Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2297 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 2297 conjugacy classes of subgroups — too many to draw. Only its 296 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 | #60643 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | #23434 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #23414 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #21240 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #10699 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#575 | C22 | 1 | 6 | yes | — |
| 7 | C22×#135 | C22 | 1 | 3 | yes | — |
| 8 | #3449 | C22 | 1 | 6 | yes | — |
| 9 | #3156 | — | 2 | 6 | no | — |
| 10 | #3155 | — | 2 | 6 | no | — |
| 11 | #3154 | — | 2 | 12 | no | — |
| 12 | #3150 | — | 2 | 6 | no | — |
| 13 | #3147 | — | 2 | 6 | no | — |
| 14 | C22×#108 | C22 | 1 | 1 | yes | characteristic |
| 15 | C2×#421 | C22 | 1 | 6 | yes | — |
| 16 | #2879 | C22 | 1 | 6 | yes | — |
| 17 | C2×#405 | C22 | 1 | 3 | yes | — |
| 18 | #2017 | — | 2 | 8 | no | — |
| 19 | #2010 | — | 2 | 24 | no | — |
| 20 | #1975 | — | 2 | 6 | no | — |
| 21 | #1974 | — | 2 | 6 | no | — |
| 22 | C2×#338 | C22 | 1 | 1 | yes | characteristic |
| 23 | C2×H(3,ℤ/4) | C22 | 1 | 3 | yes | — |
| 24 | C22×#41 | C23 | 1 | 6 | yes | — |
| 25 | #575 | — | 2 | 12 | no | — |
| 26 | #575 | — | 2 | 12 | no | — |
| 27 | #575 | — | 4 | 12 | no | — |
| 28 | C2×#137 | — | 2 | 12 | no | — |
| 29 | C2×AGL(1,ℤ/8) | — | 2 | 12 | no | — |
| 30 | C2×#135 | — | 2 | 6 | no | — |
| 31 | C2×#135 | — | 2 | 12 | no | — |
| 32 | C2×#135 | — | 2 | 6 | no | — |
| 33 | C2×#135 | — | 2 | 6 | no | — |
| 34 | C2×#135 | — | 2 | 6 | no | — |
| 35 | C2×#135 | — | 2 | 3 | no | — |
| 36 | C2×#135 | — | 2 | 3 | no | — |
| 37 | C2×#135 | — | 2 | 3 | no | — |
| 38 | C2×#135 | — | 2 | 3 | no | — |
| 39 | C22×Q16 | D4 | 1 | 3 | yes | complemented |
| 40 | C22×SD16 | D4 | 1 | 6 | yes | complemented |
| 41 | C22×D8 | D4 | 1 | 3 | yes | complemented |
| 42 | C22×M16 | C23 | 1 | 2 | yes | — |
| 43 | C22×M16 | C23 | 1 | 3 | yes | — |
| 44 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 45 | #537 | — | 4 | 6 | no | — |
| 46 | #536 | — | 2 | 12 | no | — |
| 47 | #535 | — | 2 | 12 | no | — |
| 48 | #534 | — | 4 | 6 | no | — |
| 49 | #532 | — | 2 | 12 | no | — |
| 50 | #525 | — | 2 | 24 | no | — |
| 51 | C2×#117 | C23 | 1 | 3 | yes | — |
| 52 | #497 | — | 2 | 24 | no | — |
| 53 | #496 | — | 2 | 24 | no | — |
| 54 | #495 | — | 2 | 12 | no | — |
| 55 | #495 | — | 2 | 12 | no | — |
| 56 | #494 | — | 2 | 6 | no | — |
| 57 | #493 | — | 2 | 6 | no | — |
| 58 | #492 | — | 2 | 12 | no | — |
| 59 | C2×#108 | — | 2 | 3 | no | — |
| 60 | C2×#108 | — | 2 | 3 | no | — |
| 61 | #421 | — | 2 | 24 | no | — |
| 62 | C2×#104 | — | 2 | 12 | no | — |
| 63 | #408 | — | 2 | 12 | no | — |
| 64 | #408 | — | 2 | 12 | no | — |
| 65 | #407 | — | 2 | 12 | no | — |
| 66 | #405 | — | 2 | 6 | no | — |
| 67 | #405 | — | 2 | 6 | no | — |
| 68 | C2×#97 | D4 | 1 | 6 | yes | complemented |
| 69 | C2×C4×C8 | D4 | 1 | 6 | yes | complemented |
| 70 | #338 | D4 | 1 | 4 | yes | complemented |
| 71 | H(3,ℤ/4) | D4 | 1 | 12 | yes | complemented |
| 72 | C2×#41 | — | 2 | 12 | no | — |
| 73 | C2×#41 | — | 2 | 12 | no | — |
| 74 | C2×#41 | — | 2 | 12 | no | — |
| 75 | C2×#41 | — | 2 | 6 | no | — |
| 76 | C2×#41 | — | 2 | 6 | no | — |
| 77 | C2×#41 | — | 2 | 6 | no | — |
| 78 | C2×#41 | — | 2 | 12 | no | — |
| 79 | C2×#41 | — | 2 | 6 | no | — |
| 80 | C22×Q8 | C2×D4 | 1 | 6 | yes | — |
| 81 | C22×D4 | — | 2 | 6 | no | — |
| 82 | C22×D4 | C2×D4 | 1 | 6 | yes | — |
| 83 | C23×C4 | — | 2 | 6 | no | — |
| 84 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 85 | #137 | — | 4 | 24 | no | — |
| 86 | #137 | — | 4 | 24 | no | — |
| 87 | AGL(1,ℤ/8) | — | 4 | 24 | no | — |
| 88 | AGL(1,ℤ/8) | — | 4 | 24 | no | — |
| 89 | #135 | — | 4 | 24 | no | — |
| 90 | #135 | — | 4 | 3 | no | — |
| 91 | #135 | — | 4 | 3 | no | — |
| 92 | #135 | — | 4 | 6 | no | — |
| 93 | #135 | — | 4 | 6 | no | — |
| 94 | #135 | — | 4 | 6 | no | — |
| 95 | #135 | — | 4 | 6 | no | — |
| 96 | #135 | — | 4 | 6 | no | — |
| 97 | #135 | — | 4 | 3 | no | — |
| 98 | #135 | — | 4 | 6 | no | — |
| 99 | #135 | — | 4 | 3 | no | — |
| 100 | #135 | — | 4 | 24 | no | — |
| 101 | C2×Q16 | — | 2 | 12 | no | — |
| 102 | C2×Q16 | — | 2 | 6 | no | — |
| 103 | C2×Q16 | — | 2 | 6 | no | — |
| 104 | C2×Q16 | — | 2 | 3 | no | — |
| 105 | C2×Q16 | — | 2 | 3 | no | — |
| 106 | C2×SD16 | — | 2 | 24 | no | — |
| 107 | C2×SD16 | — | 2 | 12 | no | — |
| 108 | C2×SD16 | — | 2 | 12 | no | — |
| 109 | C2×SD16 | — | 2 | 6 | no | — |
| 110 | C2×SD16 | — | 2 | 6 | no | — |
| 111 | C2×D8 | — | 2 | 12 | no | — |
| 112 | C2×D8 | — | 2 | 6 | no | — |
| 113 | C2×D8 | — | 2 | 6 | no | — |
| 114 | C2×D8 | — | 2 | 3 | no | — |
| 115 | C2×D8 | — | 2 | 3 | no | — |
| 116 | C2×M16 | — | 2 | 6 | no | — |
| 117 | C2×M16 | — | 2 | 6 | no | — |
| 118 | C2×M16 | — | 2 | 6 | no | — |
| 119 | C2×M16 | — | 2 | 6 | no | — |
| 120 | C2×M16 | C2×D4 | 1 | 6 | yes | — |
| 121 | C2×M16 | C2×D4 | 1 | 6 | yes | — |
| 122 | C22×C8 | — | 2 | 2 | no | — |
| 123 | C22×C8 | C2×D4 | 1 | 6 | yes | — |
| 124 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 125 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 126 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 127 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 128 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 129 | C22×C8 | #41 | 1 | 1 | yes | characteristic |
| 130 | C4⋊Q8 | — | 2 | 12 | no | — |
| 131 | #127 | — | 2 | 12 | no | — |
| 132 | #124 | — | 2 | 24 | no | — |
| 133 | #123 | — | 4 | 24 | no | — |
| 134 | C22⋊Q8 | — | 4 | 24 | no | — |
| 135 | #121 | — | 4 | 24 | no | — |
| 136 | #121 | — | 4 | 24 | no | — |
| 137 | #120 | — | 4 | 24 | no | — |
| 138 | C4×D4 | — | 4 | 24 | no | — |
| 139 | #117 | — | 2 | 6 | no | — |
| 140 | #117 | C2×D4 | 1 | 12 | yes | — |
| 141 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 142 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 143 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 144 | #108 | — | 4 | 1 | no | — |
| 145 | #108 | — | 4 | 3 | no | — |
| 146 | #104 | — | 4 | 24 | no | — |
| 147 | C22⋊C8 | — | 4 | 24 | no | — |
| 148 | #97 | — | 2 | 12 | no | — |
| 149 | C4×C8 | — | 2 | 6 | no | — |
| 150 | C4×C8 | — | 2 | 6 | no | — |
| 151 | C24 | — | 2 | 6 | no | — |
| 152 | C24 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 153 | #41 | — | 4 | 6 | no | — |
| 154 | #41 | — | 4 | 12 | no | — |
| 155 | #41 | — | 4 | 6 | no | — |
| 156 | #41 | — | 4 | 6 | no | — |
| 157 | #41 | — | 4 | 12 | no | — |
| 158 | #41 | — | 4 | 12 | no | — |
| 159 | #41 | — | 4 | 6 | no | — |
| 160 | #41 | — | 4 | 12 | no | — |
| 161 | #41 | — | 4 | 12 | no | — |
| 162 | #41 | — | 4 | 12 | no | — |
| 163 | C2×Q8 | — | 2 | 12 | no | — |
| 164 | C2×Q8 | — | 2 | 6 | no | — |
| 165 | C2×Q8 | — | 2 | 12 | no | — |
| 166 | C2×Q8 | — | 2 | 6 | no | — |
| 167 | C2×D4 | — | 2 | 12 | no | — |
| 168 | C2×D4 | — | 2 | 6 | no | — |
| 169 | C2×D4 | — | 2 | 12 | no | — |
| 170 | C2×D4 | — | 2 | 6 | no | — |
| 171 | C2×D4 | — | 4 | 24 | no | — |
| 172 | C2×D4 | — | 4 | 12 | no | — |
| 173 | C2×D4 | — | 4 | 6 | no | — |
| 174 | C2×D4 | — | 4 | 12 | no | — |
| 175 | C2×D4 | — | 4 | 6 | no | — |
| 176 | C22×C4 | — | 2 | 6 | no | — |
| 177 | C22×C4 | — | 2 | 6 | no | — |
| 178 | C22×C4 | — | 4 | 6 | no | — |
| 179 | C22×C4 | — | 4 | 12 | no | — |
| 180 | C22×C4 | — | 4 | 12 | no | — |
| 181 | C22×C4 | — | 4 | 6 | no | — |
| 182 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 183 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 184 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 185 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 186 | C22×C4 | C2×#41 | 1 | 1 | yes | characteristic |
| 187 | C22×C4 | H(5,𝔽2) | 1 | 3 | yes | — |
| 188 | Q16 | — | 4 | 24 | no | — |
| 189 | Q16 | — | 4 | 3 | no | — |
| 190 | Q16 | — | 4 | 6 | no | — |
| 191 | Q16 | — | 4 | 3 | no | — |
| 192 | SD16 | — | 4 | 48 | no | — |
| 193 | SD16 | — | 4 | 6 | no | — |
| 194 | SD16 | — | 4 | 12 | no | — |
| 195 | SD16 | — | 4 | 6 | no | — |
| 196 | D8 | — | 4 | 24 | no | — |
| 197 | D8 | — | 4 | 3 | no | — |
| 198 | D8 | — | 4 | 6 | no | — |
| 199 | D8 | — | 4 | 3 | no | — |
| 200 | M16 | — | 2 | 12 | no | — |
| 201 | M16 | — | 2 | 12 | no | — |
| 202 | M16 | — | 4 | 6 | no | — |
| 203 | M16 | — | 4 | 2 | no | — |
| 204 | C2×C8 | — | 2 | 6 | no | — |
| 205 | C2×C8 | — | 2 | 6 | no | — |
| 206 | C2×C8 | — | 4 | 6 | no | — |
| 207 | C2×C8 | #120 | 1 | 3 | yes | — |
| 208 | C2×C8 | #120 | 1 | 3 | yes | — |
| 209 | C2×C8 | #120 | 1 | 6 | yes | — |
| 210 | C2×C8 | #121 | 1 | 3 | yes | — |
| 211 | C2×C8 | #121 | 1 | 3 | yes | — |
| 212 | C2×C8 | #121 | 1 | 3 | yes | — |
| 213 | C2×C8 | #121 | 1 | 3 | yes | — |
| 214 | C2×C8 | #127 | 1 | 12 | yes | — |
| 215 | C2×C8 | #127 | 1 | 1 | yes | characteristic |
| 216 | C2×C8 | #127 | 1 | 3 | yes | — |
| 217 | C4⋊C4 | — | 2 | 12 | no | — |
| 218 | C4⋊C4 | — | 4 | 24 | no | — |
| 219 | C22⋊C4 | — | 2 | 12 | no | — |
| 220 | C22⋊C4 | — | 4 | 48 | no | — |
| 221 | C42 | — | 2 | 6 | no | — |
| 222 | C42 | #127 | 1 | 12 | yes | — |
| 223 | C23 | — | 4 | 12 | no | — |
| 224 | C23 | — | 4 | 6 | no | — |
| 225 | C23 | — | 8 | 1 | no | — |
| 226 | C23 | — | 16 | 6 | no | — |
| 227 | C23 | #535 | 1 | 3 | yes | — |
| 228 | C23 | D42 | 1 | 3 | yes | — |
| 229 | C23 | #550 | 1 | 1 | yes | characteristic |
| 230 | Q8 | — | 4 | 6 | no | — |
| 231 | Q8 | — | 4 | 12 | no | — |
| 232 | Q8 | — | 4 | 6 | no | — |
| 233 | D4 | — | 4 | 6 | no | — |
| 234 | D4 | — | 4 | 12 | no | — |
| 235 | D4 | — | 4 | 6 | no | — |
| 236 | D4 | — | 8 | 6 | no | — |
| 237 | D4 | — | 8 | 6 | no | — |
| 238 | D4 | — | 8 | 12 | no | — |
| 239 | D4 | — | 16 | 24 | no | — |
| 240 | C2×C4 | — | 2 | 12 | no | — |
| 241 | C2×C4 | — | 4 | 12 | no | — |
| 242 | C2×C4 | — | 4 | 6 | no | — |
| 243 | C2×C4 | — | 8 | 6 | no | — |
| 244 | C2×C4 | — | 8 | 6 | no | — |
| 245 | C2×C4 | — | 8 | 6 | no | — |
| 246 | C2×C4 | — | 8 | 12 | no | — |
| 247 | C2×C4 | C2×#120 | 1 | 3 | yes | — |
| 248 | C2×C4 | C2×#121 | 1 | 3 | yes | — |
| 249 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 250 | C2×C4 | C2×#127 | 1 | 3 | yes | — |
| 251 | C2×C4 | C2×#127 | 1 | 3 | yes | — |
| 252 | C2×C4 | #535 | 1 | 3 | yes | — |
| 253 | C2×C4 | D42 | 1 | 3 | yes | — |
| 254 | C2×C4 | D42 | 1 | 6 | yes | — |
| 255 | C2×C4 | #550 | 1 | 3 | yes | — |
| 256 | C8 | — | 2 | 12 | no | — |
| 257 | C8 | — | 8 | 2 | no | — |
| 258 | C8 | #392 | 1 | 1 | yes | characteristic |
| 259 | C8 | #392 | 1 | 3 | yes | — |
| 260 | C8 | #392 | 1 | 3 | yes | — |
| 261 | C8 | #392 | 1 | 1 | yes | characteristic |
| 262 | C22 | — | 4 | 3 | no | — |
| 263 | C22 | — | 4 | 3 | no | — |
| 264 | C22 | — | 4 | 1 | no | — |
| 265 | C22 | — | 8 | 24 | no | — |
| 266 | C22 | — | 8 | 12 | no | — |
| 267 | C22 | — | 8 | 6 | no | — |
| 268 | C22 | #2433 | 1 | 3 | yes | — |
| 269 | C22 | #2541 | 1 | 3 | yes | — |
| 270 | C22 | #2657 | 1 | 1 | yes | characteristic |
| 271 | C4 | — | 4 | 12 | no | — |
| 272 | C4 | — | 8 | 6 | no | — |
| 273 | C4 | C2×#392 | 1 | 1 | yes | characteristic |
| 274 | C4 | #2433 | 1 | 3 | yes | — |
| 275 | C4 | #2541 | 1 | 3 | yes | — |
| 276 | C4 | #2867 | 1 | 1 | yes | characteristic |
| 277 | C2 | — | 2 | 3 | no | — |
| 278 | C2 | — | 2 | 3 | no | — |
| 279 | C2 | — | 2 | 1 | no | — |
| 280 | C2 | — | 16 | 6 | no | — |
| 281 | C2 | #15142 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2867 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15142 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ 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 |
|---|---|---|
| 1. #60643 | C2 | semidirect |
| 2. #23434 | C2 | semidirect |
| 3. #23414 | C2 | semidirect |
| 4. #21240 | C2 | semidirect |
| 5. #10699 | C2 | semidirect |
| 39. C22×Q16 | D4 | semidirect |
| 40. C22×SD16 | D4 | semidirect |
| 41. C22×D8 | D4 | semidirect |
| 68. C2×#97 | D4 | semidirect |
| 69. C2×C4×C8 | D4 | semidirect |
| 70. #338 | D4 | semidirect |
| 71. H(3,ℤ/4) | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #60643 | 2 |
| #23434 | 2 |
| #23414 | 2 |
| #21240 | 2 |
| #10699 | 2 |