Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3294 conjugacy classes of subgroups; only the 2839 normal subgroups are shown.
This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 2839 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 | #3559 | C2 | 1 | 30 | yes | maximal, complemented |
| 2 | #3548 | C2 | 1 | 15 | yes | maximal, complemented |
| 3 | #3531 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #3517 | C2 | 1 | 10 | yes | maximal, complemented |
| 5 | #3504 | C2 | 1 | 5 | yes | maximal, complemented |
| 6 | #564 | C22 | 1 | 1 | yes | characteristic |
| 7 | #563 | C22 | 1 | 30 | yes | — |
| 8 | #562 | C22 | 1 | 15 | yes | — |
| 9 | #562 | C22 | 1 | 30 | yes | — |
| 10 | #557 | C22 | 1 | 10 | yes | — |
| 11 | #554 | C22 | 1 | 60 | yes | — |
| 12 | #553 | C22 | 1 | 60 | yes | — |
| 13 | #550 | C22 | 1 | 10 | yes | — |
| 14 | #550 | C22 | 1 | 30 | yes | — |
| 15 | #547 | C22 | 1 | 30 | yes | — |
| 16 | #547 | C22 | 1 | 60 | yes | — |
| 17 | #544 | C22 | 1 | 30 | yes | — |
| 18 | #540 | C22 | 1 | 60 | yes | — |
| 19 | #539 | C22 | 1 | 60 | yes | — |
| 20 | #537 | C22 | 1 | 60 | yes | — |
| 21 | #533 | C22 | 1 | 15 | yes | — |
| 22 | #532 | C22 | 1 | 30 | yes | — |
| 23 | #529 | C22 | 1 | 30 | yes | — |
| 24 | C2×#125 | C22 | 1 | 15 | yes | — |
| 25 | #520 | C22 | 1 | 10 | yes | — |
| 26 | C4×#41 | C22 | 1 | 5 | yes | — |
| 27 | C2×#41 | C23 | 1 | 5 | yes | — |
| 28 | C2×#41 | C23 | 1 | 60 | yes | — |
| 29 | C4⋊Q8 | C23 | 1 | 30 | yes | — |
| 30 | C4⋊Q8 | C23 | 1 | 30 | yes | — |
| 31 | #127 | C23 | 1 | 30 | yes | — |
| 32 | #126 | C23 | 1 | 60 | yes | — |
| 33 | #125 | C23 | 1 | 15 | yes | — |
| 34 | #125 | C23 | 1 | 30 | yes | — |
| 35 | #125 | C23 | 1 | 60 | yes | — |
| 36 | #125 | C23 | 1 | 10 | yes | — |
| 37 | #125 | C23 | 1 | 5 | yes | — |
| 38 | #124 | C23 | 1 | 30 | yes | — |
| 39 | #123 | C23 | 1 | 60 | yes | — |
| 40 | #123 | C23 | 1 | 120 | yes | — |
| 41 | C22⋊Q8 | C23 | 1 | 30 | yes | — |
| 42 | C22⋊Q8 | C23 | 1 | 60 | yes | — |
| 43 | C22⋊Q8 | C23 | 1 | 120 | yes | — |
| 44 | #121 | C23 | 1 | 30 | yes | — |
| 45 | #121 | C23 | 1 | 60 | yes | — |
| 46 | #121 | C23 | 1 | 60 | yes | — |
| 47 | #121 | C23 | 1 | 120 | yes | — |
| 48 | C4×Q8 | C23 | 1 | 10 | yes | — |
| 49 | C4×Q8 | C23 | 1 | 30 | yes | — |
| 50 | C4×D4 | C23 | 1 | 30 | yes | — |
| 51 | C4×D4 | C23 | 1 | 60 | yes | — |
| 52 | C4×D4 | C23 | 1 | 30 | yes | — |
| 53 | C4×D4 | C23 | 1 | 60 | yes | — |
| 54 | #117 | C23 | 1 | 15 | yes | — |
| 55 | #117 | C23 | 1 | 30 | yes | — |
| 56 | C2×C4⋊C4 | C23 | 1 | 30 | yes | — |
| 57 | C2×C4⋊C4 | C23 | 1 | 60 | yes | — |
| 58 | C2×C42 | C23 | 1 | 15 | yes | — |
| 59 | #41 | — | 2 | 20 | no | — |
| 60 | #41 | — | 8 | 60 | no | — |
| 61 | C2×Q8 | C24 | 1 | 30 | yes | — |
| 62 | C2×Q8 | C24 | 1 | 1 | yes | characteristic |
| 63 | C2×D4 | C24 | 1 | 15 | yes | — |
| 64 | C2×D4 | C24 | 1 | 60 | yes | — |
| 65 | C2×D4 | C24 | 1 | 30 | yes | — |
| 66 | C22×C4 | C24 | 1 | 15 | yes | — |
| 67 | C22×C4 | C24 | 1 | 30 | yes | — |
| 68 | C22×C4 | C24 | 1 | 60 | yes | — |
| 69 | C4⋊C4 | C24 | 1 | 30 | yes | — |
| 70 | C4⋊C4 | C24 | 1 | 30 | yes | — |
| 71 | C4⋊C4 | C24 | 1 | 60 | yes | — |
| 72 | C4⋊C4 | C24 | 1 | 60 | yes | — |
| 73 | C4⋊C4 | C24 | 1 | 10 | yes | — |
| 74 | C4⋊C4 | C24 | 1 | 10 | yes | — |
| 75 | C4⋊C4 | C24 | 1 | 10 | yes | — |
| 76 | C22⋊C4 | C24 | 1 | 60 | yes | — |
| 77 | C22⋊C4 | C24 | 1 | 30 | yes | — |
| 78 | C22⋊C4 | C24 | 1 | 60 | yes | — |
| 79 | C42 | C24 | 1 | 5 | yes | — |
| 80 | C42 | C24 | 1 | 30 | yes | — |
| 81 | C42 | C24 | 1 | 15 | yes | — |
| 82 | C23 | C25 | 1 | 15 | yes | — |
| 83 | Q8 | — | 4 | 30 | no | — |
| 84 | Q8 | 2−1+4 | 1 | 4 | yes | — |
| 85 | D4 | — | 2 | 30 | no | — |
| 86 | D4 | — | 4 | 60 | no | — |
| 87 | D4 | — | 4 | 30 | no | — |
| 88 | C2×C4 | — | 2 | 30 | no | — |
| 89 | C2×C4 | — | 4 | 60 | no | — |
| 90 | C2×C4 | — | 4 | 30 | no | — |
| 91 | C2×C4 | C25 | 1 | 5 | yes | — |
| 92 | C2×C4 | C25 | 1 | 10 | yes | — |
| 93 | C2×C4 | C25 | 1 | 30 | yes | — |
| 94 | C2×C4 | C25 | 1 | 3 | yes | — |
| 95 | C22 | — | 2 | 15 | no | — |
| 96 | C22 | — | 2 | 30 | no | — |
| 97 | C22 | C26 | 1 | 1 | yes | characteristic |
| 98 | C4 | — | 2 | 5 | no | — |
| 99 | C4 | — | 2 | 30 | no | — |
| 100 | C4 | — | 2 | 10 | no | — |
| 101 | C4 | C2×2−1+4 | 1 | 6 | yes | — |
| 102 | C2 | — | 4 | 15 | no | — |
| 103 | C2 | C22×2−1+4 | 1 | 1 | yes | characteristic |
| 104 | C2 | H(7,𝔽2) | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3559 | C2 | semidirect |
| 2. #3548 | C2 | semidirect |
| 3. #3531 | C2 | semidirect |
| 4. #3517 | C2 | semidirect |
| 5. #3504 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3559 | 2 |
| #3548 | 2 |
| #3531 | 2 |
| #3517 | 2 |
| #3504 | 2 |