Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
665 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 665 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#496 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #1655 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 4 | C2×#329 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C22×SD16 | — | 2 | 3 | no | — |
| 6 | C22×D8 | — | 2 | 3 | no | — |
| 7 | H(3,𝔽4) | — | 2 | 6 | no | — |
| 8 | H(3,𝔽4) | C22 | 1 | 4 | yes | complemented |
| 9 | C2×#127 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C2×#124 | — | 2 | 3 | no | — |
| 11 | C2×#124 | C22 | 1 | 3 | yes | — |
| 12 | C2×#120 | — | 2 | 3 | no | — |
| 13 | #496 | — | 2 | 6 | no | — |
| 14 | #496 | C22 | 1 | 12 | yes | complemented |
| 15 | C2×#97 | C22 | 1 | 3 | yes | complemented |
| 16 | #329 | C22 | 1 | 12 | yes | complemented |
| 17 | C25 | — | 2 | 1 | no | — |
| 18 | C22×Q8 | D4 | 1 | 3 | yes | — |
| 19 | C22×D4 | — | 2 | 3 | no | — |
| 20 | C22×D4 | — | 2 | 3 | no | — |
| 21 | C22×D4 | D4 | 1 | 3 | yes | — |
| 22 | C2×SD16 | — | 2 | 12 | no | — |
| 23 | C2×SD16 | — | 4 | 12 | no | — |
| 24 | C2×D8 | — | 2 | 12 | no | — |
| 25 | C2×D8 | — | 4 | 12 | no | — |
| 26 | C22×C8 | — | 2 | 3 | no | — |
| 27 | #127 | — | 2 | 3 | no | — |
| 28 | #127 | C23 | 1 | 2 | yes | complemented |
| 29 | #124 | — | 2 | 12 | no | — |
| 30 | #124 | — | 2 | 12 | no | — |
| 31 | #124 | — | 2 | 6 | no | — |
| 32 | #124 | D4 | 1 | 6 | yes | — |
| 33 | #124 | C23 | 1 | 6 | yes | — |
| 34 | #120 | — | 2 | 12 | no | — |
| 35 | #120 | — | 2 | 12 | no | — |
| 36 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 37 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 38 | C2×C42 | — | 2 | 1 | no | — |
| 39 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 40 | #97 | — | 2 | 3 | no | — |
| 41 | #97 | C23 | 1 | 6 | yes | complemented |
| 42 | C24 | — | 2 | 4 | no | — |
| 43 | C24 | — | 2 | 3 | no | — |
| 44 | C24 | — | 4 | 1 | no | — |
| 45 | C24 | — | 8 | 6 | no | — |
| 46 | C2×Q8 | — | 2 | 3 | no | — |
| 47 | C2×Q8 | — | 4 | 6 | no | — |
| 48 | C2×Q8 | C2×D4 | 1 | 6 | yes | — |
| 49 | C2×D4 | — | 2 | 12 | no | — |
| 50 | C2×D4 | — | 2 | 3 | no | — |
| 51 | C2×D4 | — | 2 | 6 | no | — |
| 52 | C2×D4 | — | 4 | 6 | no | — |
| 53 | C2×D4 | — | 4 | 6 | no | — |
| 54 | C2×D4 | — | 4 | 6 | no | — |
| 55 | C2×D4 | — | 4 | 3 | no | — |
| 56 | C2×D4 | — | 8 | 6 | no | — |
| 57 | C2×D4 | C2×D4 | 1 | 6 | yes | — |
| 58 | C22×C4 | — | 2 | 3 | no | — |
| 59 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 60 | SD16 | — | 4 | 24 | no | — |
| 61 | D8 | — | 4 | 24 | no | — |
| 62 | C2×C8 | — | 2 | 12 | no | — |
| 63 | C2×C8 | — | 2 | 6 | no | — |
| 64 | C22⋊C4 | — | 2 | 12 | no | — |
| 65 | C22⋊C4 | — | 2 | 12 | no | — |
| 66 | C22⋊C4 | — | 2 | 12 | no | — |
| 67 | C42 | — | 2 | 4 | no | — |
| 68 | C42 | C2×D4 | 1 | 3 | yes | — |
| 69 | C42 | C24 | 1 | 1 | yes | characteristic |
| 70 | C23 | — | 2 | 6 | no | — |
| 71 | C23 | — | 4 | 6 | no | — |
| 72 | C23 | — | 4 | 12 | no | — |
| 73 | C23 | — | 4 | 2 | no | — |
| 74 | C23 | — | 8 | 12 | no | — |
| 75 | C23 | — | 8 | 16 | no | — |
| 76 | C23 | — | 8 | 6 | no | — |
| 77 | C23 | #120 | 1 | 1 | yes | characteristic |
| 78 | Q8 | — | 4 | 6 | no | — |
| 79 | Q8 | — | 8 | 3 | no | — |
| 80 | D4 | — | 4 | 6 | no | — |
| 81 | D4 | — | 4 | 6 | no | — |
| 82 | D4 | — | 4 | 6 | no | — |
| 83 | D4 | — | 8 | 3 | no | — |
| 84 | D4 | — | 8 | 6 | no | — |
| 85 | D4 | — | 8 | 12 | no | — |
| 86 | C2×C4 | — | 2 | 6 | no | — |
| 87 | C2×C4 | — | 2 | 6 | no | — |
| 88 | C2×C4 | — | 4 | 6 | no | — |
| 89 | C2×C4 | #120 | 1 | 3 | yes | — |
| 90 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 91 | C8 | — | 2 | 12 | no | — |
| 92 | C22 | — | 4 | 6 | no | — |
| 93 | C22 | — | 4 | 12 | no | — |
| 94 | C22 | — | 8 | 16 | no | — |
| 95 | C22 | — | 8 | 12 | no | — |
| 96 | C22 | — | 8 | 6 | no | — |
| 97 | C22 | — | 16 | 4 | no | — |
| 98 | C22 | #453 | 1 | 6 | yes | — |
| 99 | C22 | C2×#120 | 1 | 1 | yes | characteristic |
| 100 | C4 | — | 2 | 3 | no | — |
| 101 | C4 | — | 2 | 3 | no | — |
| 102 | C4 | — | 4 | 6 | no | — |
| 103 | C2 | — | 8 | 6 | no | — |
| 104 | C2 | — | 16 | 2 | no | — |
| 105 | C2 | #1655 | 1 | 4 | yes | direct product factor |
| 106 | C2 | C2×#453 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×H(3,𝔽4) | C2 | semidirect |
| 2. C2×#496 | C2 | semidirect |
| 3. #1655 | C2 | direct (N × H) |
| 4. C2×#329 | C2 | semidirect |
| 8. H(3,𝔽4) | C22 | semidirect |
| 9. C2×#127 | C22 | semidirect |
| 14. #496 | C22 | semidirect |
| 15. C2×#97 | C22 | semidirect |
| 16. #329 | C22 | semidirect |
| 28. #127 | C23 | semidirect |
| 41. #97 | C23 | semidirect |
| 105. C2 | #1655 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×H(3,𝔽4) | 2 |
| C2×#496 | 2 |
| #1655 | 2 |
| C2×#329 | 2 |