Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
407 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 407 conjugacy classes of subgroups — too many to draw. Only its 115 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 | #3207 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #3205 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #1721 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #1717 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #1712 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #1711 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #1454 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #532 | C22 | 1 | 3 | yes | — |
| 9 | C2×#127 | C22 | 1 | 1 | yes | characteristic |
| 10 | #493 | C22 | 1 | 3 | yes | — |
| 11 | #493 | C22 | 1 | 3 | yes | — |
| 12 | #493 | C22 | 1 | 3 | yes | complemented |
| 13 | #492 | C22 | 1 | 3 | yes | complemented |
| 14 | #487 | C22 | 1 | 3 | yes | complemented |
| 15 | #486 | C22 | 1 | 3 | yes | — |
| 16 | #486 | C22 | 1 | 3 | yes | — |
| 17 | #486 | C22 | 1 | 3 | yes | complemented |
| 18 | #463 | — | 2 | 6 | no | — |
| 19 | #459 | — | 2 | 6 | no | — |
| 20 | #449 | — | 2 | 6 | no | — |
| 21 | #447 | — | 2 | 6 | no | — |
| 22 | #431 | C22 | 1 | 3 | yes | — |
| 23 | #322 | C22 | 1 | 3 | yes | complemented |
| 24 | C82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | C2×#41 | — | 2 | 3 | no | — |
| 26 | C22×D4 | — | 2 | 3 | no | — |
| 27 | C2×SD16 | — | 2 | 6 | no | — |
| 28 | C2×D8 | — | 2 | 6 | no | — |
| 29 | C2×D8 | — | 2 | 6 | no | — |
| 30 | C2×D8 | — | 2 | 6 | no | — |
| 31 | C4⋊Q8 | C23 | 1 | 3 | yes | — |
| 32 | #127 | — | 2 | 3 | no | — |
| 33 | #127 | C23 | 1 | 1 | yes | characteristic |
| 34 | #127 | C23 | 1 | 1 | yes | characteristic |
| 35 | #127 | C23 | 1 | 3 | yes | — |
| 36 | #124 | — | 2 | 3 | no | — |
| 37 | #121 | — | 2 | 3 | no | — |
| 38 | #121 | — | 2 | 3 | no | — |
| 39 | C4×D4 | — | 2 | 3 | no | — |
| 40 | C4×D4 | — | 2 | 3 | no | — |
| 41 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 42 | C4⋊C8 | D4 | 1 | 6 | yes | — |
| 43 | #103 | — | 2 | 6 | no | — |
| 44 | #102 | — | 2 | 6 | no | — |
| 45 | #102 | — | 2 | 6 | no | — |
| 46 | #102 | — | 2 | 6 | no | — |
| 47 | C22⋊C8 | D4 | 1 | 6 | yes | — |
| 48 | C4×C8 | C23 | 1 | 3 | yes | — |
| 49 | C4×C8 | C23 | 1 | 3 | yes | — |
| 50 | C24 | — | 4 | 1 | no | — |
| 51 | #41 | — | 8 | 6 | no | — |
| 52 | C2×Q8 | — | 2 | 3 | no | — |
| 53 | C2×D4 | — | 2 | 3 | no | — |
| 54 | C2×D4 | — | 2 | 3 | no | — |
| 55 | C2×D4 | — | 2 | 3 | no | — |
| 56 | C2×D4 | — | 2 | 3 | no | — |
| 57 | C2×D4 | — | 2 | 3 | no | — |
| 58 | C2×D4 | — | 4 | 3 | no | — |
| 59 | C2×D4 | — | 4 | 3 | no | — |
| 60 | C2×D4 | — | 8 | 6 | no | — |
| 61 | C22×C4 | — | 4 | 3 | no | — |
| 62 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 63 | SD16 | — | 8 | 6 | no | — |
| 64 | D8 | — | 8 | 6 | no | — |
| 65 | D8 | — | 8 | 6 | no | — |
| 66 | D8 | — | 8 | 6 | no | — |
| 67 | C2×C8 | C2×D4 | 1 | 6 | yes | — |
| 68 | C2×C8 | C2×D4 | 1 | 6 | yes | — |
| 69 | C4⋊C4 | — | 2 | 3 | no | — |
| 70 | C4⋊C4 | — | 2 | 3 | no | — |
| 71 | C22⋊C4 | — | 4 | 3 | no | — |
| 72 | C22⋊C4 | — | 4 | 3 | no | — |
| 73 | C42 | C2×D4 | 1 | 3 | yes | — |
| 74 | C42 | C24 | 1 | 1 | yes | characteristic |
| 75 | C23 | — | 4 | 3 | no | — |
| 76 | C23 | — | 4 | 1 | no | — |
| 77 | C23 | — | 4 | 1 | no | — |
| 78 | C23 | — | 8 | 6 | no | — |
| 79 | C23 | #127 | 1 | 1 | yes | characteristic |
| 80 | Q8 | — | 4 | 6 | no | — |
| 81 | D4 | — | 4 | 6 | no | — |
| 82 | D4 | — | 4 | 6 | no | — |
| 83 | D4 | — | 4 | 6 | no | — |
| 84 | D4 | — | 4 | 6 | no | — |
| 85 | D4 | — | 4 | 6 | no | — |
| 86 | D4 | — | 8 | 6 | no | — |
| 87 | D4 | — | 8 | 6 | no | — |
| 88 | C2×C4 | — | 2 | 6 | no | — |
| 89 | C2×C4 | — | 4 | 3 | no | — |
| 90 | C2×C4 | — | 8 | 6 | no | — |
| 91 | C2×C4 | #127 | 1 | 3 | yes | — |
| 92 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 93 | C8 | — | 2 | 6 | no | — |
| 94 | C8 | — | 2 | 6 | no | — |
| 95 | C22 | — | 2 | 3 | no | — |
| 96 | C22 | — | 8 | 3 | no | — |
| 97 | C22 | — | 8 | 3 | no | — |
| 98 | C22 | — | 8 | 3 | no | — |
| 99 | C22 | — | 8 | 3 | no | — |
| 100 | C22 | — | 8 | 3 | no | — |
| 101 | C22 | — | 16 | 4 | no | — |
| 102 | C22 | C2×#127 | 1 | 1 | yes | characteristic |
| 103 | C4 | — | 2 | 3 | no | — |
| 104 | C4 | — | 8 | 3 | no | — |
| 105 | C4 | #576 | 1 | 6 | yes | — |
| 106 | C2 | — | 4 | 1 | no | — |
| 107 | C2 | — | 16 | 3 | no | — |
| 108 | C2 | — | 16 | 1 | no | — |
| 109 | C2 | — | 16 | 1 | no | — |
| 110 | C2 | #3155 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3207 | C2 | semidirect |
| 2. #3205 | C2 | semidirect |
| 3. #1721 | C2 | semidirect |
| 4. #1717 | C2 | semidirect |
| 5. #1712 | C2 | semidirect |
| 6. #1711 | C2 | semidirect |
| 7. #1454 | C2 | semidirect |
| 12. #493 | C22 | semidirect |
| 13. #492 | C22 | semidirect |
| 14. #487 | C22 | semidirect |
| 17. #486 | C22 | semidirect |
| 23. #322 | C22 | semidirect |
| 24. C82 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3207 | 2 |
| #3205 | 2 |
| #1721 | 2 |
| #1717 | 2 |
| #1712 | 2 |
| #1711 | 2 |
| #1454 | 2 |