Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
747 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 747 conjugacy classes of subgroups — too many to draw. Only its 178 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 | #11387 | C2 | 1 | 4 | yes | maximal |
| 2 | #11386 | C2 | 1 | 2 | yes | maximal |
| 3 | #11370 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #11367 | C2 | 1 | 2 | yes | maximal |
| 5 | #10901 | C2 | 1 | 4 | yes | maximal |
| 6 | #10893 | C2 | 1 | 2 | yes | maximal |
| 7 | C2×#533 | C22 | 1 | 2 | yes | — |
| 8 | C2×#430 | C22 | 1 | 2 | yes | — |
| 9 | #2873 | C22 | 1 | 2 | yes | — |
| 10 | #2285 | C22 | 1 | 1 | yes | characteristic |
| 11 | C2×#344 | C22 | 1 | 4 | yes | — |
| 12 | #1740 | C22 | 1 | 2 | yes | — |
| 13 | #1733 | C22 | 1 | 4 | yes | — |
| 14 | C2×#339 | C22 | 1 | 4 | yes | — |
| 15 | C2×#339 | C22 | 1 | 4 | yes | — |
| 16 | C2×#339 | C22 | 1 | 4 | yes | — |
| 17 | C2×H(3,ℤ/4) | C22 | 1 | 2 | yes | — |
| 18 | #1728 | C22 | 1 | 1 | yes | characteristic |
| 19 | #1728 | C22 | 1 | 2 | yes | — |
| 20 | C4×#95 | C22 | 1 | 1 | yes | characteristic |
| 21 | C22×M16 | C23 | 1 | 4 | yes | — |
| 22 | C22×M16 | C23 | 1 | 2 | yes | — |
| 23 | #533 | — | 2 | 8 | no | — |
| 24 | #533 | — | 4 | 4 | no | — |
| 25 | C2×C4⋊Q8 | — | 2 | 2 | no | — |
| 26 | C2×#125 | — | 2 | 2 | no | — |
| 27 | C2×C22⋊Q8 | — | 4 | 2 | no | — |
| 28 | C2×C4×Q8 | — | 4 | 2 | no | — |
| 29 | C2×#117 | C23 | 1 | 4 | yes | — |
| 30 | C2×#117 | C23 | 1 | 2 | yes | — |
| 31 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 32 | C22×C42 | C23 | 1 | 2 | yes | — |
| 33 | #430 | — | 4 | 2 | no | — |
| 34 | #430 | Q8 | 1 | 8 | yes | complemented |
| 35 | C2×#108 | — | 4 | 2 | no | — |
| 36 | #405 | — | 4 | 4 | no | — |
| 37 | C4×M16 | — | 4 | 2 | no | — |
| 38 | #384 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 39 | #384 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 40 | #382 | Q8 | 1 | 2 | yes | complemented |
| 41 | #380 | — | 4 | 1 | no | — |
| 42 | C4×C4⋊C4 | — | 4 | 1 | no | — |
| 43 | C2×#95 | — | 4 | 1 | no | — |
| 44 | #344 | — | 2 | 8 | no | — |
| 45 | #342 | — | 4 | 2 | no | — |
| 46 | #339 | — | 2 | 8 | no | — |
| 47 | #339 | — | 2 | 4 | no | — |
| 48 | #339 | — | 2 | 4 | no | — |
| 49 | #339 | — | 4 | 4 | no | — |
| 50 | #339 | — | 4 | 4 | no | — |
| 51 | H(3,ℤ/4) | — | 4 | 2 | no | — |
| 52 | #336 | — | 4 | 2 | no | — |
| 53 | #336 | — | 4 | 1 | no | — |
| 54 | C22×Q8 | — | 4 | 2 | no | — |
| 55 | C23×C4 | — | 2 | 2 | no | — |
| 56 | C23×C4 | — | 2 | 1 | no | — |
| 57 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 58 | C2×M16 | — | 2 | 4 | no | — |
| 59 | C2×M16 | — | 2 | 2 | no | — |
| 60 | C2×M16 | — | 2 | 4 | no | — |
| 61 | C2×M16 | — | 2 | 4 | no | — |
| 62 | C2×M16 | — | 2 | 4 | no | — |
| 63 | C2×M16 | — | 2 | 4 | no | — |
| 64 | C2×M16 | — | 2 | 4 | no | — |
| 65 | C2×M16 | — | 2 | 4 | no | — |
| 66 | C2×M16 | C2×Q8 | 1 | 4 | yes | — |
| 67 | C2×M16 | C2×Q8 | 1 | 8 | yes | — |
| 68 | C22×C8 | — | 2 | 4 | no | — |
| 69 | C22×C8 | — | 2 | 2 | no | — |
| 70 | C4⋊Q8 | — | 4 | 8 | no | — |
| 71 | #125 | — | 4 | 8 | no | — |
| 72 | C22⋊Q8 | — | 4 | 8 | no | — |
| 73 | C22⋊Q8 | — | 4 | 8 | no | — |
| 74 | C4×Q8 | — | 4 | 16 | no | — |
| 75 | #117 | — | 2 | 4 | no | — |
| 76 | #117 | — | 2 | 4 | no | — |
| 77 | #117 | — | 2 | 4 | no | — |
| 78 | #117 | — | 2 | 4 | no | — |
| 79 | #117 | — | 2 | 4 | no | — |
| 80 | #117 | #41 | 1 | 8 | yes | — |
| 81 | C2×C4⋊C4 | — | 2 | 4 | no | — |
| 82 | C2×C4⋊C4 | — | 4 | 4 | no | — |
| 83 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 84 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 85 | C2×C4⋊C4 | — | 4 | 1 | no | — |
| 86 | C2×C4⋊C4 | C2×Q8 | 1 | 2 | yes | — |
| 87 | C2×C4⋊C4 | C2×Q8 | 1 | 2 | yes | — |
| 88 | C2×C22⋊C4 | — | 2 | 4 | no | — |
| 89 | C2×C22⋊C4 | — | 2 | 2 | no | — |
| 90 | C2×C42 | — | 2 | 4 | no | — |
| 91 | C2×C42 | — | 2 | 4 | no | — |
| 92 | C2×C42 | — | 2 | 4 | no | — |
| 93 | C2×C42 | — | 2 | 4 | no | — |
| 94 | C2×C42 | — | 2 | 2 | no | — |
| 95 | C2×C42 | — | 2 | 2 | no | — |
| 96 | C2×C42 | — | 2 | 1 | no | — |
| 97 | C2×C42 | C2×Q8 | 1 | 1 | yes | characteristic |
| 98 | C2×C42 | C2×Q8 | 1 | 1 | yes | characteristic |
| 99 | C2×C42 | #41 | 1 | 2 | yes | — |
| 100 | C2×C42 | #41 | 1 | 2 | yes | — |
| 101 | #108 | — | 4 | 8 | no | — |
| 102 | #97 | — | 4 | 4 | no | — |
| 103 | C4×C8 | — | 4 | 4 | no | — |
| 104 | #95 | — | 4 | 2 | no | — |
| 105 | #95 | — | 4 | 4 | no | — |
| 106 | C24 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 107 | C2×Q8 | — | 4 | 8 | no | — |
| 108 | C2×Q8 | — | 8 | 4 | no | — |
| 109 | C2×Q8 | — | 8 | 4 | no | — |
| 110 | C22×C4 | — | 2 | 2 | no | — |
| 111 | C22×C4 | — | 2 | 1 | no | — |
| 112 | C22×C4 | — | 2 | 2 | no | — |
| 113 | C22×C4 | — | 2 | 1 | no | — |
| 114 | C22×C4 | — | 2 | 2 | no | — |
| 115 | C22×C4 | — | 2 | 2 | no | — |
| 116 | C22×C4 | — | 4 | 2 | no | — |
| 117 | C22×C4 | — | 4 | 4 | no | — |
| 118 | C22×C4 | — | 4 | 4 | no | — |
| 119 | C22×C4 | — | 4 | 2 | no | — |
| 120 | C22×C4 | — | 4 | 2 | no | — |
| 121 | C22×C4 | — | 4 | 2 | no | — |
| 122 | C22×C4 | — | 8 | 1 | no | — |
| 123 | C22×C4 | — | 8 | 2 | no | — |
| 124 | C22×C4 | C22×Q8 | 1 | 2 | yes | — |
| 125 | C22×C4 | C22×Q8 | 1 | 1 | yes | characteristic |
| 126 | C22×C4 | C2×#41 | 1 | 4 | yes | — |
| 127 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 128 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 129 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 130 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 131 | M16 | — | 2 | 8 | no | — |
| 132 | M16 | — | 2 | 8 | no | — |
| 133 | M16 | — | 2 | 8 | no | — |
| 134 | M16 | — | 4 | 4 | no | — |
| 135 | M16 | — | 4 | 4 | no | — |
| 136 | M16 | — | 4 | 4 | no | — |
| 137 | C2×C8 | — | 2 | 4 | no | — |
| 138 | C2×C8 | — | 2 | 8 | no | — |
| 139 | C2×C8 | — | 4 | 4 | no | — |
| 140 | C2×C8 | — | 4 | 2 | no | — |
| 141 | C2×C8 | — | 4 | 2 | no | — |
| 142 | C2×C8 | — | 4 | 4 | no | — |
| 143 | C4⋊C4 | — | 2 | 8 | no | — |
| 144 | C4⋊C4 | — | 2 | 4 | no | — |
| 145 | C4⋊C4 | — | 2 | 4 | no | — |
| 146 | C4⋊C4 | — | 4 | 16 | no | — |
| 147 | C4⋊C4 | — | 4 | 8 | no | — |
| 148 | C4⋊C4 | — | 4 | 4 | no | — |
| 149 | C4⋊C4 | — | 8 | 2 | no | — |
| 150 | C4⋊C4 | — | 8 | 2 | no | — |
| 151 | C22⋊C4 | — | 2 | 8 | no | — |
| 152 | C22⋊C4 | — | 4 | 4 | no | — |
| 153 | C22⋊C4 | — | 4 | 4 | no | — |
| 154 | C42 | — | 2 | 2 | no | — |
| 155 | C42 | — | 2 | 2 | no | — |
| 156 | C42 | — | 2 | 8 | no | — |
| 157 | C42 | — | 2 | 4 | no | — |
| 158 | C42 | — | 2 | 4 | no | — |
| 159 | C42 | — | 4 | 4 | no | — |
| 160 | C42 | — | 4 | 4 | no | — |
| 161 | C42 | — | 4 | 2 | no | — |
| 162 | C42 | — | 4 | 2 | no | — |
| 163 | C42 | — | 4 | 2 | no | — |
| 164 | C23 | — | 4 | 1 | no | — |
| 165 | C23 | — | 4 | 1 | no | — |
| 166 | C23 | #538 | 1 | 2 | yes | — |
| 167 | C23 | #543 | 1 | 1 | yes | characteristic |
| 168 | C23 | #551 | 1 | 2 | yes | — |
| 169 | C23 | Q82 | 1 | 1 | yes | characteristic |
| 170 | C23 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 171 | Q8 | — | 8 | 16 | no | — |
| 172 | C2×C4 | — | 2 | 2 | no | — |
| 173 | C2×C4 | — | 2 | 4 | no | — |
| 174 | C2×C4 | — | 4 | 1 | no | — |
| 175 | C2×C4 | — | 4 | 2 | no | — |
| 176 | C2×C4 | — | 4 | 1 | no | — |
| 177 | C2×C4 | — | 4 | 2 | no | — |
| 178 | C2×C4 | — | 4 | 2 | no | — |
| 179 | C2×C4 | — | 4 | 1 | no | — |
| 180 | C2×C4 | — | 4 | 2 | no | — |
| 181 | C2×C4 | — | 4 | 1 | no | — |
| 182 | C2×C4 | — | 4 | 1 | no | — |
| 183 | C2×C4 | — | 4 | 2 | no | — |
| 184 | C2×C4 | — | 4 | 8 | no | — |
| 185 | C2×C4 | — | 4 | 2 | no | — |
| 186 | C2×C4 | — | 4 | 2 | no | — |
| 187 | C2×C4 | — | 8 | 4 | no | — |
| 188 | C2×C4 | — | 8 | 4 | no | — |
| 189 | C2×C4 | — | 8 | 4 | no | — |
| 190 | C2×C4 | — | 8 | 4 | no | — |
| 191 | C2×C4 | — | 8 | 4 | no | — |
| 192 | C2×C4 | — | 8 | 2 | no | — |
| 193 | C2×C4 | — | 8 | 2 | no | — |
| 194 | C2×C4 | #533 | 1 | 2 | yes | — |
| 195 | C2×C4 | #533 | 1 | 4 | yes | — |
| 196 | C2×C4 | #538 | 1 | 4 | yes | — |
| 197 | C2×C4 | #543 | 1 | 2 | yes | — |
| 198 | C2×C4 | #551 | 1 | 4 | yes | — |
| 199 | C2×C4 | #554 | 1 | 2 | yes | — |
| 200 | C2×C4 | #554 | 1 | 4 | yes | — |
| 201 | C2×C4 | Q82 | 1 | 2 | yes | — |
| 202 | C2×C4 | #559 | 1 | 1 | yes | characteristic |
| 203 | C2×C4 | #559 | 1 | 2 | yes | — |
| 204 | C2×C4 | H(3,𝔽4) | 1 | 1 | yes | characteristic |
| 205 | C8 | — | 4 | 4 | no | — |
| 206 | C8 | — | 4 | 8 | no | — |
| 207 | C22 | — | 2 | 2 | no | — |
| 208 | C22 | — | 2 | 1 | no | — |
| 209 | C22 | — | 2 | 2 | no | — |
| 210 | C22 | — | 2 | 1 | no | — |
| 211 | C22 | — | 4 | 2 | no | — |
| 212 | C22 | — | 4 | 1 | no | — |
| 213 | C22 | — | 4 | 1 | no | — |
| 214 | C22 | #2586 | 1 | 2 | yes | — |
| 215 | C22 | #2607 | 1 | 1 | yes | characteristic |
| 216 | C22 | #2736 | 1 | 2 | yes | — |
| 217 | C22 | #2804 | 1 | 1 | yes | characteristic |
| 218 | C22 | #2811 | 1 | 1 | yes | characteristic |
| 219 | C4 | — | 4 | 2 | no | — |
| 220 | C4 | — | 4 | 4 | no | — |
| 221 | C4 | — | 8 | 2 | no | — |
| 222 | C4 | — | 8 | 1 | no | — |
| 223 | C4 | — | 8 | 8 | no | — |
| 224 | C4 | — | 8 | 2 | no | — |
| 225 | C4 | — | 8 | 2 | no | — |
| 226 | C4 | #2586 | 1 | 4 | yes | — |
| 227 | C4 | #2607 | 1 | 2 | yes | — |
| 228 | C4 | #2804 | 1 | 2 | yes | — |
| 229 | C2 | — | 2 | 2 | no | — |
| 230 | C2 | — | 2 | 1 | no | — |
| 231 | C2 | — | 2 | 2 | no | — |
| 232 | C2 | — | 2 | 1 | no | — |
| 233 | C2 | #16248 | 1 | 1 | yes | characteristic |
| 234 | C2 | #32704 | 1 | 1 | yes | characteristic |
| 235 | C2 | #32705 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 34. #430 | Q8 | semidirect |
| 38. #384 | Q8 | semidirect |
| 39. #384 | Q8 | semidirect |
| 40. #382 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11387 | 2 |
| #11386 | 2 |
| #11370 | 2 |
| #11367 | 2 |
| #10901 | 2 |
| #10893 | 2 |