Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
345 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 99 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 | #3325 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #1709 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #1697 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #1634 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #1619 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #1589 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #555 | C22 | 1 | 2 | yes | — |
| 8 | D42 | C22 | 1 | 2 | yes | complemented |
| 9 | #494 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #493 | C22 | 1 | 1 | yes | characteristic |
| 11 | #486 | C22 | 1 | 4 | yes | complemented |
| 12 | #482 | — | 2 | 4 | no | — |
| 13 | #476 | C22 | 1 | 4 | yes | complemented |
| 14 | #469 | — | 2 | 4 | no | — |
| 15 | #459 | C22 | 1 | 4 | yes | — |
| 16 | #447 | — | 2 | 4 | no | — |
| 17 | C4×D8 | C22 | 1 | 2 | yes | — |
| 18 | #436 | C22 | 1 | 4 | yes | complemented |
| 19 | #418 | — | 2 | 4 | no | — |
| 20 | #335 | C22 | 1 | 2 | yes | complemented |
| 21 | #333 | C22 | 1 | 2 | yes | complemented |
| 22 | #331 | C22 | 1 | 2 | yes | complemented |
| 23 | D4⋊C8 | C22 | 1 | 4 | yes | complemented |
| 24 | C82 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | C22×D4 | — | 2 | 4 | no | — |
| 26 | C2×Q16 | — | 4 | 1 | no | — |
| 27 | C2×D8 | — | 2 | 4 | no | — |
| 28 | C2×D8 | — | 4 | 1 | no | — |
| 29 | C2×D8 | D4 | 1 | 2 | yes | — |
| 30 | C2×M16 | — | 2 | 4 | no | — |
| 31 | C4⋊Q8 | C23 | 1 | 2 | yes | — |
| 32 | #127 | C23 | 1 | 2 | yes | — |
| 33 | #124 | — | 4 | 2 | no | — |
| 34 | #121 | — | 2 | 4 | no | — |
| 35 | #121 | — | 2 | 4 | no | — |
| 36 | #120 | — | 4 | 2 | no | — |
| 37 | C4×D4 | C23 | 1 | 4 | yes | — |
| 38 | #117 | — | 2 | 4 | no | — |
| 39 | #107 | D4 | 1 | 2 | yes | — |
| 40 | #106 | — | 2 | 4 | no | — |
| 41 | C4⋊C8 | C23 | 1 | 4 | yes | — |
| 42 | #102 | — | 2 | 4 | no | — |
| 43 | #102 | — | 2 | 4 | no | — |
| 44 | #102 | — | 2 | 4 | no | — |
| 45 | #102 | — | 2 | 4 | no | — |
| 46 | #102 | D4 | 1 | 4 | yes | — |
| 47 | C22⋊C8 | — | 2 | 4 | no | — |
| 48 | C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 49 | C4×C8 | C23 | 1 | 2 | yes | — |
| 50 | C24 | — | 4 | 2 | no | — |
| 51 | C2×Q8 | — | 2 | 2 | no | — |
| 52 | C2×D4 | — | 2 | 4 | no | — |
| 53 | C2×D4 | — | 2 | 2 | no | — |
| 54 | C2×D4 | — | 4 | 4 | no | — |
| 55 | C2×D4 | — | 4 | 8 | no | — |
| 56 | C2×D4 | — | 4 | 2 | no | — |
| 57 | C2×D4 | — | 8 | 4 | no | — |
| 58 | C2×D4 | C2×D4 | 1 | 4 | yes | — |
| 59 | C22×C4 | — | 2 | 4 | no | — |
| 60 | Q16 | — | 8 | 2 | no | — |
| 61 | D8 | — | 4 | 2 | no | — |
| 62 | D8 | — | 8 | 2 | no | — |
| 63 | D8 | — | 8 | 4 | no | — |
| 64 | M16 | — | 8 | 4 | no | — |
| 65 | C2×C8 | — | 2 | 1 | no | — |
| 66 | C2×C8 | — | 2 | 4 | no | — |
| 67 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 68 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 69 | C4⋊C4 | — | 2 | 4 | no | — |
| 70 | C4⋊C4 | C2×D4 | 1 | 4 | yes | — |
| 71 | C22⋊C4 | — | 2 | 4 | no | — |
| 72 | C22⋊C4 | — | 4 | 2 | no | — |
| 73 | C22⋊C4 | — | 4 | 4 | no | — |
| 74 | C42 | — | 4 | 2 | no | — |
| 75 | C42 | C24 | 1 | 1 | yes | characteristic |
| 76 | C23 | — | 2 | 4 | no | — |
| 77 | C23 | — | 4 | 2 | no | — |
| 78 | C23 | — | 8 | 8 | no | — |
| 79 | C23 | — | 16 | 2 | no | — |
| 80 | Q8 | — | 8 | 2 | no | — |
| 81 | D4 | — | 2 | 4 | no | — |
| 82 | D4 | — | 2 | 4 | no | — |
| 83 | D4 | — | 4 | 4 | no | — |
| 84 | D4 | — | 4 | 4 | no | — |
| 85 | D4 | — | 8 | 2 | no | — |
| 86 | D4 | — | 8 | 8 | no | — |
| 87 | D4 | — | 16 | 2 | no | — |
| 88 | C2×C4 | — | 2 | 4 | no | — |
| 89 | C2×C4 | — | 4 | 4 | no | — |
| 90 | C2×C4 | — | 4 | 4 | no | — |
| 91 | C2×C4 | — | 4 | 2 | no | — |
| 92 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 93 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 94 | C8 | — | 2 | 2 | no | — |
| 95 | C8 | — | 2 | 2 | no | — |
| 96 | C8 | — | 2 | 2 | no | — |
| 97 | C8 | — | 4 | 4 | no | — |
| 98 | C22 | — | 4 | 4 | no | — |
| 99 | C22 | — | 4 | 4 | no | — |
| 100 | C22 | — | 4 | 4 | no | — |
| 101 | C22 | — | 8 | 4 | no | — |
| 102 | C22 | — | 8 | 2 | no | — |
| 103 | C22 | — | 16 | 8 | no | — |
| 104 | C22 | D42 | 1 | 1 | yes | characteristic |
| 105 | C4 | — | 2 | 1 | no | — |
| 106 | C4 | — | 4 | 4 | no | — |
| 107 | C4 | — | 8 | 2 | no | — |
| 108 | C4 | #575 | 1 | 2 | yes | — |
| 109 | C4 | #576 | 1 | 2 | yes | — |
| 110 | C2 | — | 8 | 4 | no | — |
| 111 | C2 | — | 16 | 2 | no | — |
| 112 | C2 | #3273 | 1 | 2 | yes | — |
| 113 | C2 | #3292 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3325 | C2 | semidirect |
| 2. #1709 | C2 | semidirect |
| 3. #1697 | C2 | semidirect |
| 4. #1634 | C2 | semidirect |
| 5. #1619 | C2 | semidirect |
| 6. #1589 | C2 | semidirect |
| 8. D42 | C22 | semidirect |
| 9. #494 | C22 | semidirect |
| 11. #486 | C22 | semidirect |
| 13. #476 | C22 | semidirect |
| 18. #436 | C22 | semidirect |
| 20. #335 | C22 | semidirect |
| 21. #333 | C22 | semidirect |
| 22. #331 | C22 | semidirect |
| 23. D4⋊C8 | 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 |
|---|---|
| #3325 | 2 |
| #1709 | 2 |
| #1697 | 2 |
| #1634 | 2 |
| #1619 | 2 |
| #1589 | 2 |