | C22×C4 | 1 | 2 | yes | — |
| 27 | C22×C4 | #41 | 1 | 12 | yes | — |
| 28 | C4⋊C4 | — | 4 | 16 | no | — |
| 29 | C42 | — | 2 | 24 | no | — |
| 30 | C23 | C23×C4 | 1 | 1 | yes | characteristic |
| 31 | C23 | C2×#41 | 1 | 6 | yes | — |
| 32 | C23 | H(5,𝔽2) | 1 | 4 | yes | — |
| 33 | C23 | 2−1+4 | 1 | 4 | yes | — |
| 34 | C2×C4 | — | 2 | 12 | no | — |
| 35 | C2×C4 | — | 2 | 24 | no | — |
| 36 | C2×C4 | — | 4 | 6 | no | — |
| 37 | C2×C4 | — | 4 | 6 | no | — |
| 38 | C2×C4 | — | 4 | 12 | no | — |
| 39 | C2×C4 | — | 4 | 8 | no | — |
| 40 | C2×C4 | — | 4 | 24 | no | — |
| 41 | C2×C4 | — | 4 | 24 | no | — |
| 42 | C2×C4 | #117 | 1 | 12 | yes | — |
| 43 | C2×C4 | #126 | 1 | 16 | yes | — |
| 44 | C22 | C2×#117 | 1 | 3 | yes | — |
| 45 | C22 | #520 | 1 | 4 | yes | — |
| 46 | C22 | C2×#126 | 1 | 4 | yes | — |
| 47 | C22 | #540 | 1 | 6 | yes | — |
| 48 | C22 | #541 | 1 | 6 | yes | — |
| 49 | C22 | #542 | 1 | 12 | yes | — |
| 50 | C4 | — | 4 | 12 | no | — |
| 51 | C4 | — | 8 | 8 | no | — |
| 52 | C4 | #383 | 1 | 8 | yes | — |
| 53 | C2 | C2×#383 | 1 | 1 | yes | characteristic |
| 54 | C2 | #2314 | 1 | 6 | yes | — |
| 55 | C2 | #2644 | 1 | 4 | yes | — |
| 56 | C2 | #2645 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. #376 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#376 | 2 |
| #1443 | 2 |
| C4⋊#95 | 2 |