Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
701 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 701 conjugacy classes of subgroups — too many to draw. Only its 167 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 | #3458 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #2309 | C2 | 1 | 3 | yes | maximal |
| 3 | #2029 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #1740 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C23×D4 | C22 | 1 | 1 | yes | characteristic |
| 6 | #544 | — | 2 | 6 | no | — |
| 7 | #543 | C22 | 1 | 4 | yes | — |
| 8 | #533 | — | 2 | 6 | no | — |
| 9 | C2×#117 | C22 | 1 | 3 | yes | — |
| 10 | C2×#117 | C22 | 1 | 3 | yes | — |
| 11 | #390 | — | 2 | 6 | no | — |
| 12 | #390 | — | 2 | 6 | no | — |
| 13 | #379 | C22 | 1 | 12 | yes | — |
| 14 | #342 | C22 | 1 | 12 | yes | — |
| 15 | C25 | C23 | 1 | 2 | yes | — |
| 16 | C22×D4 | — | 2 | 3 | no | — |
| 17 | C22×D4 | — | 2 | 3 | no | — |
| 18 | C22×D4 | — | 2 | 6 | no | — |
| 19 | C22×D4 | Q8 | 1 | 4 | yes | complemented |
| 20 | C23×C4 | C23 | 1 | 1 | yes | characteristic |
| 21 | C4⋊Q8 | — | 2 | 12 | no | — |
| 22 | #125 | — | 2 | 12 | no | — |
| 23 | C22⋊Q8 | — | 2 | 24 | no | — |
| 24 | C4×Q8 | — | 2 | 8 | no | — |
| 25 | #117 | — | 2 | 6 | no | — |
| 26 | #117 | — | 2 | 6 | no | — |
| 27 | #117 | — | 2 | 12 | no | — |
| 28 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 29 | C2×C4⋊C4 | D4 | 1 | 6 | yes | — |
| 30 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 31 | C2×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 32 | C2×C22⋊C4 | C23 | 1 | 6 | yes | — |
| 33 | C2×C42 | — | 2 | 3 | no | — |
| 34 | C2×C42 | D4 | 1 | 6 | yes | — |
| 35 | C24 | — | 2 | 6 | no | — |
| 36 | C24 | — | 2 | 6 | no | — |
| 37 | C24 | — | 8 | 4 | no | — |
| 38 | C24 | C2×Q8 | 1 | 4 | yes | — |
| 39 | C24 | C24 | 1 | 1 | yes | characteristic |
| 40 | C2×Q8 | — | 2 | 8 | no | — |
| 41 | C2×D4 | — | 2 | 12 | no | — |
| 42 | C2×D4 | — | 2 | 12 | no | — |
| 43 | C2×D4 | — | 4 | 1 | no | — |
| 44 | C2×D4 | — | 4 | 3 | no | — |
| 45 | C2×D4 | — | 4 | 6 | no | — |
| 46 | C2×D4 | — | 4 | 6 | no | — |
| 47 | C22×C4 | — | 2 | 6 | no | — |
| 48 | C22×C4 | C2×D4 | 1 | 12 | yes | — |
| 49 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 50 | C22×C4 | C2×D4 | 1 | 3 | yes | — |
| 51 | C22×C4 | C2×Q8 | 1 | 2 | yes | — |
| 52 | C22×C4 | #41 | 1 | 6 | yes | — |
| 53 | C4⋊C4 | — | 2 | 24 | no | — |
| 54 | C4⋊C4 | — | 2 | 12 | no | — |
| 55 | C4⋊C4 | — | 4 | 6 | no | — |
| 56 | C22⋊C4 | — | 2 | 12 | no | — |
| 57 | C22⋊C4 | — | 2 | 12 | no | — |
| 58 | C22⋊C4 | — | 8 | 24 | no | — |
| 59 | C42 | — | 2 | 6 | no | — |
| 60 | C42 | — | 2 | 6 | no | — |
| 61 | C42 | — | 4 | 6 | no | — |
| 62 | C23 | — | 2 | 12 | no | — |
| 63 | C23 | — | 4 | 2 | no | — |
| 64 | C23 | — | 4 | 24 | no | — |
| 65 | C23 | — | 4 | 2 | no | — |
| 66 | C23 | — | 4 | 6 | no | — |
| 67 | C23 | — | 8 | 4 | no | — |
| 68 | C23 | — | 8 | 12 | no | — |
| 69 | C23 | C22×D4 | 1 | 3 | yes | — |
| 70 | C23 | C22×Q8 | 1 | 1 | yes | characteristic |
| 71 | C23 | C2×#41 | 1 | 3 | yes | — |
| 72 | Q8 | — | 8 | 8 | no | — |
| 73 | D4 | — | 4 | 4 | no | — |
| 74 | D4 | — | 4 | 12 | no | — |
| 75 | C2×C4 | — | 2 | 12 | no | — |
| 76 | C2×C4 | — | 4 | 12 | no | — |
| 77 | C2×C4 | — | 4 | 12 | no | — |
| 78 | C2×C4 | — | 4 | 6 | no | — |
| 79 | C2×C4 | #120 | 1 | 1 | yes | characteristic |
| 80 | C2×C4 | #120 | 1 | 3 | yes | — |
| 81 | C2×C4 | C22⋊Q8 | 1 | 6 | yes | — |
| 82 | C2×C4 | C22⋊Q8 | 1 | 6 | yes | — |
| 83 | C2×C4 | #124 | 1 | 6 | yes | — |
| 84 | C2×C4 | #124 | 1 | 6 | yes | — |
| 85 | C22 | — | 2 | 6 | no | — |
| 86 | C22 | — | 4 | 1 | no | — |
| 87 | C22 | — | 4 | 3 | no | — |
| 88 | C22 | — | 4 | 24 | no | — |
| 89 | C22 | — | 4 | 4 | no | — |
| 90 | C22 | — | 8 | 12 | no | — |
| 91 | C22 | — | 8 | 4 | no | — |
| 92 | C22 | C2×#120 | 1 | 1 | yes | characteristic |
| 93 | C22 | C2×C22⋊Q8 | 1 | 3 | yes | — |
| 94 | C22 | C2×#124 | 1 | 3 | yes | — |
| 95 | C4 | — | 4 | 12 | no | — |
| 96 | C4 | — | 8 | 6 | no | — |
| 97 | C4 | #393 | 1 | 2 | yes | — |
| 98 | C4 | #393 | 1 | 6 | yes | — |
| 99 | C2 | — | 2 | 3 | no | — |
| 100 | C2 | — | 2 | 3 | no | — |
| 101 | C2 | — | 8 | 4 | no | — |
| 102 | C2 | C2×#393 | 1 | 1 | yes | characteristic |
| 103 | C2 | #3027 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3458 | C2 | semidirect |
| 3. #2029 | C2 | semidirect |
| 4. #1740 | C2 | semidirect |
| 19. C22×D4 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3458 | 2 |
| #2309 | 2 |
| #2029 | 2 |
| #1740 | 2 |