Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
643 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 643 conjugacy classes of subgroups — too many to draw. Only its 142 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 | #52305 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #29225 | C2 | 1 | 3 | yes | maximal |
| 3 | #7832 | C2 | 1 | 2 | yes | maximal |
| 4 | #7827 | C2 | 1 | 6 | yes | maximal |
| 5 | #7636 | C2 | 1 | 3 | yes | maximal |
| 6 | #2868 | C22 | 1 | 1 | yes | characteristic |
| 7 | #2810 | C22 | 1 | 1 | yes | characteristic |
| 8 | #2767 | — | 2 | 3 | no | — |
| 9 | #2726 | C22 | 1 | 3 | yes | — |
| 10 | #2719 | — | 2 | 3 | no | — |
| 11 | #2402 | — | 2 | 3 | no | — |
| 12 | C4×#125 | — | 2 | 3 | no | — |
| 13 | C4×#125 | C22 | 1 | 3 | yes | — |
| 14 | #2100 | C22 | 1 | 6 | yes | — |
| 15 | #2100 | C22 | 1 | 6 | yes | — |
| 16 | #1992 | — | 2 | 3 | no | — |
| 17 | #1988 | — | 2 | 3 | no | — |
| 18 | #1988 | — | 2 | 3 | no | — |
| 19 | #1836 | — | 2 | 3 | no | — |
| 20 | #1767 | C22 | 1 | 3 | yes | — |
| 21 | #1271 | C22 | 1 | 6 | yes | — |
| 22 | #1271 | C22 | 1 | 6 | yes | — |
| 23 | C2×C4⋊Q8 | — | 2 | 3 | no | — |
| 24 | C2×#125 | — | 2 | 3 | no | — |
| 25 | C2×#125 | D4 | 1 | 3 | yes | — |
| 26 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 27 | C2×#97 | — | 2 | 3 | no | — |
| 28 | #400 | — | 2 | 3 | no | — |
| 29 | #400 | — | 2 | 3 | no | — |
| 30 | #400 | — | 2 | 3 | no | — |
| 31 | #400 | — | 2 | 3 | no | — |
| 32 | #398 | — | 2 | 3 | no | — |
| 33 | #398 | — | 2 | 3 | no | — |
| 34 | #398 | — | 2 | 3 | no | — |
| 35 | #398 | — | 2 | 3 | no | — |
| 36 | #389 | — | 2 | 3 | no | — |
| 37 | #389 | — | 2 | 3 | no | — |
| 38 | #389 | C23 | 1 | 3 | yes | — |
| 39 | #389 | C23 | 1 | 3 | yes | — |
| 40 | #387 | — | 2 | 3 | no | — |
| 41 | #387 | — | 2 | 3 | no | — |
| 42 | #387 | — | 2 | 3 | no | — |
| 43 | #387 | — | 2 | 3 | no | — |
| 44 | #387 | — | 2 | 3 | no | — |
| 45 | #387 | — | 2 | 3 | no | — |
| 46 | #384 | — | 2 | 3 | no | — |
| 47 | #384 | C23 | 1 | 2 | yes | — |
| 48 | #382 | — | 2 | 3 | no | — |
| 49 | #382 | D4 | 1 | 3 | yes | — |
| 50 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 51 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 52 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 53 | C4×C4⋊C4 | — | 2 | 3 | no | — |
| 54 | C4×C4⋊C4 | D4 | 1 | 3 | yes | — |
| 55 | C4×C4⋊C4 | D4 | 1 | 3 | yes | — |
| 56 | #376 | — | 2 | 1 | no | — |
| 57 | C43 | — | 2 | 1 | no | — |
| 58 | C43 | C23 | 1 | 1 | yes | characteristic |
| 59 | #340 | — | 2 | 6 | no | — |
| 60 | #340 | — | 2 | 6 | no | — |
| 61 | #340 | — | 2 | 6 | no | — |
| 62 | #340 | — | 2 | 6 | no | — |
| 63 | #336 | C23 | 1 | 6 | yes | — |
| 64 | C22×Q8 | — | 4 | 1 | no | — |
| 65 | C22×C8 | — | 2 | 6 | no | — |
| 66 | C4⋊Q8 | — | 4 | 12 | no | — |
| 67 | #125 | — | 2 | 6 | no | — |
| 68 | #125 | — | 4 | 3 | no | — |
| 69 | #125 | — | 4 | 6 | no | — |
| 70 | #125 | — | 4 | 6 | no | — |
| 71 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 72 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 73 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 74 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 75 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 76 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 77 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 78 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 79 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 80 | C2×C4⋊C4 | — | 4 | 3 | no | — |
| 81 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 82 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 83 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 84 | C2×C4⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 85 | C2×C42 | — | 2 | 3 | no | — |
| 86 | C2×C42 | — | 2 | 3 | no | — |
| 87 | C2×C42 | — | 2 | 3 | no | — |
| 88 | C2×C42 | — | 2 | 1 | no | — |
| 89 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 90 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 91 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 92 | C4⋊C8 | — | 4 | 6 | no | — |
| 93 | #97 | — | 4 | 6 | no | — |
| 94 | #95 | — | 2 | 1 | no | — |
| 95 | #95 | — | 2 | 3 | no | — |
| 96 | #95 | — | 2 | 3 | no | — |
| 97 | #95 | — | 2 | 3 | no | — |
| 98 | #95 | — | 2 | 3 | no | — |
| 99 | #95 | — | 2 | 3 | no | — |
| 100 | #95 | — | 2 | 3 | no | — |
| 101 | #95 | — | 2 | 3 | no | — |
| 102 | #95 | — | 2 | 3 | no | — |
| 103 | #95 | — | 2 | 3 | no | — |
| 104 | #95 | — | 2 | 3 | no | — |
| 105 | #95 | — | 2 | 3 | no | — |
| 106 | C2×Q8 | — | 8 | 6 | no | — |
| 107 | C22×C4 | — | 2 | 3 | no | — |
| 108 | C22×C4 | — | 2 | 3 | no | — |
| 109 | C22×C4 | — | 2 | 3 | no | — |
| 110 | C22×C4 | — | 2 | 3 | no | — |
| 111 | C22×C4 | — | 4 | 2 | no | — |
| 112 | C22×C4 | #120 | 1 | 3 | yes | — |
| 113 | C22×C4 | #120 | 1 | 1 | yes | characteristic |
| 114 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 115 | C2×C8 | — | 4 | 6 | no | — |
| 116 | C2×C8 | — | 4 | 6 | no | — |
| 117 | C2×C8 | — | 4 | 6 | no | — |
| 118 | C4⋊C4 | — | 2 | 3 | no | — |
| 119 | C4⋊C4 | — | 2 | 3 | no | — |
| 120 | C4⋊C4 | — | 2 | 3 | no | — |
| 121 | C4⋊C4 | — | 4 | 6 | no | — |
| 122 | C4⋊C4 | — | 4 | 6 | no | — |
| 123 | C4⋊C4 | — | 4 | 3 | no | — |
| 124 | C4⋊C4 | — | 4 | 3 | no | — |
| 125 | C4⋊C4 | — | 4 | 6 | no | — |
| 126 | C4⋊C4 | — | 4 | 6 | no | — |
| 127 | C4⋊C4 | — | 4 | 6 | no | — |
| 128 | C4⋊C4 | — | 4 | 12 | no | — |
| 129 | C4⋊C4 | — | 4 | 3 | no | — |
| 130 | C4⋊C4 | — | 4 | 3 | no | — |
| 131 | C4⋊C4 | — | 4 | 3 | no | — |
| 132 | C4⋊C4 | — | 4 | 3 | no | — |
| 133 | C4⋊C4 | — | 8 | 6 | no | — |
| 134 | C4⋊C4 | — | 8 | 3 | no | — |
| 135 | C4⋊C4 | #137 | 1 | 6 | yes | — |
| 136 | C42 | — | 2 | 3 | no | — |
| 137 | C42 | — | 2 | 3 | no | — |
| 138 | C42 | — | 2 | 6 | no | — |
| 139 | C42 | — | 4 | 1 | no | — |
| 140 | C42 | — | 4 | 6 | no | — |
| 141 | C42 | — | 4 | 6 | no | — |
| 142 | C42 | — | 4 | 6 | no | — |
| 143 | C42 | — | 8 | 1 | no | — |
| 144 | C23 | C2×#120 | 1 | 1 | yes | characteristic |
| 145 | Q8 | — | 16 | 4 | no | — |
| 146 | C2×C4 | — | 2 | 3 | no | — |
| 147 | C2×C4 | — | 2 | 3 | no | — |
| 148 | C2×C4 | — | 2 | 3 | no | — |
| 149 | C2×C4 | — | 2 | 3 | no | — |
| 150 | C2×C4 | — | 2 | 6 | no | — |
| 151 | C2×C4 | — | 2 | 3 | no | — |
| 152 | C2×C4 | — | 4 | 3 | no | — |
| 153 | C2×C4 | — | 4 | 3 | no | — |
| 154 | C2×C4 | — | 4 | 3 | no | — |
| 155 | C2×C4 | — | 4 | 3 | no | — |
| 156 | C2×C4 | — | 4 | 3 | no | — |
| 157 | C2×C4 | — | 4 | 3 | no | — |
| 158 | C2×C4 | — | 4 | 3 | no | — |
| 159 | C2×C4 | — | 4 | 3 | no | — |
| 160 | C2×C4 | — | 4 | 3 | no | — |
| 161 | C2×C4 | — | 4 | 3 | no | — |
| 162 | C2×C4 | — | 4 | 3 | no | — |
| 163 | C2×C4 | — | 8 | 6 | no | — |
| 164 | C2×C4 | #454 | 1 | 6 | yes | — |
| 165 | C2×C4 | C2×#137 | 1 | 3 | yes | — |
| 166 | C2×C4 | #575 | 1 | 3 | yes | — |
| 167 | C8 | — | 8 | 6 | no | — |
| 168 | C22 | #3007 | 1 | 3 | yes | — |
| 169 | C22 | C2×#454 | 1 | 3 | yes | — |
| 170 | C22 | #3028 | 1 | 1 | yes | characteristic |
| 171 | C4 | — | 2 | 3 | no | — |
| 172 | C4 | — | 2 | 3 | no | — |
| 173 | C4 | — | 4 | 3 | no | — |
| 174 | C4 | — | 4 | 6 | no | — |
| 175 | C4 | — | 8 | 3 | no | — |
| 176 | C4 | — | 8 | 3 | no | — |
| 177 | C4 | — | 8 | 3 | no | — |
| 178 | C4 | — | 16 | 2 | no | — |
| 179 | C4 | #1658 | 1 | 4 | yes | — |
| 180 | C2 | #18345 | 1 | 3 | yes | — |
| 181 | C2 | #18538 | 1 | 3 | yes | — |
| 182 | C2 | C2×#1658 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×#120 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #52305 | 2 |
| #29225 | 2 |
| #7832 | 2 |
| #7827 | 2 |
| #7636 | 2 |