Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1227 conjugacy classes of subgroups; only the 190 normal subgroups are shown.
This group has 1227 conjugacy classes of subgroups — too many to draw. Only its 190 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 | C8×H(5,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #32975 | C2 | 1 | 2 | yes | maximal |
| 3 | #24268 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #9467 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #8609 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | C23×Q16 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | C4×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#467 | — | 2 | 3 | no | — |
| 9 | #2988 | — | 2 | 3 | no | — |
| 10 | #2972 | — | 2 | 3 | no | — |
| 11 | #2972 | C22 | 1 | 3 | yes | — |
| 12 | C8×#41 | — | 2 | 3 | no | — |
| 13 | C2×C8×D4 | — | 2 | 3 | no | — |
| 14 | C2×C8×D4 | C22 | 1 | 3 | yes | — |
| 15 | #2409 | C22 | 1 | 6 | yes | — |
| 16 | #2056 | — | 2 | 3 | no | — |
| 17 | #2001 | — | 2 | 3 | no | — |
| 18 | #1969 | — | 2 | 3 | no | — |
| 19 | #1867 | — | 2 | 6 | no | — |
| 20 | #1808 | C22 | 1 | 3 | yes | complemented |
| 21 | #1799 | C22 | 1 | 6 | yes | — |
| 22 | #1790 | — | 2 | 2 | no | — |
| 23 | #1787 | C22 | 1 | 6 | yes | complemented |
| 24 | #1289 | C22 | 1 | 6 | yes | complemented |
| 25 | C2×H(5,𝔽2) | D4 | 1 | 1 | yes | characteristic |
| 26 | C23×Q8 | C23 | 1 | 2 | yes | — |
| 27 | C22×Q16 | — | 2 | 3 | no | — |
| 28 | C22×Q16 | — | 2 | 3 | no | — |
| 29 | C22×Q16 | — | 4 | 4 | no | — |
| 30 | C23×C8 | — | 2 | 1 | no | — |
| 31 | C23×C8 | D4 | 1 | 3 | yes | — |
| 32 | C23×C8 | C23 | 1 | 1 | yes | characteristic |
| 33 | C2×C22⋊Q8 | — | 2 | 6 | no | — |
| 34 | #520 | — | 2 | 3 | no | — |
| 35 | #518 | — | 2 | 3 | no | — |
| 36 | #518 | D4 | 1 | 3 | yes | — |
| 37 | C4×#41 | — | 2 | 3 | no | — |
| 38 | C2×C4×D4 | — | 2 | 3 | no | — |
| 39 | C2×C4×D4 | C23 | 1 | 3 | yes | — |
| 40 | #467 | — | 4 | 12 | no | — |
| 41 | C8×Q8 | — | 2 | 1 | no | — |
| 42 | C8×Q8 | — | 2 | 3 | no | — |
| 43 | C8×D4 | — | 2 | 3 | no | — |
| 44 | C8×D4 | — | 2 | 6 | no | — |
| 45 | C8×D4 | — | 2 | 3 | no | — |
| 46 | C8×D4 | — | 2 | 6 | no | — |
| 47 | C8×D4 | — | 2 | 6 | no | — |
| 48 | C8×D4 | — | 2 | 6 | no | — |
| 49 | C8×D4 | — | 2 | 6 | no | — |
| 50 | #431 | — | 2 | 6 | no | — |
| 51 | #431 | — | 2 | 3 | no | — |
| 52 | C2×#107 | — | 2 | 3 | no | — |
| 53 | C2×C4⋊C8 | — | 2 | 3 | no | — |
| 54 | C2×C4⋊C8 | D4 | 1 | 3 | yes | — |
| 55 | C2×#103 | — | 2 | 6 | no | — |
| 56 | C2×#103 | — | 2 | 6 | no | — |
| 57 | C2×#99 | — | 2 | 6 | no | — |
| 58 | C2×C22⋊C8 | — | 2 | 3 | no | — |
| 59 | C2×C22⋊C8 | — | 2 | 3 | no | — |
| 60 | C2×C22⋊C8 | D4 | 1 | 3 | yes | — |
| 61 | C2×C22⋊C8 | C23 | 1 | 3 | yes | — |
| 62 | C2×C4×C8 | — | 2 | 3 | no | — |
| 63 | C2×C4×C8 | D4 | 1 | 3 | yes | — |
| 64 | #395 | — | 2 | 6 | no | — |
| 65 | #393 | — | 2 | 6 | no | — |
| 66 | #391 | — | 2 | 6 | no | — |
| 67 | #386 | — | 2 | 6 | no | — |
| 68 | #385 | — | 2 | 6 | no | — |
| 69 | #380 | C23 | 1 | 6 | yes | — |
| 70 | #342 | — | 2 | 6 | no | — |
| 71 | #340 | — | 2 | 3 | no | — |
| 72 | H(5,𝔽2) | — | 2 | 6 | no | — |
| 73 | H(5,𝔽2) | Q16 | 1 | 4 | yes | complemented |
| 74 | C2×#41 | — | 2 | 3 | no | — |
| 75 | C22×Q8 | — | 2 | 6 | no | — |
| 76 | C22×Q8 | — | 2 | 6 | no | — |
| 77 | C22×Q8 | — | 4 | 4 | no | — |
| 78 | C22×Q8 | — | 8 | 2 | no | — |
| 79 | C22×D4 | — | 2 | 3 | no | — |
| 80 | C22×D4 | C2×D4 | 1 | 3 | yes | — |
| 81 | C23×C4 | — | 2 | 1 | no | — |
| 82 | C23×C4 | — | 2 | 2 | no | — |
| 83 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 84 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 85 | C2×Q16 | — | 2 | 12 | no | — |
| 86 | C2×Q16 | — | 4 | 6 | no | — |
| 87 | C2×Q16 | — | 4 | 12 | no | — |
| 88 | C2×Q16 | — | 4 | 3 | no | — |
| 89 | C2×Q16 | — | 4 | 1 | no | — |
| 90 | C22×C8 | — | 2 | 3 | no | — |
| 91 | C22×C8 | — | 2 | 3 | no | — |
| 92 | C22×C8 | — | 2 | 3 | no | — |
| 93 | C22×C8 | — | 2 | 3 | no | — |
| 94 | C22×C8 | — | 4 | 2 | no | — |
| 95 | C22×C8 | — | 4 | 6 | no | — |
| 96 | C22×C8 | — | 8 | 2 | no | — |
| 97 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 98 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 99 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 100 | C22×C8 | C2×D4 | 1 | 3 | yes | — |
| 101 | C22⋊Q8 | — | 4 | 12 | no | — |
| 102 | C22⋊Q8 | — | 8 | 6 | no | — |
| 103 | C4×Q8 | — | 2 | 1 | no | — |
| 104 | C4×Q8 | — | 2 | 3 | no | — |
| 105 | C4×D4 | — | 2 | 3 | no | — |
| 106 | C4×D4 | — | 2 | 6 | no | — |
| 107 | C4×D4 | — | 2 | 3 | no | — |
| 108 | C4×D4 | — | 2 | 6 | no | — |
| 109 | C4×D4 | — | 2 | 6 | no | — |
| 110 | C4×D4 | — | 2 | 6 | no | — |
| 111 | C4×D4 | — | 2 | 6 | no | — |
| 112 | #117 | — | 2 | 6 | no | — |
| 113 | #117 | — | 2 | 3 | no | — |
| 114 | C2×C4⋊C4 | — | 2 | 3 | no | — |
| 115 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 116 | C2×C4⋊C4 | — | 4 | 6 | no | — |
| 117 | C2×C4⋊C4 | #41 | 1 | 3 | yes | — |
| 118 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 119 | C2×C22⋊C4 | — | 2 | 3 | no | — |
| 120 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 121 | C2×C22⋊C4 | — | 4 | 6 | no | — |
| 122 | C2×C22⋊C4 | C2×D4 | 1 | 3 | yes | — |
| 123 | C2×C22⋊C4 | #41 | 1 | 3 | yes | — |
| 124 | C2×C42 | — | 2 | 3 | no | — |
| 125 | C2×C42 | C2×D4 | 1 | 3 | yes | — |
| 126 | #107 | — | 4 | 6 | no | — |
| 127 | C4⋊C8 | — | 2 | 3 | no | — |
| 128 | C4⋊C8 | — | 2 | 3 | no | — |
| 129 | C4⋊C8 | — | 2 | 6 | no | — |
| 130 | #103 | — | 4 | 12 | no | — |
| 131 | #103 | — | 8 | 6 | no | — |
| 132 | #99 | — | 4 | 12 | no | — |
| 133 | C22⋊C8 | — | 2 | 6 | no | — |
| 134 | C22⋊C8 | — | 2 | 6 | no | — |
| 135 | C22⋊C8 | — | 2 | 6 | no | — |
| 136 | C4×C8 | — | 2 | 6 | no | — |
| 137 | C4×C8 | — | 2 | 3 | no | — |
| 138 | C4×C8 | — | 2 | 3 | no | — |
| 139 | #95 | — | 2 | 6 | no | — |
| 140 | #95 | — | 4 | 6 | no | — |
| 141 | #95 | — | 4 | 6 | no | — |
| 142 | C24 | — | 2 | 1 | no | — |
| 143 | C24 | #121 | 1 | 3 | yes | — |
| 144 | C24 | C22×D4 | 1 | 1 | yes | characteristic |
| 145 | #41 | — | 2 | 12 | no | — |
| 146 | #41 | — | 2 | 12 | no | — |
| 147 | C2×Q8 | — | 2 | 1 | no | — |
| 148 | C2×Q8 | — | 2 | 12 | no | — |
| 149 | C2×Q8 | — | 4 | 6 | no | — |
| 150 | C2×Q8 | — | 4 | 6 | no | — |
| 151 | C2×Q8 | — | 4 | 6 | no | — |
| 152 | C2×Q8 | — | 4 | 2 | no | — |
| 153 | C2×Q8 | — | 4 | 6 | no | — |
| 154 | C2×Q8 | — | 8 | 12 | no | — |
| 155 | C2×D4 | — | 2 | 3 | no | — |
| 156 | C2×D4 | — | 2 | 12 | no | — |
| 157 | C2×D4 | — | 2 | 6 | no | — |
| 158 | C2×D4 | — | 2 | 6 | no | — |
| 159 | C2×D4 | — | 2 | 12 | no | — |
| 160 | C2×D4 | C2×Q16 | 1 | 6 | yes | — |
| 161 | C2×D4 | #135 | 1 | 6 | yes | — |
| 162 | C22×C4 | — | 2 | 3 | no | — |
| 163 | C22×C4 | — | 2 | 3 | no | — |
| 164 | C22×C4 | — | 2 | 3 | no | — |
| 165 | C22×C4 | — | 2 | 3 | no | — |
| 166 | C22×C4 | — | 2 | 3 | no | — |
| 167 | C22×C4 | — | 4 | 1 | no | — |
| 168 | C22×C4 | — | 4 | 1 | no | — |
| 169 | C22×C4 | — | 4 | 3 | no | — |
| 170 | C22×C4 | — | 4 | 3 | no | — |
| 171 | C22×C4 | — | 4 | 6 | no | — |
| 172 | C22×C4 | — | 4 | 6 | no | — |
| 173 | C22×C4 | — | 8 | 4 | no | — |
| 174 | C22×C4 | — | 8 | 1 | no | — |
| 175 | C22×C4 | — | 8 | 1 | no | — |
| 176 | C22×C4 | #121 | 1 | 3 | yes | — |
| 177 | C22×C4 | #121 | 1 | 3 | yes | — |
| 178 | C22×C4 | #121 | 1 | 3 | yes | — |
| 179 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 180 | C22×C4 | C2×#41 | 1 | 3 | yes | — |
| 181 | Q16 | — | 4 | 12 | no | — |
| 182 | Q16 | — | 4 | 4 | no | — |
| 183 | C2×C8 | — | 2 | 6 | no | — |
| 184 | C2×C8 | — | 2 | 3 | no | — |
| 185 | C2×C8 | — | 2 | 3 | no | — |
| 186 | C2×C8 | — | 4 | 3 | no | — |
| 187 | C2×C8 | — | 4 | 6 | no | — |
| 188 | C2×C8 | — | 4 | 3 | no | — |
| 189 | C2×C8 | — | 4 | 3 | no | — |
| 190 | C2×C8 | — | 8 | 3 | no | — |
| 191 | C2×C8 | — | 8 | 3 | no | — |
| 192 | C2×C8 | — | 8 | 3 | no | — |
| 193 | C2×C8 | #120 | 1 | 3 | yes | — |
| 194 | C2×C8 | #120 | 1 | 1 | yes | characteristic |
| 195 | C4⋊C4 | — | 2 | 3 | no | — |
| 196 | C4⋊C4 | — | 2 | 3 | no | — |
| 197 | C4⋊C4 | — | 2 | 6 | no | — |
| 198 | C4⋊C4 | — | 4 | 6 | no | — |
| 199 | C4⋊C4 | — | 4 | 6 | no | — |
| 200 | C4⋊C4 | — | 8 | 12 | no | — |
| 201 | C22⋊C4 | — | 2 | 6 | no | — |
| 202 | C22⋊C4 | — | 2 | 6 | no | — |
| 203 | C22⋊C4 | — | 2 | 6 | no | — |
| 204 | C22⋊C4 | — | 4 | 12 | no | — |
| 205 | C22⋊C4 | — | 8 | 12 | no | — |
| 206 | C42 | — | 2 | 6 | no | — |
| 207 | C42 | — | 2 | 3 | no | — |
| 208 | C42 | — | 2 | 3 | no | — |
| 209 | C23 | — | 2 | 4 | no | — |
| 210 | C23 | — | 2 | 3 | no | — |
| 211 | C23 | — | 2 | 3 | no | — |
| 212 | C23 | — | 4 | 1 | no | — |
| 213 | C23 | — | 4 | 1 | no | — |
| 214 | C23 | — | 4 | 3 | no | — |
| 215 | C23 | — | 4 | 3 | no | — |
| 216 | C23 | — | 8 | 1 | no | — |
| 217 | C23 | — | 8 | 1 | no | — |
| 218 | C23 | #467 | 1 | 6 | yes | complemented |
| 219 | C23 | C2×#121 | 1 | 3 | yes | — |
| 220 | C23 | C22×Q16 | 1 | 1 | yes | characteristic |
| 221 | C23 | C2×#135 | 1 | 3 | yes | — |
| 222 | Q8 | — | 2 | 4 | no | — |
| 223 | Q8 | — | 4 | 12 | no | — |
| 224 | Q8 | — | 4 | 4 | no | — |
| 225 | Q8 | — | 8 | 6 | no | — |
| 226 | Q8 | — | 8 | 2 | no | — |
| 227 | D4 | — | 2 | 12 | no | — |
| 228 | D4 | — | 2 | 12 | no | — |
| 229 | D4 | — | 2 | 12 | no | — |
| 230 | C2×C4 | — | 2 | 12 | no | — |
| 231 | C2×C4 | — | 2 | 3 | no | — |
| 232 | C2×C4 | — | 2 | 3 | no | — |
| 233 | C2×C4 | — | 2 | 3 | no | — |
| 234 | C2×C4 | — | 2 | 3 | no | — |
| 235 | C2×C4 | — | 2 | 3 | no | — |
| 236 | C2×C4 | — | 2 | 3 | no | — |
| 237 | C2×C4 | — | 4 | 3 | no | — |
| 238 | C2×C4 | — | 4 | 3 | no | — |
| 239 | C2×C4 | — | 4 | 3 | no | — |
| 240 | C2×C4 | — | 4 | 3 | no | — |
| 241 | C2×C4 | — | 4 | 3 | no | — |
| 242 | C2×C4 | — | 4 | 12 | no | — |
| 243 | C2×C4 | — | 8 | 3 | no | — |
| 244 | C2×C4 | — | 8 | 3 | no | — |
| 245 | C2×C4 | — | 8 | 3 | no | — |
| 246 | C2×C4 | — | 8 | 6 | no | — |
| 247 | C2×C4 | — | 8 | 2 | no | — |
| 248 | C2×C4 | — | 8 | 6 | no | — |
| 249 | C2×C4 | — | 8 | 6 | no | — |
| 250 | C2×C4 | — | 8 | 6 | no | — |
| 251 | C2×C4 | #467 | 1 | 6 | yes | complemented |
| 252 | C2×C4 | C2×#120 | 1 | 1 | yes | characteristic |
| 253 | C2×C4 | #525 | 1 | 3 | yes | — |
| 254 | C8 | — | 2 | 3 | no | — |
| 255 | C8 | — | 4 | 3 | no | — |
| 256 | C8 | — | 4 | 3 | no | — |
| 257 | C8 | B(4,𝔽2) | 1 | 2 | yes | — |
| 258 | C22 | — | 2 | 3 | no | — |
| 259 | C22 | — | 2 | 3 | no | — |
| 260 | C22 | — | 2 | 6 | no | — |
| 261 | C22 | — | 4 | 1 | no | — |
| 262 | C22 | — | 4 | 3 | no | — |
| 263 | C22 | — | 4 | 6 | no | — |
| 264 | C22 | — | 4 | 3 | no | — |
| 265 | C22 | — | 4 | 3 | no | — |
| 266 | C22 | — | 8 | 3 | no | — |
| 267 | C22 | — | 8 | 4 | no | — |
| 268 | C22 | #2847 | 1 | 1 | yes | characteristic |
| 269 | C22 | C2×#467 | 1 | 3 | yes | — |
| 270 | C22 | #3050 | 1 | 3 | yes | — |
| 271 | C4 | — | 2 | 3 | no | — |
| 272 | C4 | — | 2 | 6 | no | — |
| 273 | C4 | — | 4 | 3 | no | — |
| 274 | C4 | — | 4 | 3 | no | — |
| 275 | C4 | — | 8 | 4 | no | — |
| 276 | C4 | — | 16 | 6 | no | — |
| 277 | C4 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 278 | C4 | #3027 | 1 | 1 | yes | characteristic |
| 279 | C2 | — | 2 | 3 | no | — |
| 280 | C2 | — | 2 | 3 | no | — |
| 281 | C2 | — | 4 | 6 | no | — |
| 282 | C2 | #32926 | 1 | 1 | yes | characteristic |
| 283 | C2 | #32931 | 1 | 1 | yes | characteristic |
| 284 | C2 | #32949 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×H(5,𝔽2) | D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #24268 | C2 | semidirect |
| 4. #9467 | C2 | semidirect |
| 5. #8609 | C2 | semidirect |
| 6. C23×Q16 | C22 | semidirect |
| 20. #1808 | C22 | semidirect |
| 23. #1787 | C22 | semidirect |
| 24. #1289 | C22 | semidirect |
| 73. H(5,𝔽2) | Q16 | semidirect |
| 218. C23 | #467 | semidirect |
| 251. C2×C4 | #467 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C8×H(5,𝔽2) | 2 |
| #32975 | 2 |
| #24268 | 2 |
| #9467 | 2 |
| #8609 | 2 |