Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
306 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 62 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 | #20765 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11513 | C2 | 1 | 2 | yes | maximal |
| 3 | #7366 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | C2×C4×M16 | C4 | 1 | 1 | yes | characteristic |
| 5 | #2285 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×#382 | C4 | 1 | 1 | yes | characteristic |
| 7 | #1836 | — | 2 | 2 | no | — |
| 8 | #1835 | C4 | 1 | 2 | yes | — |
| 9 | #1835 | C22 | 1 | 2 | yes | — |
| 10 | #1736 | — | 2 | 2 | no | — |
| 11 | #1464 | — | 2 | 2 | no | — |
| 12 | #1325 | — | 2 | 2 | no | — |
| 13 | #1324 | C22 | 1 | 4 | yes | — |
| 14 | C22×M16 | — | 2 | 1 | no | — |
| 15 | C2×#117 | D4 | 1 | 2 | yes | — |
| 16 | C22×C4⋊C4 | — | 2 | 1 | no | — |
| 17 | C22×C42 | C2×C4 | 1 | 1 | yes | characteristic |
| 18 | #432 | — | 4 | 2 | no | — |
| 19 | C4×M16 | — | 2 | 4 | no | — |
| 20 | C2×#97 | — | 2 | 1 | no | — |
| 21 | C2×C4×C8 | C2×C4 | 1 | 2 | yes | — |
| 22 | #384 | D4 | 1 | 2 | yes | — |
| 23 | #382 | — | 2 | 1 | no | — |
| 24 | #382 | — | 4 | 1 | no | — |
| 25 | #382 | C2×C4 | 1 | 2 | yes | — |
| 26 | #382 | C2×C4 | 1 | 1 | yes | characteristic |
| 27 | #382 | C23 | 1 | 1 | yes | characteristic |
| 28 | #380 | — | 4 | 1 | no | — |
| 29 | C4×C4⋊C4 | — | 4 | 1 | no | — |
| 30 | C2×#95 | — | 4 | 1 | no | — |
| 31 | #340 | — | 2 | 4 | no | — |
| 32 | #340 | — | 2 | 4 | no | — |
| 33 | #324 | — | 2 | 4 | no | — |
| 34 | #323 | — | 4 | 2 | no | — |
| 35 | C23×C4 | — | 2 | 1 | no | — |
| 36 | C23×C4 | — | 4 | 1 | no | — |
| 37 | C23×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 38 | C2×M16 | — | 2 | 4 | no | — |
| 39 | C2×M16 | — | 4 | 2 | no | — |
| 40 | C2×M16 | — | 4 | 1 | no | — |
| 41 | C2×M16 | — | 4 | 1 | no | — |
| 42 | C22×C8 | — | 2 | 2 | no | — |
| 43 | #117 | — | 4 | 2 | no | — |
| 44 | #117 | — | 4 | 2 | no | — |
| 45 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 46 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 47 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 48 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 49 | C2×C4⋊C4 | C22⋊C4 | 1 | 2 | yes | — |
| 50 | C2×C4⋊C4 | C2×D4 | 1 | 2 | yes | — |
| 51 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 52 | C2×C42 | — | 2 | 1 | no | — |
| 53 | C2×C42 | — | 2 | 2 | no | — |
| 54 | C2×C42 | — | 2 | 2 | no | — |
| 55 | C2×C42 | — | 2 | 2 | no | — |
| 56 | C2×C42 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 57 | C2×C42 | C22×C4 | 1 | 1 | yes | characteristic |
| 58 | C4⋊C8 | — | 8 | 2 | no | — |
| 59 | C22⋊C8 | — | 4 | 4 | no | — |
| 60 | #97 | — | 2 | 4 | no | — |
| 61 | #97 | — | 8 | 2 | no | — |
| 62 | C4×C8 | — | 2 | 2 | no | — |
| 63 | C4×C8 | M16 | 1 | 4 | yes | — |
| 64 | #95 | — | 4 | 1 | no | — |
| 65 | #95 | — | 4 | 1 | no | — |
| 66 | #95 | — | 4 | 2 | no | — |
| 67 | #95 | — | 4 | 2 | no | — |
| 68 | C24 | #99 | 1 | 1 | yes | characteristic |
| 69 | C22×C4 | — | 2 | 2 | no | — |
| 70 | C22×C4 | — | 2 | 1 | no | — |
| 71 | C22×C4 | — | 2 | 1 | no | — |
| 72 | C22×C4 | — | 2 | 1 | no | — |
| 73 | C22×C4 | — | 2 | 1 | no | — |
| 74 | C22×C4 | — | 2 | 1 | no | — |
| 75 | C22×C4 | — | 2 | 1 | no | — |
| 76 | C22×C4 | — | 2 | 4 | no | — |
| 77 | C22×C4 | — | 2 | 2 | no | — |
| 78 | C22×C4 | — | 2 | 2 | no | — |
| 79 | C22×C4 | — | 4 | 2 | no | — |
| 80 | C22×C4 | — | 8 | 2 | no | — |
| 81 | C22×C4 | — | 8 | 4 | no | — |
| 82 | C22×C4 | — | 8 | 2 | no | — |
| 83 | C22×C4 | #99 | 1 | 1 | yes | characteristic |
| 84 | C22×C4 | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 85 | M16 | — | 4 | 4 | no | — |
| 86 | M16 | — | 4 | 4 | no | — |
| 87 | C2×C8 | — | 2 | 4 | no | — |
| 88 | C2×C8 | — | 4 | 2 | no | — |
| 89 | C2×C8 | — | 4 | 2 | no | — |
| 90 | C2×C8 | — | 8 | 4 | no | — |
| 91 | C4⋊C4 | — | 4 | 2 | no | — |
| 92 | C4⋊C4 | — | 4 | 2 | no | — |
| 93 | C4⋊C4 | — | 4 | 4 | no | — |
| 94 | C4⋊C4 | — | 8 | 2 | no | — |
| 95 | C4⋊C4 | — | 8 | 4 | no | — |
| 96 | C22⋊C4 | — | 8 | 2 | no | — |
| 97 | C42 | — | 2 | 4 | no | — |
| 98 | C42 | — | 2 | 1 | no | — |
| 99 | C42 | — | 2 | 1 | no | — |
| 100 | C42 | — | 4 | 2 | no | — |
| 101 | C42 | — | 4 | 2 | no | — |
| 102 | C42 | #101 | 1 | 2 | yes | — |
| 103 | C42 | C2×M16 | 1 | 2 | yes | — |
| 104 | C23 | — | 2 | 1 | no | — |
| 105 | C23 | — | 2 | 1 | no | — |
| 106 | C23 | — | 4 | 2 | no | — |
| 107 | C23 | #355 | 1 | 2 | yes | — |
| 108 | C23 | C2×#99 | 1 | 1 | yes | characteristic |
| 109 | C2×C4 | — | 2 | 2 | no | — |
| 110 | C2×C4 | — | 2 | 2 | no | — |
| 111 | C2×C4 | — | 2 | 2 | no | — |
| 112 | C2×C4 | — | 2 | 2 | no | — |
| 113 | C2×C4 | — | 2 | 1 | no | — |
| 114 | C2×C4 | — | 2 | 1 | no | — |
| 115 | C2×C4 | — | 2 | 1 | no | — |
| 116 | C2×C4 | — | 2 | 1 | no | — |
| 117 | C2×C4 | — | 2 | 2 | no | — |
| 118 | C2×C4 | — | 2 | 2 | no | — |
| 119 | C2×C4 | — | 4 | 2 | no | — |
| 120 | C2×C4 | — | 4 | 2 | no | — |
| 121 | C2×C4 | — | 4 | 2 | no | — |
| 122 | C2×C4 | — | 4 | 2 | no | — |
| 123 | C2×C4 | — | 8 | 4 | no | — |
| 124 | C2×C4 | — | 8 | 2 | no | — |
| 125 | C2×C4 | — | 8 | 2 | no | — |
| 126 | C2×C4 | — | 8 | 4 | no | — |
| 127 | C2×C4 | — | 8 | 2 | no | — |
| 128 | C2×C4 | — | 16 | 4 | no | — |
| 129 | C2×C4 | #354 | 1 | 1 | yes | characteristic |
| 130 | C2×C4 | #354 | 1 | 1 | yes | characteristic |
| 131 | C2×C4 | #407 | 1 | 1 | yes | characteristic |
| 132 | C2×C4 | C2×#101 | 1 | 1 | yes | characteristic |
| 133 | C8 | — | 4 | 4 | no | — |
| 134 | C8 | — | 16 | 4 | no | — |
| 135 | C22 | — | 2 | 2 | no | — |
| 136 | C22 | — | 2 | 2 | no | — |
| 137 | C22 | — | 4 | 2 | no | — |
| 138 | C22 | — | 4 | 2 | no | — |
| 139 | C22 | — | 4 | 2 | no | — |
| 140 | C22 | #1463 | 1 | 1 | yes | characteristic |
| 141 | C22 | C2×#354 | 1 | 1 | yes | characteristic |
| 142 | C22 | C2×#355 | 1 | 1 | yes | characteristic |
| 143 | C4 | — | 2 | 1 | no | — |
| 144 | C4 | — | 2 | 1 | no | — |
| 145 | C4 | — | 2 | 1 | no | — |
| 146 | C4 | — | 2 | 1 | no | — |
| 147 | C4 | — | 4 | 2 | no | — |
| 148 | C4 | — | 4 | 2 | no | — |
| 149 | C4 | — | 8 | 2 | no | — |
| 150 | C4 | — | 8 | 2 | no | — |
| 151 | C4 | — | 16 | 2 | no | — |
| 152 | C2 | — | 2 | 1 | no | — |
| 153 | C2 | — | 2 | 1 | no | — |
| 154 | C2 | — | 4 | 2 | no | — |
| 155 | C2 | #11511 | 1 | 1 | yes | characteristic |
| 156 | C2 | #13604 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #382 | C23 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #7366 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20765 | 2 |
| #11513 | 2 |
| #7366 | 2 |