Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3136 conjugacy classes of subgroups; only the 322 normal subgroups are shown.
This group has 3136 conjugacy classes of subgroups — too many to draw. Only its 322 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 | C22×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×#355 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#1888 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #12774 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 5 | C2×H(3,𝔽4) | — | 2 | 12 | no | — |
| 6 | C2×H(3,𝔽4) | — | 2 | 6 | no | — |
| 7 | C2×H(3,𝔽4) | C4 | 1 | 8 | yes | complemented |
| 8 | C2×H(3,𝔽4) | C22 | 1 | 4 | yes | — |
| 9 | C22×#124 | — | 2 | 3 | no | — |
| 10 | C22×#124 | C22 | 1 | 3 | yes | — |
| 11 | C22×#120 | — | 2 | 3 | no | — |
| 12 | C2×#519 | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×#418 | — | 2 | 3 | no | — |
| 14 | C22×#101 | C22 | 1 | 3 | yes | complemented |
| 15 | C2×#355 | — | 2 | 12 | no | — |
| 16 | C2×#355 | C22 | 1 | 12 | yes | complemented |
| 17 | #1888 | — | 2 | 6 | no | — |
| 18 | #1888 | C22 | 1 | 12 | yes | complemented |
| 19 | C2×#339 | — | 2 | 3 | no | — |
| 20 | C26 | — | 2 | 1 | no | — |
| 21 | C23×Q8 | — | 2 | 1 | no | — |
| 22 | C23×Q8 | C23 | 1 | 1 | yes | characteristic |
| 23 | C23×D4 | — | 2 | 3 | no | — |
| 24 | C23×D4 | D4 | 1 | 3 | yes | — |
| 25 | C22×M16 | — | 2 | 3 | no | — |
| 26 | H(3,𝔽4) | — | 2 | 48 | no | — |
| 27 | H(3,𝔽4) | — | 2 | 48 | no | — |
| 28 | H(3,𝔽4) | — | 2 | 24 | no | — |
| 29 | H(3,𝔽4) | C2×C4 | 1 | 16 | yes | complemented |
| 30 | C2×#124 | — | 2 | 24 | no | — |
| 31 | C2×#124 | — | 2 | 12 | no | — |
| 32 | C2×#124 | — | 2 | 12 | no | — |
| 33 | C2×#124 | — | 2 | 6 | no | — |
| 34 | C2×#124 | — | 2 | 24 | no | — |
| 35 | C2×#124 | — | 2 | 12 | no | — |
| 36 | C2×#124 | C2×C4 | 1 | 12 | yes | — |
| 37 | C2×#124 | D4 | 1 | 12 | yes | — |
| 38 | C2×#124 | D4 | 1 | 6 | yes | — |
| 39 | C2×#124 | C23 | 1 | 6 | yes | — |
| 40 | C2×#120 | — | 2 | 12 | no | — |
| 41 | C2×#120 | — | 2 | 24 | no | — |
| 42 | C2×#120 | — | 2 | 24 | no | — |
| 43 | C2×#120 | — | 2 | 12 | no | — |
| 44 | #519 | — | 2 | 3 | no | — |
| 45 | #519 | — | 4 | 2 | no | — |
| 46 | #519 | C23 | 1 | 2 | yes | complemented |
| 47 | C2×C4×Q8 | — | 4 | 2 | no | — |
| 48 | C2×#117 | — | 2 | 3 | no | — |
| 49 | C22×C22⋊C4 | — | 2 | 3 | no | — |
| 50 | C22×C22⋊C4 | — | 2 | 6 | no | — |
| 51 | C22×C42 | D4 | 1 | 3 | yes | — |
| 52 | #418 | — | 2 | 12 | no | — |
| 53 | #418 | — | 4 | 6 | no | — |
| 54 | C2×#102 | — | 4 | 6 | no | — |
| 55 | C2×#101 | — | 2 | 12 | no | — |
| 56 | C2×#101 | — | 2 | 3 | no | — |
| 57 | C2×#101 | C23 | 1 | 6 | yes | complemented |
| 58 | #355 | — | 2 | 24 | no | — |
| 59 | #339 | — | 2 | 12 | no | — |
| 60 | C25 | — | 2 | 4 | no | — |
| 61 | C25 | — | 2 | 8 | no | — |
| 62 | C25 | — | 2 | 3 | no | — |
| 63 | C25 | — | 8 | 8 | no | — |
| 64 | C25 | — | 8 | 1 | no | — |
| 65 | C25 | — | 8 | 1 | no | — |
| 66 | C25 | — | 8 | 2 | no | — |
| 67 | C22×Q8 | — | 2 | 4 | no | — |
| 68 | C22×Q8 | — | 2 | 8 | no | — |
| 69 | C22×Q8 | — | 4 | 1 | no | — |
| 70 | C22×Q8 | — | 4 | 1 | no | — |
| 71 | C22×Q8 | — | 4 | 2 | no | — |
| 72 | C22×Q8 | — | 8 | 4 | no | — |
| 73 | C22×Q8 | C22×C4 | 1 | 2 | yes | — |
| 74 | C22×Q8 | C2×D4 | 1 | 6 | yes | — |
| 75 | C22×Q8 | C2×D4 | 1 | 3 | yes | — |
| 76 | C22×Q8 | C24 | 1 | 1 | yes | characteristic |
| 77 | C22×D4 | — | 2 | 12 | no | — |
| 78 | C22×D4 | — | 2 | 24 | no | — |
| 79 | C22×D4 | — | 2 | 6 | no | — |
| 80 | C22×D4 | — | 2 | 3 | no | — |
| 81 | C22×D4 | — | 4 | 6 | no | — |
| 82 | C22×D4 | — | 4 | 3 | no | — |
| 83 | C22×D4 | — | 4 | 3 | no | — |
| 84 | C22×D4 | — | 8 | 12 | no | — |
| 85 | C22×D4 | C22⋊C4 | 1 | 12 | yes | — |
| 86 | C22×D4 | C2×D4 | 1 | 6 | yes | — |
| 87 | C23×C4 | — | 2 | 3 | no | — |
| 88 | C23×C4 | C2×D4 | 1 | 3 | yes | — |
| 89 | C2×M16 | — | 2 | 6 | no | — |
| 90 | C2×M16 | — | 4 | 3 | no | — |
| 91 | C2×M16 | — | 4 | 6 | no | — |
| 92 | C2×M16 | — | 4 | 6 | no | — |
| 93 | C22×C8 | — | 4 | 3 | no | — |
| 94 | #124 | — | 2 | 48 | no | — |
| 95 | #124 | — | 2 | 96 | no | — |
| 96 | #124 | — | 2 | 24 | no | — |
| 97 | #124 | — | 2 | 12 | no | — |
| 98 | #124 | — | 2 | 24 | no | — |
| 99 | #124 | — | 2 | 12 | no | — |
| 100 | #124 | — | 2 | 48 | no | — |
| 101 | #124 | C22⋊C4 | 1 | 24 | yes | — |
| 102 | #124 | C22×C4 | 1 | 12 | yes | — |
| 103 | #124 | C2×D4 | 1 | 12 | yes | — |
| 104 | #120 | — | 2 | 48 | no | — |
| 105 | #120 | — | 2 | 96 | no | — |
| 106 | #120 | — | 2 | 48 | no | — |
| 107 | C4×Q8 | — | 4 | 4 | no | — |
| 108 | C4×Q8 | — | 4 | 12 | no | — |
| 109 | #117 | — | 2 | 6 | no | — |
| 110 | #117 | — | 2 | 6 | no | — |
| 111 | #117 | — | 4 | 3 | no | — |
| 112 | #117 | — | 4 | 3 | no | — |
| 113 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 114 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 115 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 116 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 117 | C2×C22⋊C4 | — | 2 | 24 | no | — |
| 118 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 119 | C2×C22⋊C4 | — | 2 | 12 | no | — |
| 120 | C2×C22⋊C4 | — | 4 | 1 | no | — |
| 121 | C2×C42 | — | 2 | 6 | no | — |
| 122 | C2×C42 | — | 2 | 3 | no | — |
| 123 | C2×C42 | — | 4 | 3 | no | — |
| 124 | C2×C42 | C22⋊C4 | 1 | 12 | yes | — |
| 125 | C2×C42 | C2×D4 | 1 | 6 | yes | — |
| 126 | #102 | — | 4 | 24 | no | — |
| 127 | #101 | — | 2 | 12 | no | — |
| 128 | #101 | — | 4 | 6 | no | — |
| 129 | C24 | — | 2 | 16 | no | — |
| 130 | C24 | — | 2 | 6 | no | — |
| 131 | C24 | — | 2 | 12 | no | — |
| 132 | C24 | — | 4 | 24 | no | — |
| 133 | C24 | — | 4 | 3 | no | — |
| 134 | C24 | — | 4 | 3 | no | — |
| 135 | C24 | — | 4 | 6 | no | — |
| 136 | C24 | — | 8 | 16 | no | — |
| 137 | C24 | — | 8 | 32 | no | — |
| 138 | C24 | — | 8 | 4 | no | — |
| 139 | C24 | — | 8 | 4 | no | — |
| 140 | C24 | — | 8 | 8 | no | — |
| 141 | C24 | — | 8 | 8 | no | — |
| 142 | C24 | — | 8 | 16 | no | — |
| 143 | C24 | — | 8 | 32 | no | — |
| 144 | C24 | — | 8 | 16 | no | — |
| 145 | C24 | #120 | 1 | 1 | yes | characteristic |
| 146 | C2×Q8 | — | 2 | 16 | no | — |
| 147 | C2×Q8 | — | 4 | 3 | no | — |
| 148 | C2×Q8 | — | 4 | 1 | no | — |
| 149 | C2×Q8 | — | 4 | 2 | no | — |
| 150 | C2×Q8 | — | 4 | 2 | no | — |
| 151 | C2×Q8 | — | 4 | 3 | no | — |
| 152 | C2×Q8 | — | 4 | 6 | no | — |
| 153 | C2×Q8 | — | 4 | 6 | no | — |
| 154 | C2×Q8 | — | 4 | 1 | no | — |
| 155 | C2×Q8 | — | 8 | 8 | no | — |
| 156 | C2×Q8 | — | 8 | 16 | no | — |
| 157 | C2×Q8 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 158 | C2×Q8 | #120 | 1 | 6 | yes | — |
| 159 | C2×Q8 | C23×C4 | 1 | 1 | yes | characteristic |
| 160 | C2×Q8 | C22×D4 | 1 | 3 | yes | — |
| 161 | C2×D4 | — | 2 | 48 | no | — |
| 162 | C2×D4 | — | 2 | 6 | no | — |
| 163 | C2×D4 | — | 2 | 12 | no | — |
| 164 | C2×D4 | — | 4 | 6 | no | — |
| 165 | C2×D4 | — | 4 | 12 | no | — |
| 166 | C2×D4 | — | 4 | 12 | no | — |
| 167 | C2×D4 | — | 4 | 6 | no | — |
| 168 | C2×D4 | — | 8 | 3 | no | — |
| 169 | C2×D4 | — | 8 | 3 | no | — |
| 170 | C2×D4 | — | 8 | 6 | no | — |
| 171 | C2×D4 | — | 8 | 6 | no | — |
| 172 | C2×D4 | — | 8 | 24 | no | — |
| 173 | C2×D4 | — | 8 | 48 | no | — |
| 174 | C2×D4 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 175 | C22×C4 | — | 2 | 3 | no | — |
| 176 | C22×C4 | — | 2 | 3 | no | — |
| 177 | C22×C4 | — | 2 | 6 | no | — |
| 178 | C22×C4 | — | 2 | 6 | no | — |
| 179 | C22×C4 | — | 2 | 12 | no | — |
| 180 | C22×C4 | — | 4 | 12 | no | — |
| 181 | C22×C4 | — | 4 | 2 | no | — |
| 182 | C22×C4 | C2×C22⋊C4 | 1 | 6 | yes | — |
| 183 | C22×C4 | #120 | 1 | 6 | yes | — |
| 184 | C22×C4 | #120 | 1 | 3 | yes | — |
| 185 | C22×C4 | C22×D4 | 1 | 3 | yes | — |
| 186 | M16 | — | 4 | 6 | no | — |
| 187 | M16 | — | 4 | 6 | no | — |
| 188 | M16 | — | 8 | 6 | no | — |
| 189 | C2×C8 | — | 4 | 6 | no | — |
| 190 | C2×C8 | — | 8 | 6 | no | — |
| 191 | C4⋊C4 | — | 2 | 12 | no | — |
| 192 | C4⋊C4 | — | 4 | 6 | no | — |
| 193 | C22⋊C4 | — | 2 | 48 | no | — |
| 194 | C22⋊C4 | — | 2 | 24 | no | — |
| 195 | C22⋊C4 | — | 2 | 24 | no | — |
| 196 | C22⋊C4 | — | 2 | 48 | no | — |
| 197 | C22⋊C4 | — | 4 | 2 | no | — |
| 198 | C22⋊C4 | — | 4 | 2 | no | — |
| 199 | C42 | — | 2 | 6 | no | — |
| 200 | C42 | — | 2 | 12 | no | — |
| 201 | C42 | — | 4 | 6 | no | — |
| 202 | C42 | — | 4 | 6 | no | — |
| 203 | C42 | C2×C22⋊C4 | 1 | 12 | yes | — |
| 204 | C23 | — | 2 | 4 | no | — |
| 205 | C23 | — | 2 | 12 | no | — |
| 206 | C23 | — | 4 | 24 | no | — |
| 207 | C23 | — | 4 | 48 | no | — |
| 208 | C23 | — | 4 | 6 | no | — |
| 209 | C23 | — | 4 | 6 | no | — |
| 210 | C23 | — | 4 | 12 | no | — |
| 211 | C23 | — | 4 | 12 | no | — |
| 212 | C23 | — | 8 | 12 | no | — |
| 213 | C23 | — | 8 | 24 | no | — |
| 214 | C23 | — | 8 | 32 | no | — |
| 215 | C23 | — | 8 | 16 | no | — |
| 216 | C23 | — | 8 | 32 | no | — |
| 217 | C23 | — | 8 | 64 | no | — |
| 218 | C23 | — | 8 | 32 | no | — |
| 219 | C23 | — | 8 | 64 | no | — |
| 220 | C23 | — | 8 | 12 | no | — |
| 221 | C23 | #379 | 1 | 2 | yes | — |
| 222 | C23 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 223 | C23 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 224 | C23 | #458 | 1 | 2 | yes | — |
| 225 | C23 | C2×#120 | 1 | 1 | yes | characteristic |
| 226 | Q8 | — | 4 | 6 | no | — |
| 227 | Q8 | — | 4 | 6 | no | — |
| 228 | Q8 | — | 4 | 1 | no | — |
| 229 | Q8 | — | 4 | 3 | no | — |
| 230 | Q8 | — | 8 | 16 | no | — |
| 231 | D4 | — | 4 | 12 | no | — |
| 232 | D4 | — | 8 | 6 | no | — |
| 233 | D4 | — | 8 | 12 | no | — |
| 234 | D4 | — | 8 | 48 | no | — |
| 235 | C2×C4 | — | 2 | 3 | no | — |
| 236 | C2×C4 | — | 2 | 3 | no | — |
| 237 | C2×C4 | — | 2 | 3 | no | — |
| 238 | C2×C4 | — | 2 | 3 | no | — |
| 239 | C2×C4 | — | 2 | 6 | no | — |
| 240 | C2×C4 | — | 2 | 6 | no | — |
| 241 | C2×C4 | — | 2 | 6 | no | — |
| 242 | C2×C4 | — | 2 | 6 | no | — |
| 243 | C2×C4 | — | 2 | 12 | no | — |
| 244 | C2×C4 | — | 4 | 12 | no | — |
| 245 | C2×C4 | — | 4 | 24 | no | — |
| 246 | C2×C4 | — | 4 | 4 | no | — |
| 247 | C2×C4 | — | 8 | 4 | no | — |
| 248 | C2×C4 | #379 | 1 | 6 | yes | — |
| 249 | C2×C4 | C22×C22⋊C4 | 1 | 3 | yes | — |
| 250 | C2×C4 | C2×#120 | 1 | 3 | yes | — |
| 251 | C8 | — | 8 | 6 | no | — |
| 252 | C22 | — | 2 | 2 | no | — |
| 253 | C22 | — | 2 | 2 | no | — |
| 254 | C22 | — | 2 | 4 | no | — |
| 255 | C22 | — | 2 | 4 | no | — |
| 256 | C22 | — | 4 | 24 | no | — |
| 257 | C22 | — | 4 | 12 | no | — |
| 258 | C22 | — | 8 | 12 | no | — |
| 259 | C22 | — | 8 | 24 | no | — |
| 260 | C22 | — | 8 | 64 | no | — |
| 261 | C22 | — | 8 | 12 | no | — |
| 262 | C22 | — | 8 | 24 | no | — |
| 263 | C22 | #1790 | 1 | 2 | yes | — |
| 264 | C22 | #1790 | 1 | 2 | yes | — |
| 265 | C22 | #2202 | 1 | 4 | yes | — |
| 266 | C22 | C2×#379 | 1 | 1 | yes | characteristic |
| 267 | C22 | C2×B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 268 | C22 | C2×#458 | 1 | 1 | yes | characteristic |
| 269 | C4 | — | 2 | 3 | no | — |
| 270 | C4 | — | 2 | 3 | no | — |
| 271 | C4 | — | 2 | 6 | no | — |
| 272 | C4 | — | 4 | 12 | no | — |
| 273 | C4 | — | 8 | 4 | no | — |
| 274 | C2 | — | 2 | 1 | no | — |
| 275 | C2 | — | 2 | 1 | no | — |
| 276 | C2 | — | 2 | 2 | no | — |
| 277 | C2 | — | 8 | 12 | no | — |
| 278 | C2 | #12774 | 1 | 4 | yes | direct product factor |
| 279 | C2 | C2×#1790 | 1 | 1 | yes | characteristic |
| 280 | C2 | C2×#2202 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C22×H(3,𝔽4) | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#355 | C2 | semidirect |
| 3. C2×#1888 | C2 | semidirect |
| 4. #12774 | C2 | direct (N × H) |
| 7. C2×H(3,𝔽4) | C4 | semidirect |
| 12. C2×#519 | C22 | semidirect |
| 14. C22×#101 | C22 | semidirect |
| 16. C2×#355 | C22 | semidirect |
| 18. #1888 | C22 | semidirect |
| 29. H(3,𝔽4) | C2×C4 | semidirect |
| 46. #519 | C23 | semidirect |
| 57. C2×#101 | C23 | semidirect |
| 278. C2 | #12774 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×H(3,𝔽4) | 2 |
| C22×#355 | 2 |
| C2×#1888 | 2 |
| #12774 | 2 |