Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2038 conjugacy classes of subgroups; only the 426 normal subgroups are shown.
This group has 2038 conjugacy classes of subgroups — too many to draw. Only its 426 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 | #34641 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#1980 | C2 | 1 | 12 | yes | maximal, complemented |
| 3 | #20329 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #13856 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C2×#532 | C22 | 1 | 6 | yes | — |
| 6 | C22×#104 | C22 | 1 | 16 | yes | — |
| 7 | C2×C4×M16 | C22 | 1 | 6 | yes | — |
| 8 | #2658 | C4 | 1 | 4 | yes | complemented |
| 9 | #2657 | C4 | 1 | 4 | yes | complemented |
| 10 | #2656 | — | 2 | 4 | no | — |
| 11 | #2397 | — | 2 | 4 | no | — |
| 12 | C2×C4×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 13 | C2×#376 | C22 | 1 | 1 | yes | characteristic |
| 14 | #1980 | — | 2 | 48 | no | — |
| 15 | #1726 | — | 2 | 4 | no | — |
| 16 | #1444 | — | 2 | 4 | no | — |
| 17 | C22×#41 | C23 | 1 | 4 | yes | — |
| 18 | C22×M16 | C23 | 1 | 4 | yes | — |
| 19 | #532 | — | 2 | 24 | no | — |
| 20 | #532 | — | 2 | 24 | no | — |
| 21 | C2×C4⋊Q8 | C2×C4 | 1 | 6 | yes | — |
| 22 | C2×#127 | C2×C4 | 1 | 6 | yes | — |
| 23 | C2×#124 | C2×C4 | 1 | 12 | yes | — |
| 24 | C2×#121 | — | 2 | 12 | no | — |
| 25 | C2×C4×D4 | — | 2 | 12 | no | — |
| 26 | C22×C4⋊C4 | — | 2 | 6 | no | — |
| 27 | C22×C42 | C23 | 1 | 4 | yes | — |
| 28 | C22×C42 | C23 | 1 | 3 | yes | — |
| 29 | C2×#104 | — | 2 | 96 | no | — |
| 30 | C4×M16 | — | 2 | 12 | no | — |
| 31 | C4×M16 | D4 | 1 | 24 | yes | complemented |
| 32 | C2×#97 | — | 2 | 6 | no | — |
| 33 | C2×C4×C8 | — | 2 | 6 | no | — |
| 34 | #399 | — | 2 | 16 | no | — |
| 35 | #392 | — | 2 | 16 | no | — |
| 36 | C4×C4⋊C4 | — | 2 | 24 | no | — |
| 37 | #376 | C2×C4 | 1 | 4 | yes | — |
| 38 | #376 | D4 | 1 | 4 | yes | — |
| 39 | C2×#95 | — | 2 | 6 | no | — |
| 40 | C2×#95 | D4 | 1 | 4 | yes | — |
| 41 | C2×#41 | — | 2 | 24 | no | — |
| 42 | C2×#41 | — | 2 | 24 | no | — |
| 43 | C22×Q8 | C22×C4 | 1 | 4 | yes | — |
| 44 | C22×D4 | — | 2 | 4 | no | — |
| 45 | C22×D4 | — | 2 | 12 | no | — |
| 46 | C22×D4 | C22×C4 | 1 | 4 | yes | — |
| 47 | C23×C4 | — | 2 | 4 | no | — |
| 48 | C23×C4 | — | 2 | 4 | no | — |
| 49 | C23×C4 | C2×D4 | 1 | 6 | yes | — |
| 50 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 51 | C2×M16 | — | 2 | 12 | no | — |
| 52 | C2×M16 | C2×D4 | 1 | 24 | yes | — |
| 53 | C22×C8 | — | 2 | 4 | no | — |
| 54 | C4⋊Q8 | — | 2 | 24 | no | — |
| 55 | #127 | — | 2 | 24 | no | — |
| 56 | #124 | — | 2 | 48 | no | — |
| 57 | #121 | — | 4 | 24 | no | — |
| 58 | #121 | — | 4 | 24 | no | — |
| 59 | C4×D4 | — | 4 | 24 | no | — |
| 60 | C4×D4 | — | 4 | 24 | no | — |
| 61 | C2×C4⋊C4 | — | 2 | 12 | no | — |
| 62 | C2×C4⋊C4 | — | 4 | 24 | no | — |
| 63 | C2×C4⋊C4 | — | 4 | 12 | no | — |
| 64 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 65 | C2×C42 | — | 2 | 24 | no | — |
| 66 | C2×C42 | C22×C4 | 1 | 6 | yes | — |
| 67 | C2×C42 | C2×D4 | 1 | 12 | yes | — |
| 68 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 69 | C2×C42 | #41 | 1 | 12 | yes | — |
| 70 | #104 | — | 4 | 64 | no | — |
| 71 | #97 | — | 2 | 24 | no | — |
| 72 | C4×C8 | — | 2 | 24 | no | — |
| 73 | #95 | — | 4 | 12 | no | — |
| 74 | #95 | C22⋊C4 | 1 | 16 | yes | — |
| 75 | C24 | — | 2 | 4 | no | — |
| 76 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 77 | #41 | — | 4 | 16 | no | — |
| 78 | #41 | — | 4 | 48 | no | — |
| 79 | C2×Q8 | — | 2 | 24 | no | — |
| 80 | C2×D4 | — | 2 | 24 | no | — |
| 81 | C2×D4 | — | 4 | 12 | no | — |
| 82 | C2×D4 | — | 4 | 12 | no | — |
| 83 | C2×D4 | — | 4 | 24 | no | — |
| 84 | C2×D4 | — | 8 | 24 | no | — |
| 85 | C22×C4 | — | 2 | 8 | no | — |
| 86 | C22×C4 | — | 2 | 4 | no | — |
| 87 | C22×C4 | — | 4 | 12 | no | — |
| 88 | C22×C4 | — | 4 | 12 | no | — |
| 89 | C22×C4 | — | 4 | 12 | no | — |
| 90 | C22×C4 | — | 4 | 24 | no | — |
| 91 | C22×C4 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 92 | C22×C4 | #124 | 1 | 12 | yes | — |
| 93 | C22×C4 | #127 | 1 | 12 | yes | — |
| 94 | C22×C4 | C23×C4 | 1 | 1 | yes | characteristic |
| 95 | C22×C4 | C22×D4 | 1 | 6 | yes | — |
| 96 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 97 | C22×C4 | C2×#41 | 1 | 6 | yes | — |
| 98 | M16 | — | 2 | 24 | no | — |
| 99 | M16 | #127 | 1 | 16 | yes | — |
| 100 | C2×C8 | — | 2 | 24 | no | — |
| 101 | C4⋊C4 | — | 4 | 48 | no | — |
| 102 | C4⋊C4 | — | 4 | 24 | no | — |
| 103 | C22⋊C4 | — | 4 | 48 | no | — |
| 104 | C42 | — | 4 | 16 | no | — |
| 105 | C42 | C4×D4 | 1 | 24 | yes | — |
| 106 | C42 | #121 | 1 | 12 | yes | — |
| 107 | C42 | #121 | 1 | 12 | yes | — |
| 108 | C23 | — | 2 | 4 | no | — |
| 109 | C23 | — | 4 | 16 | no | — |
| 110 | C23 | — | 4 | 12 | no | — |
| 111 | C23 | C22×C22⋊C4 | 1 | 1 | yes | characteristic |
| 112 | C23 | C2×#124 | 1 | 3 | yes | — |
| 113 | C23 | C2×#127 | 1 | 3 | yes | — |
| 114 | Q8 | — | 4 | 16 | no | — |
| 115 | D4 | — | 4 | 16 | no | — |
| 116 | D4 | — | 8 | 48 | no | — |
| 117 | D4 | — | 8 | 12 | no | — |
| 118 | D4 | — | 8 | 4 | no | — |
| 119 | C2×C4 | — | 2 | 24 | no | — |
| 120 | C2×C4 | — | 4 | 12 | no | — |
| 121 | C2×C4 | — | 4 | 12 | no | — |
| 122 | C2×C4 | — | 4 | 24 | no | — |
| 123 | C2×C4 | — | 4 | 12 | no | — |
| 124 | C2×C4 | — | 8 | 12 | no | — |
| 125 | C2×C4 | — | 8 | 4 | no | — |
| 126 | C2×C4 | — | 8 | 16 | no | — |
| 127 | C2×C4 | #390 | 1 | 24 | yes | — |
| 128 | C2×C4 | C2×C4×D4 | 1 | 6 | yes | — |
| 129 | C2×C4 | C2×#121 | 1 | 6 | yes | — |
| 130 | C2×C4 | C2×#127 | 1 | 4 | yes | — |
| 131 | C2×C4 | #532 | 1 | 12 | yes | — |
| 132 | C8 | — | 2 | 16 | no | — |
| 133 | C22 | — | 2 | 6 | no | — |
| 134 | C22 | — | 2 | 6 | no | — |
| 135 | C22 | — | 8 | 24 | no | — |
| 136 | C22 | — | 8 | 4 | no | — |
| 137 | C22 | C2×#390 | 1 | 3 | yes | — |
| 138 | C22 | #2858 | 1 | 3 | yes | — |
| 139 | C22 | #2867 | 1 | 1 | yes | characteristic |
| 140 | C22 | #2901 | 1 | 4 | yes | — |
| 141 | C4 | — | 2 | 24 | no | — |
| 142 | C4 | — | 8 | 8 | no | — |
| 143 | C4 | — | 8 | 4 | no | — |
| 144 | C4 | C4×#127 | 1 | 4 | yes | — |
| 145 | C4 | #2856 | 1 | 4 | yes | — |
| 146 | C2 | — | 2 | 1 | no | — |
| 147 | C2 | — | 2 | 3 | no | — |
| 148 | C2 | — | 16 | 4 | no | — |
| 149 | C2 | #14713 | 1 | 1 | yes | characteristic |
| 150 | C2 | #21253 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #34641 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#1980 | C2 | semidirect |
| 3. #20329 | C2 | semidirect |
| 4. #13856 | C2 | semidirect |
| 8. #2658 | C4 | semidirect |
| 9. #2657 | C4 | semidirect |
| 31. C4×M16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #34641 | 2 |
| C2×#1980 | 2 |
| #20329 | 2 |
| #13856 | 2 |