Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1016 conjugacy classes of subgroups; only the 218 normal subgroups are shown.
This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 218 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 | #33255 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #33235 | C2 | 1 | 3 | yes | maximal |
| 3 | C2×#1720 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C2×#1711 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #7846 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#498 | C22 | 1 | 3 | yes | — |
| 7 | C2×#494 | C22 | 1 | 3 | yes | — |
| 8 | C2×#487 | C22 | 1 | 6 | yes | — |
| 9 | C2×#486 | C22 | 1 | 6 | yes | — |
| 10 | C2×#486 | C22 | 1 | 6 | yes | — |
| 11 | #2867 | C22 | 1 | 1 | yes | characteristic |
| 12 | #2862 | C22 | 1 | 3 | yes | — |
| 13 | #1971 | — | 2 | 12 | no | — |
| 14 | #1961 | — | 2 | 12 | no | — |
| 15 | C4×#97 | C22 | 1 | 3 | yes | — |
| 16 | #1720 | — | 2 | 24 | no | — |
| 17 | #1711 | — | 2 | 8 | no | — |
| 18 | C2×#322 | C22 | 1 | 3 | yes | — |
| 19 | C2×C82 | C22 | 1 | 1 | yes | characteristic |
| 20 | C22×Q16 | — | 2 | 6 | no | — |
| 21 | C2×C4⋊Q8 | C23 | 1 | 6 | yes | — |
| 22 | C2×#127 | — | 2 | 6 | no | — |
| 23 | C2×#127 | C23 | 1 | 2 | yes | — |
| 24 | #498 | — | 2 | 12 | no | — |
| 25 | #494 | — | 2 | 12 | no | — |
| 26 | #487 | — | 2 | 12 | no | — |
| 27 | #487 | — | 2 | 12 | no | — |
| 28 | #486 | — | 2 | 12 | no | — |
| 29 | #486 | — | 2 | 12 | no | — |
| 30 | #486 | — | 2 | 12 | no | — |
| 31 | #486 | — | 2 | 12 | no | — |
| 32 | C2×#106 | — | 2 | 6 | no | — |
| 33 | C2×#103 | — | 2 | 12 | no | — |
| 34 | C2×#102 | — | 2 | 12 | no | — |
| 35 | C2×#102 | — | 2 | 12 | no | — |
| 36 | C2×#97 | D4 | 1 | 12 | yes | — |
| 37 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 38 | C2×C4×C8 | C23 | 1 | 3 | yes | — |
| 39 | #391 | — | 2 | 12 | no | — |
| 40 | #382 | — | 2 | 6 | no | — |
| 41 | C43 | C23 | 1 | 1 | yes | characteristic |
| 42 | #322 | D4 | 1 | 12 | yes | complemented |
| 43 | C82 | D4 | 1 | 4 | yes | complemented |
| 44 | C22×Q8 | — | 2 | 6 | no | — |
| 45 | C22×D4 | — | 2 | 6 | no | — |
| 46 | C22×D4 | — | 4 | 4 | no | — |
| 47 | C2×Q16 | — | 4 | 24 | no | — |
| 48 | C2×Q16 | — | 4 | 12 | no | — |
| 49 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 50 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 51 | C4⋊Q8 | — | 2 | 6 | no | — |
| 52 | C4⋊Q8 | — | 2 | 6 | no | — |
| 53 | C4⋊Q8 | — | 2 | 6 | no | — |
| 54 | C4⋊Q8 | — | 2 | 6 | no | — |
| 55 | #127 | — | 2 | 2 | no | — |
| 56 | #127 | — | 2 | 6 | no | — |
| 57 | #127 | — | 4 | 24 | no | — |
| 58 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 59 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 60 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 61 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 62 | #106 | — | 4 | 12 | no | — |
| 63 | #103 | — | 4 | 24 | no | — |
| 64 | #102 | — | 4 | 24 | no | — |
| 65 | #102 | — | 4 | 24 | no | — |
| 66 | #97 | — | 4 | 12 | no | — |
| 67 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 68 | C4×C8 | C2×D4 | 1 | 6 | yes | — |
| 69 | C4×C8 | C2×D4 | 1 | 12 | yes | — |
| 70 | #95 | — | 4 | 12 | no | — |
| 71 | C24 | — | 4 | 2 | no | — |
| 72 | C2×Q8 | — | 4 | 6 | no | — |
| 73 | C2×Q8 | — | 4 | 6 | no | — |
| 74 | C2×Q8 | — | 4 | 24 | no | — |
| 75 | C2×D4 | — | 4 | 6 | no | — |
| 76 | C2×D4 | — | 4 | 6 | no | — |
| 77 | C2×D4 | — | 4 | 24 | no | — |
| 78 | C2×D4 | — | 8 | 24 | no | — |
| 79 | C22×C4 | — | 4 | 6 | no | — |
| 80 | C22×C4 | #127 | 1 | 4 | yes | — |
| 81 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 82 | Q16 | — | 8 | 24 | no | — |
| 83 | C2×C8 | — | 2 | 12 | no | — |
| 84 | C2×C8 | — | 2 | 12 | no | — |
| 85 | C2×C8 | #127 | 1 | 12 | yes | — |
| 86 | C2×C8 | #127 | 1 | 12 | yes | — |
| 87 | C4⋊C4 | — | 4 | 6 | no | — |
| 88 | C4⋊C4 | — | 4 | 6 | no | — |
| 89 | C4⋊C4 | — | 4 | 6 | no | — |
| 90 | C4⋊C4 | — | 4 | 6 | no | — |
| 91 | C42 | — | 2 | 12 | no | — |
| 92 | C42 | C22×D4 | 1 | 1 | yes | characteristic |
| 93 | C42 | C22×D4 | 1 | 3 | yes | — |
| 94 | C23 | — | 8 | 2 | no | — |
| 95 | C23 | — | 8 | 12 | no | — |
| 96 | C23 | C2×#127 | 1 | 1 | yes | characteristic |
| 97 | Q8 | — | 8 | 12 | no | — |
| 98 | Q8 | — | 8 | 12 | no | — |
| 99 | D4 | — | 8 | 12 | no | — |
| 100 | D4 | — | 8 | 12 | no | — |
| 101 | D4 | — | 16 | 16 | no | — |
| 102 | C2×C4 | — | 2 | 12 | no | — |
| 103 | C2×C4 | — | 8 | 6 | no | — |
| 104 | C2×C4 | — | 8 | 12 | no | — |
| 105 | C2×C4 | C2×#127 | 1 | 3 | yes | — |
| 106 | C2×C4 | C2×#127 | 1 | 3 | yes | — |
| 107 | C2×C4 | #575 | 1 | 12 | yes | — |
| 108 | C8 | — | 2 | 12 | no | — |
| 109 | C8 | — | 2 | 12 | no | — |
| 110 | C22 | — | 16 | 6 | no | — |
| 111 | C22 | — | 16 | 8 | no | — |
| 112 | C22 | #2867 | 1 | 1 | yes | characteristic |
| 113 | C22 | #3153 | 1 | 6 | yes | — |
| 114 | C4 | — | 4 | 4 | no | — |
| 115 | C4 | — | 16 | 6 | no | — |
| 116 | C4 | #3150 | 1 | 6 | yes | — |
| 117 | C4 | #3153 | 1 | 6 | yes | — |
| 118 | C2 | — | 32 | 2 | no | — |
| 119 | C2 | #19753 | 1 | 4 | yes | — |
| 120 | C2 | #33306 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2867 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #33255 | C2 | semidirect |
| 3. C2×#1720 | C2 | semidirect |
| 4. C2×#1711 | C2 | semidirect |
| 5. #7846 | C2 | semidirect |
| 42. #322 | D4 | semidirect |
| 43. C82 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33255 | 2 |
| #33235 | 2 |
| C2×#1720 | 2 |
| C2×#1711 | 2 |
| #7846 | 2 |