Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7878 conjugacy classes of subgroups; only the 1450 normal subgroups are shown.
This group has 7878 conjugacy classes of subgroups — too many to draw. Only its 1450 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 | C2×#2867 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#2856 | C2 | 1 | 14 | yes | maximal, complemented |
| 3 | C22×C43 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #14713 | C2 | 1 | 14 | yes | maximal, complemented |
| 5 | #14692 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C22×#127 | C22 | 1 | 14 | yes | — |
| 7 | C22×#121 | C22 | 1 | 28 | yes | — |
| 8 | C23×C42 | C22 | 1 | 7 | yes | — |
| 9 | #2867 | — | 2 | 8 | no | — |
| 10 | #2856 | — | 2 | 112 | no | — |
| 11 | C2×#390 | C22 | 1 | 84 | yes | — |
| 12 | C2×#384 | C22 | 1 | 14 | yes | — |
| 13 | C2×C43 | — | 2 | 8 | no | — |
| 14 | C2×C43 | C22 | 1 | 7 | yes | — |
| 15 | C2×C43 | C22 | 1 | 1 | yes | characteristic |
| 16 | C23×D4 | C23 | 1 | 8 | yes | — |
| 17 | C23×D4 | C23 | 1 | 28 | yes | — |
| 18 | C24×C4 | C23 | 1 | 7 | yes | — |
| 19 | C2×#127 | — | 2 | 56 | no | — |
| 20 | C2×#127 | — | 2 | 56 | no | — |
| 21 | C2×#127 | — | 2 | 56 | no | — |
| 22 | C2×#121 | — | 2 | 336 | no | — |
| 23 | C22×C4⋊C4 | C23 | 1 | 28 | yes | — |
| 24 | C22×C22⋊C4 | C23 | 1 | 56 | yes | — |
| 25 | C22×C42 | — | 2 | 28 | no | — |
| 26 | C22×C42 | — | 2 | 28 | no | — |
| 27 | C22×C42 | D4 | 1 | 56 | yes | complemented |
| 28 | C22×C42 | C23 | 1 | 21 | yes | — |
| 29 | C22×C42 | C23 | 1 | 7 | yes | — |
| 30 | #390 | — | 2 | 336 | no | — |
| 31 | #384 | — | 2 | 56 | no | — |
| 32 | C43 | — | 2 | 28 | no | — |
| 33 | C43 | D4 | 1 | 8 | yes | complemented |
| 34 | C25 | C24 | 1 | 8 | yes | — |
| 35 | C25 | C24 | 1 | 1 | yes | characteristic |
| 36 | C22×D4 | — | 2 | 112 | no | — |
| 37 | C22×D4 | — | 2 | 168 | no | — |
| 38 | C22×D4 | — | 2 | 56 | no | — |
| 39 | C22×D4 | — | 2 | 168 | no | — |
| 40 | C23×C4 | — | 2 | 14 | no | — |
| 41 | C23×C4 | — | 2 | 42 | no | — |
| 42 | C23×C4 | C2×D4 | 1 | 84 | yes | — |
| 43 | C23×C4 | C24 | 1 | 8 | yes | — |
| 44 | C23×C4 | C24 | 1 | 7 | yes | — |
| 45 | C23×C4 | C24 | 1 | 7 | yes | — |
| 46 | #127 | — | 4 | 224 | no | — |
| 47 | #121 | — | 4 | 448 | no | — |
| 48 | C2×C4⋊C4 | — | 2 | 168 | no | — |
| 49 | C2×C22⋊C4 | — | 2 | 336 | no | — |
| 50 | C2×C42 | — | 2 | 42 | no | — |
| 51 | C2×C42 | — | 2 | 42 | no | — |
| 52 | C2×C42 | — | 2 | 224 | no | — |
| 53 | C2×C42 | C2×D4 | 1 | 84 | yes | — |
| 54 | C2×C42 | C2×D4 | 1 | 28 | yes | — |
| 55 | C2×C42 | C2×D4 | 1 | 28 | yes | — |
| 56 | C2×C42 | #41 | 1 | 28 | yes | — |
| 57 | C24 | — | 2 | 1 | no | — |
| 58 | C24 | — | 2 | 7 | no | — |
| 59 | C24 | — | 2 | 56 | no | — |
| 60 | C24 | — | 2 | 8 | no | — |
| 61 | C24 | — | 2 | 56 | no | — |
| 62 | C24 | C22×D4 | 1 | 14 | yes | — |
| 63 | C24 | C25 | 1 | 1 | yes | characteristic |
| 64 | C2×D4 | — | 4 | 224 | no | — |
| 65 | C2×D4 | — | 4 | 336 | no | — |
| 66 | C2×D4 | — | 4 | 112 | no | — |
| 67 | C2×D4 | — | 4 | 336 | no | — |
| 68 | C22×C4 | — | 2 | 7 | no | — |
| 69 | C22×C4 | — | 2 | 21 | no | — |
| 70 | C22×C4 | — | 2 | 168 | no | — |
| 71 | C22×C4 | — | 2 | 168 | no | — |
| 72 | C22×C4 | — | 2 | 56 | no | — |
| 73 | C22×C4 | #127 | 1 | 112 | yes | complemented |
| 74 | C22×C4 | C22×D4 | 1 | 42 | yes | — |
| 75 | C22×C4 | C22×D4 | 1 | 14 | yes | — |
| 76 | C22×C4 | C22×D4 | 1 | 42 | yes | — |
| 77 | C22×C4 | C2×#41 | 1 | 42 | yes | — |
| 78 | C4⋊C4 | — | 4 | 112 | no | — |
| 79 | C22⋊C4 | — | 4 | 224 | no | — |
| 80 | C42 | — | 2 | 168 | no | — |
| 81 | C42 | #121 | 1 | 112 | yes | complemented |
| 82 | C23 | — | 2 | 14 | no | — |
| 83 | C23 | — | 2 | 42 | no | — |
| 84 | C23 | — | 4 | 56 | no | — |
| 85 | C23 | — | 4 | 56 | no | — |
| 86 | C23 | — | 4 | 168 | no | — |
| 87 | C23 | C2×#127 | 1 | 28 | yes | — |
| 88 | C23 | C23×D4 | 1 | 1 | yes | characteristic |
| 89 | C23 | C23×D4 | 1 | 7 | yes | — |
| 90 | C23 | C22×#41 | 1 | 7 | yes | — |
| 91 | D4 | — | 8 | 64 | no | — |
| 92 | D4 | — | 8 | 224 | no | — |
| 93 | C2×C4 | — | 2 | 42 | no | — |
| 94 | C2×C4 | — | 2 | 42 | no | — |
| 95 | C2×C4 | — | 2 | 224 | no | — |
| 96 | C2×C4 | — | 4 | 56 | no | — |
| 97 | C2×C4 | C2×#121 | 1 | 84 | yes | — |
| 98 | C2×C4 | C2×#127 | 1 | 28 | yes | — |
| 99 | C2×C4 | C2×#127 | 1 | 28 | yes | — |
| 100 | C2×C4 | #532 | 1 | 84 | yes | — |
| 101 | C22 | — | 2 | 28 | no | — |
| 102 | C22 | — | 2 | 28 | no | — |
| 103 | C22 | — | 8 | 56 | no | — |
| 104 | C22 | — | 8 | 8 | no | — |
| 105 | C22 | — | 8 | 56 | no | — |
| 106 | C22 | #2867 | 1 | 8 | yes | complemented |
| 107 | C22 | C22×#121 | 1 | 7 | yes | — |
| 108 | C22 | C22×#127 | 1 | 7 | yes | — |
| 109 | C22 | C2×#532 | 1 | 21 | yes | — |
| 110 | C4 | — | 2 | 28 | no | — |
| 111 | C4 | — | 8 | 8 | no | — |
| 112 | C4 | #2856 | 1 | 56 | yes | complemented |
| 113 | C2 | — | 2 | 8 | no | — |
| 114 | C2 | — | 16 | 8 | no | — |
| 115 | C2 | C2×#2856 | 1 | 7 | yes | — |
| 116 | C2 | C2×#2867 | 1 | 1 | yes | characteristic |
| 117 | C2 | #36644 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2867 | C2 | semidirect |
| 2. C2×#2856 | C2 | semidirect |
| 3. C22×C43 | C2 | semidirect |
| 4. #14713 | C2 | semidirect |
| 5. #14692 | C2 | semidirect |
| 27. C22×C42 | D4 | semidirect |
| 33. C43 | D4 | semidirect |
| 73. C22×C4 | #127 | semidirect |
| 81. C42 | #121 | semidirect |
| 106. C22 | #2867 | semidirect |
| 112. C4 | #2856 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2867 | 2 |
| C2×#2856 | 2 |
| C22×C43 | 2 |
| #14713 | 2 |
| #14692 | 2 |