Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
406 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 126 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 | #21737 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #21677 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #10688 | C2 | 1 | 2 | yes | maximal |
| 4 | #10630 | C2 | 1 | 2 | yes | maximal |
| 5 | #10548 | C2 | 1 | 2 | yes | maximal |
| 6 | #10373 | C2 | 1 | 2 | yes | maximal |
| 7 | #8254 | C2 | 1 | 4 | yes | maximal |
| 8 | #8139 | C2 | 1 | 1 | yes | maximal, characteristic |
| 9 | #3162 | C22 | 1 | 1 | yes | characteristic |
| 10 | #2960 | C22 | 1 | 2 | yes | — |
| 11 | #2402 | C22 | 1 | 1 | yes | characteristic |
| 12 | #1995 | C22 | 1 | 2 | yes | — |
| 13 | #1945 | — | 2 | 2 | no | — |
| 14 | #1940 | C22 | 1 | 2 | yes | — |
| 15 | #1917 | — | 2 | 2 | no | — |
| 16 | #1849 | C22 | 1 | 1 | yes | characteristic |
| 17 | #1832 | C22 | 1 | 1 | yes | characteristic |
| 18 | #1773 | — | 2 | 2 | no | — |
| 19 | #1771 | — | 2 | 2 | no | — |
| 20 | #1770 | C22 | 1 | 2 | yes | — |
| 21 | #1726 | C22 | 1 | 1 | yes | characteristic |
| 22 | #1542 | C22 | 1 | 4 | yes | — |
| 23 | #1503 | C22 | 1 | 4 | yes | — |
| 24 | #1452 | C22 | 1 | 2 | yes | — |
| 25 | #1303 | C22 | 1 | 4 | yes | — |
| 26 | #1301 | C22 | 1 | 4 | yes | — |
| 27 | #1275 | C22 | 1 | 2 | yes | — |
| 28 | #1274 | C22 | 1 | 2 | yes | — |
| 29 | #533 | C23 | 1 | 2 | yes | — |
| 30 | #501 | — | 2 | 4 | no | — |
| 31 | #446 | — | 2 | 4 | no | — |
| 32 | C8×Q8 | — | 2 | 4 | no | — |
| 33 | #431 | C23 | 1 | 4 | yes | — |
| 34 | C2×#108 | — | 2 | 2 | no | — |
| 35 | #427 | — | 2 | 2 | no | — |
| 36 | #424 | — | 2 | 2 | no | — |
| 37 | #423 | C23 | 1 | 2 | yes | — |
| 38 | C2×C4⋊C8 | — | 2 | 2 | no | — |
| 39 | C2×C4⋊C8 | C23 | 1 | 2 | yes | — |
| 40 | #405 | — | 2 | 2 | no | — |
| 41 | C2×#97 | — | 2 | 2 | no | — |
| 42 | C2×#97 | D4 | 1 | 2 | yes | — |
| 43 | C2×#97 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 45 | C2×C4×C8 | C23 | 1 | 1 | yes | characteristic |
| 46 | #400 | — | 2 | 2 | no | — |
| 47 | #398 | — | 2 | 2 | no | — |
| 48 | C4×C4⋊C4 | — | 2 | 2 | no | — |
| 49 | C4×C4⋊C4 | C23 | 1 | 2 | yes | — |
| 50 | #376 | C23 | 1 | 1 | yes | characteristic |
| 51 | #339 | — | 2 | 4 | no | — |
| 52 | #336 | — | 2 | 2 | no | — |
| 53 | #336 | — | 2 | 2 | no | — |
| 54 | #336 | — | 2 | 2 | no | — |
| 55 | #333 | — | 2 | 4 | no | — |
| 56 | #332 | — | 2 | 4 | no | — |
| 57 | #328 | — | 2 | 4 | no | — |
| 58 | #328 | — | 2 | 4 | no | — |
| 59 | #322 | — | 2 | 4 | no | — |
| 60 | C2×M16 | — | 2 | 2 | no | — |
| 61 | C22×C8 | — | 2 | 2 | no | — |
| 62 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 63 | C22×C8 | C2×D4 | 1 | 1 | yes | characteristic |
| 64 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 65 | C4⋊Q8 | #41 | 1 | 4 | yes | — |
| 66 | #125 | — | 2 | 2 | no | — |
| 67 | C22⋊Q8 | — | 2 | 4 | no | — |
| 68 | C4×Q8 | — | 2 | 4 | no | — |
| 69 | #117 | — | 2 | 2 | no | — |
| 70 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 71 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 72 | C2×C4⋊C4 | — | 2 | 2 | no | — |
| 73 | C2×C42 | — | 2 | 2 | no | — |
| 74 | C2×C42 | C2×D4 | 1 | 2 | yes | — |
| 75 | C2×C42 | C24 | 1 | 1 | yes | characteristic |
| 76 | #108 | — | 4 | 4 | no | — |
| 77 | #107 | — | 2 | 4 | no | — |
| 78 | #106 | — | 2 | 4 | no | — |
| 79 | C4⋊C8 | — | 2 | 4 | no | — |
| 80 | C4⋊C8 | — | 2 | 4 | no | — |
| 81 | C4⋊C8 | — | 2 | 4 | no | — |
| 82 | C4⋊C8 | — | 4 | 4 | no | — |
| 83 | C4⋊C8 | — | 4 | 4 | no | — |
| 84 | C22⋊C8 | — | 2 | 4 | no | — |
| 85 | #97 | — | 2 | 4 | no | — |
| 86 | #97 | — | 2 | 2 | no | — |
| 87 | #97 | — | 4 | 4 | no | — |
| 88 | #97 | — | 4 | 2 | no | — |
| 89 | C4×C8 | — | 2 | 4 | no | — |
| 90 | C4×C8 | — | 4 | 2 | no | — |
| 91 | C4×C8 | — | 4 | 2 | no | — |
| 92 | C4×C8 | #41 | 1 | 4 | yes | — |
| 93 | #95 | — | 2 | 2 | no | — |
| 94 | #95 | — | 2 | 2 | no | — |
| 95 | #95 | — | 2 | 2 | no | — |
| 96 | #95 | #41 | 1 | 2 | yes | — |
| 97 | #95 | #41 | 1 | 2 | yes | — |
| 98 | C2×Q8 | — | 2 | 4 | no | — |
| 99 | C22×C4 | — | 2 | 2 | no | — |
| 100 | C22×C4 | — | 2 | 2 | no | — |
| 101 | C22×C4 | — | 2 | 2 | no | — |
| 102 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 103 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 104 | M16 | — | 4 | 4 | no | — |
| 105 | C2×C8 | — | 2 | 2 | no | — |
| 106 | C2×C8 | — | 2 | 2 | no | — |
| 107 | C2×C8 | — | 2 | 2 | no | — |
| 108 | C2×C8 | — | 2 | 2 | no | — |
| 109 | C2×C8 | — | 4 | 2 | no | — |
| 110 | C2×C8 | — | 4 | 2 | no | — |
| 111 | C2×C8 | — | 4 | 4 | no | — |
| 112 | C2×C8 | — | 4 | 2 | no | — |
| 113 | C2×C8 | — | 4 | 2 | no | — |
| 114 | C2×C8 | #124 | 1 | 2 | yes | — |
| 115 | C2×C8 | #124 | 1 | 2 | yes | — |
| 116 | C2×C8 | #137 | 1 | 4 | yes | — |
| 117 | C4⋊C4 | — | 2 | 4 | no | — |
| 118 | C4⋊C4 | — | 2 | 4 | no | — |
| 119 | C4⋊C4 | — | 4 | 4 | no | — |
| 120 | C4⋊C4 | — | 8 | 2 | no | — |
| 121 | C4⋊C4 | — | 8 | 2 | no | — |
| 122 | C4⋊C4 | — | 8 | 2 | no | — |
| 123 | C22⋊C4 | — | 4 | 2 | no | — |
| 124 | C42 | — | 2 | 2 | no | — |
| 125 | C42 | — | 2 | 2 | no | — |
| 126 | C42 | — | 4 | 2 | no | — |
| 127 | C42 | — | 4 | 2 | no | — |
| 128 | C42 | — | 4 | 2 | no | — |
| 129 | C42 | C2×#41 | 1 | 2 | yes | — |
| 130 | C42 | C2×#41 | 1 | 2 | yes | — |
| 131 | C23 | #525 | 1 | 1 | yes | characteristic |
| 132 | Q8 | — | 8 | 4 | no | — |
| 133 | C2×C4 | — | 4 | 2 | no | — |
| 134 | C2×C4 | — | 4 | 2 | no | — |
| 135 | C2×C4 | — | 4 | 2 | no | — |
| 136 | C2×C4 | — | 4 | 2 | no | — |
| 137 | C2×C4 | — | 4 | 2 | no | — |
| 138 | C2×C4 | — | 4 | 2 | no | — |
| 139 | C2×C4 | — | 4 | 2 | no | — |
| 140 | C2×C4 | — | 4 | 2 | no | — |
| 141 | C2×C4 | — | 4 | 2 | no | — |
| 142 | C2×C4 | — | 4 | 2 | no | — |
| 143 | C2×C4 | — | 4 | 4 | no | — |
| 144 | C2×C4 | #525 | 1 | 1 | yes | characteristic |
| 145 | C2×C4 | C2×#124 | 1 | 1 | yes | characteristic |
| 146 | C2×C4 | #529 | 1 | 4 | yes | — |
| 147 | C2×C4 | C2×#137 | 1 | 2 | yes | — |
| 148 | C2×C4 | #575 | 1 | 2 | yes | — |
| 149 | C8 | — | 2 | 4 | no | — |
| 150 | C8 | — | 2 | 2 | no | — |
| 151 | C8 | — | 2 | 2 | no | — |
| 152 | C8 | — | 8 | 4 | no | — |
| 153 | C8 | — | 8 | 2 | no | — |
| 154 | C8 | — | 8 | 2 | no | — |
| 155 | C22 | — | 2 | 2 | no | — |
| 156 | C22 | #2398 | 1 | 1 | yes | characteristic |
| 157 | C22 | #3116 | 1 | 2 | yes | — |
| 158 | C4 | — | 2 | 4 | no | — |
| 159 | C4 | — | 8 | 2 | no | — |
| 160 | C4 | — | 8 | 2 | no | — |
| 161 | C4 | — | 8 | 2 | no | — |
| 162 | C4 | — | 8 | 4 | no | — |
| 163 | C4 | #3117 | 1 | 2 | yes | — |
| 164 | C4 | #3139 | 1 | 1 | yes | characteristic |
| 165 | C4 | #3142 | 1 | 1 | yes | characteristic |
| 166 | C2 | — | 2 | 2 | no | — |
| 167 | C2 | #23825 | 1 | 1 | yes | characteristic |
| 168 | C2 | #23834 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2398 |
| prime radical | C23 | #525 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21737 | 2 |
| #21677 | 2 |
| #10688 | 2 |
| #10630 | 2 |
| #10548 | 2 |
| #10373 | 2 |
| #8254 | 2 |
| #8139 | 2 |