Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1354 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 1354 conjugacy classes of subgroups — too many to draw. Only its 202 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 | C2×#2050 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #23438 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | C2×#1974 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #11099 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #10844 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#575 | C22 | 1 | 4 | yes | — |
| 7 | C22×#135 | C22 | 1 | 2 | yes | — |
| 8 | C22×#108 | C22 | 1 | 2 | yes | — |
| 9 | C2×#423 | C22 | 1 | 2 | yes | — |
| 10 | C22×#104 | C22 | 1 | 4 | yes | — |
| 11 | C2×#413 | C22 | 1 | 8 | yes | — |
| 12 | C2×#408 | C22 | 1 | 4 | yes | — |
| 13 | C22×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 14 | #2050 | — | 2 | 16 | no | — |
| 15 | #2020 | — | 2 | 4 | no | — |
| 16 | #2019 | — | 2 | 8 | no | — |
| 17 | #2018 | — | 2 | 4 | no | — |
| 18 | #1974 | — | 2 | 4 | no | — |
| 19 | #1974 | — | 2 | 4 | no | — |
| 20 | C2×#341 | C22 | 1 | 4 | yes | — |
| 21 | C2×#338 | C22 | 1 | 4 | yes | — |
| 22 | C22×#41 | C23 | 1 | 4 | yes | — |
| 23 | #575 | — | 2 | 8 | no | — |
| 24 | #575 | — | 4 | 4 | no | — |
| 25 | #575 | — | 4 | 8 | no | — |
| 26 | C2×#137 | — | 2 | 8 | no | — |
| 27 | C2×AGL(1,ℤ/8) | — | 2 | 8 | no | — |
| 28 | C2×#135 | — | 2 | 4 | no | — |
| 29 | C2×#135 | — | 2 | 4 | no | — |
| 30 | C2×#135 | — | 4 | 4 | no | — |
| 31 | C2×#135 | — | 4 | 4 | no | — |
| 32 | C2×#135 | — | 4 | 4 | no | — |
| 33 | C22×Q16 | D4 | 1 | 2 | yes | — |
| 34 | C22×SD16 | D4 | 1 | 4 | yes | — |
| 35 | C22×D8 | D4 | 1 | 2 | yes | — |
| 36 | C22×M16 | C23 | 1 | 4 | yes | — |
| 37 | C22×M16 | C23 | 1 | 4 | yes | — |
| 38 | C23×C8 | C23 | 1 | 2 | yes | — |
| 39 | C22×C42 | C23 | 1 | 1 | yes | characteristic |
| 40 | C2×#108 | — | 2 | 4 | no | — |
| 41 | C2×#108 | — | 4 | 4 | no | — |
| 42 | #423 | — | 4 | 2 | no | — |
| 43 | #423 | D4 | 1 | 8 | yes | — |
| 44 | C2×C4⋊C8 | — | 4 | 2 | no | — |
| 45 | C2×#104 | — | 2 | 16 | no | — |
| 46 | C2×#104 | — | 2 | 4 | no | — |
| 47 | C2×#104 | — | 2 | 4 | no | — |
| 48 | #413 | — | 2 | 16 | no | — |
| 49 | #413 | — | 4 | 8 | no | — |
| 50 | C2×#101 | — | 2 | 8 | no | — |
| 51 | C2×#100 | — | 2 | 8 | no | — |
| 52 | #408 | — | 2 | 8 | no | — |
| 53 | #408 | — | 2 | 4 | no | — |
| 54 | #408 | — | 2 | 4 | no | — |
| 55 | C2×C22⋊C8 | — | 2 | 8 | no | — |
| 56 | C2×C4×C8 | — | 4 | 2 | no | — |
| 57 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 58 | C2×C4×C8 | D4 | 1 | 2 | yes | — |
| 59 | #341 | — | 2 | 4 | no | — |
| 60 | #341 | — | 2 | 4 | no | — |
| 61 | #338 | — | 2 | 8 | no | — |
| 62 | C2×#41 | — | 2 | 4 | no | — |
| 63 | C2×#41 | — | 2 | 4 | no | — |
| 64 | C2×#41 | — | 2 | 8 | no | — |
| 65 | C2×#41 | — | 2 | 4 | no | — |
| 66 | C2×#41 | — | 2 | 4 | no | — |
| 67 | C2×#41 | — | 2 | 8 | no | — |
| 68 | C2×#41 | — | 2 | 8 | no | — |
| 69 | C2×#41 | — | 2 | 8 | no | — |
| 70 | C22×Q8 | C2×D4 | 1 | 4 | yes | — |
| 71 | C22×D4 | — | 2 | 4 | no | — |
| 72 | C22×D4 | C2×D4 | 1 | 4 | yes | — |
| 73 | C23×C4 | — | 2 | 1 | no | — |
| 74 | C23×C4 | — | 2 | 4 | no | — |
| 75 | C23×C4 | C24 | 1 | 1 | yes | characteristic |
| 76 | #137 | — | 4 | 16 | no | — |
| 77 | #137 | — | 4 | 16 | no | — |
| 78 | AGL(1,ℤ/8) | — | 4 | 16 | no | — |
| 79 | AGL(1,ℤ/8) | — | 4 | 16 | no | — |
| 80 | #135 | — | 4 | 8 | no | — |
| 81 | #135 | — | 4 | 8 | no | — |
| 82 | #135 | — | 4 | 16 | no | — |
| 83 | #135 | — | 4 | 16 | no | — |
| 84 | #135 | — | 4 | 16 | no | — |
| 85 | C2×Q16 | — | 2 | 4 | no | — |
| 86 | C2×Q16 | — | 4 | 4 | no | — |
| 87 | C2×Q16 | — | 4 | 4 | no | — |
| 88 | C2×SD16 | — | 2 | 8 | no | — |
| 89 | C2×SD16 | — | 4 | 8 | no | — |
| 90 | C2×SD16 | — | 4 | 8 | no | — |
| 91 | C2×D8 | — | 2 | 4 | no | — |
| 92 | C2×D8 | — | 4 | 4 | no | — |
| 93 | C2×D8 | — | 4 | 4 | no | — |
| 94 | C2×M16 | — | 2 | 4 | no | — |
| 95 | C2×M16 | — | 2 | 8 | no | — |
| 96 | C2×M16 | — | 2 | 8 | no | — |
| 97 | C2×M16 | — | 2 | 8 | no | — |
| 98 | C2×M16 | — | 2 | 4 | no | — |
| 99 | C2×M16 | — | 2 | 4 | no | — |
| 100 | C2×M16 | — | 2 | 8 | no | — |
| 101 | C2×M16 | C2×D4 | 1 | 8 | yes | — |
| 102 | C22×C8 | — | 2 | 4 | no | — |
| 103 | C22×C8 | — | 2 | 4 | no | — |
| 104 | C22×C8 | — | 2 | 4 | no | — |
| 105 | C22×C8 | — | 2 | 4 | no | — |
| 106 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 107 | C22×C8 | C2×D4 | 1 | 2 | yes | — |
| 108 | C22×C8 | C2×D4 | 1 | 4 | yes | — |
| 109 | C22×C8 | #41 | 1 | 4 | yes | — |
| 110 | C2×C42 | — | 2 | 2 | no | — |
| 111 | C2×C42 | — | 2 | 1 | no | — |
| 112 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 113 | C2×C42 | C2×D4 | 1 | 1 | yes | characteristic |
| 114 | C2×C42 | C2×D4 | 1 | 4 | yes | — |
| 115 | #108 | — | 4 | 8 | no | — |
| 116 | C4⋊C8 | — | 4 | 8 | no | — |
| 117 | #104 | — | 4 | 8 | no | — |
| 118 | #104 | — | 4 | 8 | no | — |
| 119 | #101 | — | 4 | 16 | no | — |
| 120 | #100 | — | 4 | 16 | no | — |
| 121 | C22⋊C8 | — | 4 | 8 | no | — |
| 122 | C22⋊C8 | — | 4 | 8 | no | — |
| 123 | C4×C8 | — | 4 | 2 | no | — |
| 124 | C4×C8 | — | 4 | 2 | no | — |
| 125 | C24 | — | 2 | 4 | no | — |
| 126 | C24 | 2−1+4 | 1 | 1 | yes | characteristic |
| 127 | #41 | — | 4 | 4 | no | — |
| 128 | #41 | — | 4 | 4 | no | — |
| 129 | #41 | — | 4 | 8 | no | — |
| 130 | #41 | — | 4 | 8 | no | — |
| 131 | #41 | — | 4 | 4 | no | — |
| 132 | #41 | — | 4 | 4 | no | — |
| 133 | #41 | — | 4 | 8 | no | — |
| 134 | #41 | — | 4 | 8 | no | — |
| 135 | #41 | — | 4 | 8 | no | — |
| 136 | #41 | — | 4 | 8 | no | — |
| 137 | C2×Q8 | — | 2 | 4 | no | — |
| 138 | C2×Q8 | — | 2 | 8 | no | — |
| 139 | C2×Q8 | — | 2 | 8 | no | — |
| 140 | C2×Q8 | — | 2 | 4 | no | — |
| 141 | C2×D4 | — | 2 | 4 | no | — |
| 142 | C2×D4 | — | 2 | 8 | no | — |
| 143 | C2×D4 | — | 2 | 8 | no | — |
| 144 | C2×D4 | — | 2 | 4 | no | — |
| 145 | C2×D4 | — | 4 | 8 | no | — |
| 146 | C2×D4 | — | 4 | 4 | no | — |
| 147 | C2×D4 | — | 4 | 4 | no | — |
| 148 | C2×D4 | — | 4 | 8 | no | — |
| 149 | C22×C4 | — | 2 | 1 | no | — |
| 150 | C22×C4 | — | 2 | 4 | no | — |
| 151 | C22×C4 | — | 2 | 1 | no | — |
| 152 | C22×C4 | — | 2 | 4 | no | — |
| 153 | C22×C4 | — | 4 | 4 | no | — |
| 154 | C22×C4 | — | 4 | 4 | no | — |
| 155 | C22×C4 | — | 4 | 8 | no | — |
| 156 | C22×C4 | — | 4 | 8 | no | — |
| 157 | C22×C4 | — | 4 | 4 | no | — |
| 158 | C22×C4 | C22×D4 | 1 | 1 | yes | characteristic |
| 159 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 160 | C22×C4 | C22×D4 | 1 | 2 | yes | — |
| 161 | C22×C4 | C2×#41 | 1 | 2 | yes | — |
| 162 | C22×C4 | H(5,𝔽2) | 1 | 2 | yes | — |
| 163 | C22×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 164 | C22×C4 | H(5,𝔽2) | 1 | 4 | yes | — |
| 165 | Q16 | — | 4 | 8 | no | — |
| 166 | Q16 | — | 4 | 16 | no | — |
| 167 | SD16 | — | 4 | 16 | no | — |
| 168 | SD16 | — | 4 | 32 | no | — |
| 169 | D8 | — | 4 | 8 | no | — |
| 170 | D8 | — | 4 | 16 | no | — |
| 171 | M16 | — | 2 | 16 | no | — |
| 172 | M16 | — | 4 | 8 | no | — |
| 173 | M16 | — | 4 | 4 | no | — |
| 174 | M16 | — | 4 | 4 | no | — |
| 175 | M16 | — | 4 | 8 | no | — |
| 176 | C2×C8 | — | 2 | 4 | no | — |
| 177 | C2×C8 | — | 2 | 4 | no | — |
| 178 | C2×C8 | — | 2 | 4 | no | — |
| 179 | C2×C8 | — | 2 | 4 | no | — |
| 180 | C2×C8 | — | 2 | 8 | no | — |
| 181 | C2×C8 | — | 2 | 8 | no | — |
| 182 | C2×C8 | — | 4 | 4 | no | — |
| 183 | C2×C8 | — | 4 | 4 | no | — |
| 184 | C2×C8 | — | 4 | 4 | no | — |
| 185 | C2×C8 | — | 4 | 8 | no | — |
| 186 | C2×C8 | #121 | 1 | 4 | yes | — |
| 187 | C2×C8 | #121 | 1 | 4 | yes | — |
| 188 | C42 | — | 2 | 1 | no | — |
| 189 | C42 | — | 2 | 1 | no | — |
| 190 | C42 | — | 2 | 4 | no | — |
| 191 | C42 | #127 | 1 | 2 | yes | — |
| 192 | C42 | #127 | 1 | 2 | yes | — |
| 193 | C23 | — | 4 | 2 | no | — |
| 194 | C23 | — | 4 | 4 | no | — |
| 195 | C23 | — | 4 | 8 | no | — |
| 196 | C23 | — | 8 | 8 | no | — |
| 197 | C23 | #537 | 1 | 1 | yes | characteristic |
| 198 | C23 | #537 | 1 | 2 | yes | — |
| 199 | C23 | #547 | 1 | 2 | yes | — |
| 200 | C23 | #562 | 1 | 2 | yes | — |
| 201 | Q8 | — | 4 | 4 | no | — |
| 202 | Q8 | — | 4 | 8 | no | — |
| 203 | Q8 | — | 4 | 4 | no | — |
| 204 | D4 | — | 4 | 4 | no | — |
| 205 | D4 | — | 4 | 8 | no | — |
| 206 | D4 | — | 4 | 4 | no | — |
| 207 | D4 | — | 8 | 4 | no | — |
| 208 | D4 | — | 8 | 4 | no | — |
| 209 | D4 | — | 8 | 8 | no | — |
| 210 | C2×C4 | — | 2 | 4 | no | — |
| 211 | C2×C4 | — | 2 | 2 | no | — |
| 212 | C2×C4 | — | 2 | 2 | no | — |
| 213 | C2×C4 | — | 4 | 8 | no | — |
| 214 | C2×C4 | — | 4 | 2 | no | — |
| 215 | C2×C4 | — | 4 | 8 | no | — |
| 216 | C2×C4 | — | 4 | 4 | no | — |
| 217 | C2×C4 | — | 8 | 4 | no | — |
| 218 | C2×C4 | — | 8 | 8 | no | — |
| 219 | C2×C4 | — | 8 | 4 | no | — |
| 220 | C2×C4 | C2×#121 | 1 | 2 | yes | — |
| 221 | C2×C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | #534 | 1 | 1 | yes | characteristic |
| 223 | C2×C4 | #534 | 1 | 4 | yes | — |
| 224 | C2×C4 | #534 | 1 | 2 | yes | — |
| 225 | C2×C4 | #535 | 1 | 2 | yes | — |
| 226 | C2×C4 | #538 | 1 | 4 | yes | — |
| 227 | C2×C4 | D42 | 1 | 1 | yes | characteristic |
| 228 | C2×C4 | D42 | 1 | 4 | yes | — |
| 229 | C2×C4 | #550 | 1 | 4 | yes | — |
| 230 | C2×C4 | #555 | 1 | 1 | yes | characteristic |
| 231 | C2×C4 | H(3,𝔽4) | 1 | 2 | yes | — |
| 232 | C8 | — | 2 | 4 | no | — |
| 233 | C8 | — | 2 | 4 | no | — |
| 234 | C8 | — | 4 | 8 | no | — |
| 235 | C8 | — | 8 | 4 | no | — |
| 236 | C22 | — | 2 | 2 | no | — |
| 237 | C22 | — | 2 | 4 | no | — |
| 238 | C22 | — | 4 | 2 | no | — |
| 239 | C22 | — | 4 | 2 | no | — |
| 240 | C22 | — | 8 | 8 | no | — |
| 241 | C22 | — | 8 | 4 | no | — |
| 242 | C22 | — | 16 | 8 | no | — |
| 243 | C22 | #2537 | 1 | 1 | yes | characteristic |
| 244 | C22 | #2656 | 1 | 1 | yes | characteristic |
| 245 | C22 | #2682 | 1 | 4 | yes | — |
| 246 | C22 | #2819 | 1 | 1 | yes | characteristic |
| 247 | C4 | — | 2 | 4 | no | — |
| 248 | C4 | — | 4 | 2 | no | — |
| 249 | C4 | — | 8 | 4 | no | — |
| 250 | C4 | #2541 | 1 | 2 | yes | — |
| 251 | C4 | #2617 | 1 | 2 | yes | — |
| 252 | C4 | #2679 | 1 | 2 | yes | — |
| 253 | C4 | #2733 | 1 | 2 | yes | — |
| 254 | C2 | — | 2 | 1 | no | — |
| 255 | C2 | — | 2 | 4 | no | — |
| 256 | C2 | — | 2 | 1 | no | — |
| 257 | C2 | — | 16 | 4 | no | — |
| 258 | C2 | #17068 | 1 | 1 | yes | characteristic |
| 259 | C2 | #28249 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2537 |
| 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 |
|---|---|---|
| 1. C2×#2050 | C2 | semidirect |
| 2. #23438 | C2 | semidirect |
| 3. C2×#1974 | C2 | semidirect |
| 4. #11099 | C2 | semidirect |
| 5. #10844 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2050 | 2 |
| #23438 | 2 |
| C2×#1974 | 2 |
| #11099 | 2 |
| #10844 | 2 |