Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
851 conjugacy classes of subgroups; only the 242 normal subgroups are shown.
This group has 851 conjugacy classes of subgroups — too many to draw. Only its 242 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 | #21164 | C2 | 1 | 3 | yes | maximal |
| 2 | #21147 | C2 | 1 | 3 | yes | maximal |
| 3 | #11338 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11337 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | #11336 | C2 | 1 | 3 | yes | maximal |
| 6 | #11335 | C2 | 1 | 3 | yes | maximal |
| 7 | #10699 | C2 | 1 | 1 | yes | maximal, characteristic |
| 8 | C2×#430 | C22 | 1 | 3 | yes | — |
| 9 | #2911 | C22 | 1 | 3 | yes | — |
| 10 | C2×#424 | C22 | 1 | 3 | yes | — |
| 11 | C2×#424 | C22 | 1 | 3 | yes | — |
| 12 | C22×C4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#405 | C22 | 1 | 3 | yes | — |
| 14 | #2296 | C22 | 1 | 3 | yes | — |
| 15 | #1946 | — | 2 | 6 | no | — |
| 16 | #1945 | — | 2 | 6 | no | — |
| 17 | #1940 | — | 2 | 6 | no | — |
| 18 | #1939 | — | 2 | 6 | no | — |
| 19 | #1733 | C22 | 1 | 6 | yes | — |
| 20 | C2×#338 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×H(3,ℤ/4) | C22 | 1 | 3 | yes | — |
| 22 | #1729 | C22 | 1 | 3 | yes | — |
| 23 | #1728 | C22 | 1 | 3 | yes | — |
| 24 | C22×M16 | C23 | 1 | 3 | yes | — |
| 25 | C22×M16 | C23 | 1 | 3 | yes | — |
| 26 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 27 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 28 | C2×#117 | C23 | 1 | 3 | yes | — |
| 29 | C2×#117 | C23 | 1 | 3 | yes | — |
| 30 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 31 | #430 | — | 4 | 6 | no | — |
| 32 | C2×#108 | — | 2 | 6 | no | — |
| 33 | #428 | — | 2 | 6 | no | — |
| 34 | #428 | — | 2 | 6 | no | — |
| 35 | #427 | — | 2 | 6 | no | — |
| 36 | #424 | — | 2 | 6 | no | — |
| 37 | #424 | — | 2 | 6 | no | — |
| 38 | #424 | — | 4 | 6 | no | — |
| 39 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 40 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 41 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 42 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 43 | C2×C4⋊C8 | — | 2 | 6 | no | — |
| 44 | #405 | — | 4 | 6 | no | — |
| 45 | C2×#97 | D4 | 1 | 6 | yes | — |
| 46 | C2×C4×C8 | D4 | 1 | 6 | yes | — |
| 47 | #389 | — | 2 | 3 | no | — |
| 48 | #389 | — | 2 | 3 | no | — |
| 49 | #385 | — | 2 | 3 | no | — |
| 50 | #385 | — | 2 | 3 | no | — |
| 51 | C4×C4⋊C4 | — | 2 | 6 | no | — |
| 52 | #339 | — | 2 | 12 | no | — |
| 53 | #338 | Q8 | 1 | 4 | yes | — |
| 54 | H(3,ℤ/4) | Q8 | 1 | 12 | yes | — |
| 55 | #336 | — | 2 | 6 | no | — |
| 56 | #336 | — | 2 | 6 | no | — |
| 57 | C23×C4 | #41 | 1 | 1 | yes | characteristic |
| 58 | C23×C4 | #41 | 1 | 1 | yes | characteristic |
| 59 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 60 | C2×M16 | — | 2 | 6 | no | — |
| 61 | C2×M16 | — | 2 | 6 | no | — |
| 62 | C2×M16 | — | 2 | 6 | no | — |
| 63 | C2×M16 | — | 2 | 6 | no | — |
| 64 | C2×M16 | — | 2 | 3 | no | — |
| 65 | C2×M16 | — | 2 | 3 | no | — |
| 66 | C2×M16 | C2×Q8 | 1 | 6 | yes | — |
| 67 | C2×M16 | C2×Q8 | 1 | 6 | yes | — |
| 68 | C22×C8 | — | 2 | 3 | no | — |
| 69 | C22×C8 | — | 2 | 1 | no | — |
| 70 | C22×C8 | — | 2 | 3 | no | — |
| 71 | C22×C8 | — | 2 | 3 | no | — |
| 72 | C22×C8 | — | 2 | 3 | no | — |
| 73 | C22×C8 | — | 2 | 1 | no | — |
| 74 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 75 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 76 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 77 | #117 | — | 2 | 6 | no | — |
| 78 | #117 | — | 2 | 6 | no | — |
| 79 | #117 | — | 2 | 6 | no | — |
| 80 | #117 | C2×Q8 | 1 | 12 | yes | — |
| 81 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 82 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 83 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 84 | C2×C4⋊C4 | #41 | 1 | 3 | yes | — |
| 85 | C2×C4⋊C4 | #41 | 1 | 3 | yes | — |
| 86 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 87 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 88 | C2×C42 | — | 2 | 3 | no | — |
| 89 | C2×C42 | — | 2 | 3 | no | — |
| 90 | C2×C42 | — | 2 | 3 | no | — |
| 91 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 92 | #108 | — | 4 | 12 | no | — |
| 93 | #107 | — | 2 | 6 | no | — |
| 94 | #107 | — | 2 | 6 | no | — |
| 95 | #106 | — | 2 | 12 | no | — |
| 96 | C4⋊C8 | — | 4 | 12 | no | — |
| 97 | C4⋊C8 | — | 4 | 1 | no | — |
| 98 | C4⋊C8 | — | 4 | 3 | no | — |
| 99 | C4⋊C8 | — | 4 | 3 | no | — |
| 100 | C4⋊C8 | — | 4 | 3 | no | — |
| 101 | C4⋊C8 | — | 4 | 6 | no | — |
| 102 | #97 | — | 4 | 6 | no | — |
| 103 | C4×C8 | — | 4 | 6 | no | — |
| 104 | #95 | — | 2 | 6 | no | — |
| 105 | #95 | — | 2 | 6 | no | — |
| 106 | C24 | C2×#41 | 1 | 1 | yes | characteristic |
| 107 | C22×C4 | — | 2 | 3 | no | — |
| 108 | C22×C4 | — | 2 | 6 | no | — |
| 109 | C22×C4 | — | 2 | 3 | no | — |
| 110 | C22×C4 | — | 2 | 3 | no | — |
| 111 | C22×C4 | — | 2 | 3 | no | — |
| 112 | C22×C4 | — | 4 | 2 | no | — |
| 113 | C22×C4 | — | 8 | 1 | no | — |
| 114 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 115 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 116 | C22×C4 | C22×Q8 | 1 | 3 | yes | — |
| 117 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 118 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 119 | C22×C4 | 2−1+4 | 1 | 1 | yes | characteristic |
| 120 | M16 | — | 2 | 12 | no | — |
| 121 | M16 | — | 2 | 12 | no | — |
| 122 | M16 | — | 4 | 6 | no | — |
| 123 | M16 | — | 4 | 6 | no | — |
| 124 | C2×C8 | — | 2 | 6 | no | — |
| 125 | C2×C8 | — | 2 | 3 | no | — |
| 126 | C2×C8 | — | 2 | 6 | no | — |
| 127 | C2×C8 | — | 2 | 3 | no | — |
| 128 | C2×C8 | — | 2 | 3 | no | — |
| 129 | C2×C8 | — | 2 | 3 | no | — |
| 130 | C2×C8 | — | 4 | 1 | no | — |
| 131 | C2×C8 | — | 4 | 3 | no | — |
| 132 | C2×C8 | — | 4 | 3 | no | — |
| 133 | C2×C8 | — | 4 | 3 | no | — |
| 134 | C2×C8 | — | 4 | 3 | no | — |
| 135 | C2×C8 | — | 4 | 3 | no | — |
| 136 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 137 | C2×C8 | #127 | 1 | 3 | yes | — |
| 138 | C2×C8 | C4⋊Q8 | 1 | 12 | yes | — |
| 139 | C4⋊C4 | — | 2 | 12 | no | — |
| 140 | C4⋊C4 | — | 2 | 6 | no | — |
| 141 | C4⋊C4 | — | 2 | 6 | no | — |
| 142 | C4⋊C4 | — | 4 | 12 | no | — |
| 143 | C4⋊C4 | — | 8 | 6 | no | — |
| 144 | C22⋊C4 | — | 2 | 12 | no | — |
| 145 | C22⋊C4 | — | 4 | 6 | no | — |
| 146 | C42 | — | 2 | 6 | no | — |
| 147 | C42 | — | 4 | 3 | no | — |
| 148 | C42 | — | 4 | 3 | no | — |
| 149 | C42 | — | 4 | 1 | no | — |
| 150 | C42 | — | 4 | 3 | no | — |
| 151 | C42 | C4⋊Q8 | 1 | 12 | yes | — |
| 152 | C23 | — | 4 | 1 | no | — |
| 153 | C23 | — | 4 | 1 | no | — |
| 154 | C23 | #525 | 1 | 3 | yes | — |
| 155 | C23 | #554 | 1 | 3 | yes | — |
| 156 | C23 | #557 | 1 | 1 | yes | characteristic |
| 157 | C2×C4 | — | 2 | 6 | no | — |
| 158 | C2×C4 | — | 2 | 12 | no | — |
| 159 | C2×C4 | — | 4 | 6 | no | — |
| 160 | C2×C4 | — | 4 | 6 | no | — |
| 161 | C2×C4 | — | 4 | 3 | no | — |
| 162 | C2×C4 | — | 4 | 1 | no | — |
| 163 | C2×C4 | — | 4 | 6 | no | — |
| 164 | C2×C4 | — | 4 | 3 | no | — |
| 165 | C2×C4 | — | 4 | 6 | no | — |
| 166 | C2×C4 | — | 4 | 3 | no | — |
| 167 | C2×C4 | — | 4 | 1 | no | — |
| 168 | C2×C4 | — | 4 | 3 | no | — |
| 169 | C2×C4 | — | 4 | 3 | no | — |
| 170 | C2×C4 | #525 | 1 | 3 | yes | — |
| 171 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 172 | C2×C4 | C2×C4⋊Q8 | 1 | 3 | yes | — |
| 173 | C2×C4 | C2×C4⋊Q8 | 1 | 3 | yes | — |
| 174 | C2×C4 | #534 | 1 | 3 | yes | — |
| 175 | C2×C4 | #537 | 1 | 3 | yes | — |
| 176 | C2×C4 | #554 | 1 | 3 | yes | — |
| 177 | C2×C4 | #554 | 1 | 6 | yes | — |
| 178 | C2×C4 | #557 | 1 | 3 | yes | — |
| 179 | C8 | — | 2 | 12 | no | — |
| 180 | C8 | — | 2 | 1 | no | — |
| 181 | C8 | — | 2 | 3 | no | — |
| 182 | C8 | — | 8 | 1 | no | — |
| 183 | C8 | — | 8 | 3 | no | — |
| 184 | C22 | — | 2 | 3 | no | — |
| 185 | C22 | — | 2 | 3 | no | — |
| 186 | C22 | — | 4 | 3 | no | — |
| 187 | C22 | — | 4 | 1 | no | — |
| 188 | C22 | #2397 | 1 | 1 | yes | characteristic |
| 189 | C22 | #2583 | 1 | 3 | yes | — |
| 190 | C22 | #2584 | 1 | 3 | yes | — |
| 191 | C4 | — | 4 | 12 | no | — |
| 192 | C4 | — | 4 | 2 | no | — |
| 193 | C4 | — | 8 | 3 | no | — |
| 194 | C4 | — | 8 | 3 | no | — |
| 195 | C4 | — | 8 | 1 | no | — |
| 196 | C4 | #2583 | 1 | 3 | yes | — |
| 197 | C4 | #2584 | 1 | 3 | yes | — |
| 198 | C4 | #2656 | 1 | 1 | yes | characteristic |
| 199 | C4 | #2868 | 1 | 1 | yes | characteristic |
| 200 | C2 | — | 2 | 3 | no | — |
| 201 | C2 | — | 2 | 3 | no | — |
| 202 | C2 | #15211 | 1 | 1 | yes | characteristic |
| 203 | C2 | #23794 | 1 | 1 | yes | characteristic |
| 204 | C2 | #23795 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21164 | 2 |
| #21147 | 2 |
| #11338 | 2 |
| #11337 | 2 |
| #11336 | 2 |
| #11335 | 2 |
| #10699 | 2 |