Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1430 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 1430 conjugacy classes of subgroups — too many to draw. Only its 121 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×#5925 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C4⋊#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #85529 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #85410 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #85409 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | #85407 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | #85344 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C4×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 9 | #61093 | — | 3 | 1 | no | maximal, Sylow |
| 10 | C2×#994 | C22 | 1 | 2 | yes | complemented |
| 11 | #5925 | — | 2 | 8 | no | — |
| 12 | C2×#986 | C22 | 1 | 1 | yes | characteristic |
| 13 | C2×#986 | C22 | 1 | 4 | yes | complemented |
| 14 | C2×#985 | C22 | 1 | 2 | yes | complemented |
| 15 | C2×#985 | C22 | 1 | 4 | yes | complemented |
| 16 | #5726 | — | 4 | 1 | no | maximal |
| 17 | C4⋊(#41⋊C3) | C22 | 1 | 2 | yes | complemented |
| 18 | C4⋊(#41⋊C3) | C22 | 1 | 2 | yes | complemented |
| 19 | C4×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 20 | #5427 | C22 | 1 | 4 | yes | complemented |
| 21 | #5426 | C22 | 1 | 1 | yes | characteristic, complemented |
| 22 | #5425 | C22 | 1 | 4 | yes | complemented |
| 23 | #5395 | C22 | 1 | 2 | yes | complemented |
| 24 | #5394 | C22 | 1 | 2 | yes | complemented |
| 25 | C4⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 26 | C4×GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 27 | C4⋊#226 | C22 | 1 | 1 | yes | characteristic, complemented |
| 28 | C2×#576 | — | 3 | 2 | no | — |
| 29 | #3498 | D3 | 1 | 1 | yes | characteristic, complemented |
| 30 | #3496 | — | 3 | 1 | no | — |
| 31 | #3284 | — | 3 | 2 | no | — |
| 32 | D4×D8 | — | 3 | 2 | no | — |
| 33 | #3274 | — | 3 | 2 | no | — |
| 34 | #3272 | — | 3 | 2 | no | — |
| 35 | #3268 | — | 3 | 4 | no | — |
| 36 | #3264 | — | 3 | 4 | no | — |
| 37 | #3151 | — | 3 | 1 | no | — |
| 38 | #3062 | — | 3 | 2 | no | — |
| 39 | #3035 | — | 3 | 1 | no | — |
| 40 | C2×#459 | — | 3 | 1 | no | — |
| 41 | #3012 | — | 3 | 2 | no | — |
| 42 | #3008 | — | 3 | 2 | no | — |
| 43 | C4×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 44 | #2935 | — | 3 | 1 | no | — |
| 45 | C2×#239 | — | 4 | 2 | no | — |
| 46 | #994 | — | 2 | 4 | no | — |
| 47 | C2×#41⋊C3 | C23 | 1 | 4 | yes | — |
| 48 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 49 | #986 | — | 2 | 8 | no | — |
| 50 | #986 | D4 | 1 | 4 | yes | complemented |
| 51 | #985 | — | 2 | 4 | no | — |
| 52 | #985 | — | 2 | 8 | no | — |
| 53 | C2×GL(2,𝔽3) | C23 | 1 | 2 | yes | — |
| 54 | C2×GL(2,𝔽3) | C23 | 1 | 2 | yes | — |
| 55 | C2×#226 | C23 | 1 | 2 | yes | — |
| 56 | C3×C4⋊Q8 | — | 4 | 1 | no | — |
| 57 | #946 | — | 4 | 2 | no | — |
| 58 | #895 | — | 4 | 2 | no | — |
| 59 | #894 | — | 4 | 4 | no | — |
| 60 | D3×C4⋊C4 | — | 4 | 2 | no | — |
| 61 | C4×D12 | — | 4 | 1 | no | — |
| 62 | C4×D12 | — | 4 | 1 | no | — |
| 63 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 64 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 65 | #859 | C23 | 1 | 2 | yes | — |
| 66 | C2×H(5,𝔽2) | — | 3 | 2 | no | — |
| 67 | C2×H(5,𝔽2) | C2×D3 | 1 | 2 | yes | — |
| 68 | #576 | — | 6 | 8 | no | — |
| 69 | #576 | — | 6 | 8 | no | — |
| 70 | C2×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 71 | C2×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 72 | C2×AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 73 | C2×AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 74 | C2×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 75 | C2×#135 | — | 3 | 2 | no | — |
| 76 | C2×#135 | — | 3 | 4 | no | — |
| 77 | C22×D8 | — | 3 | 2 | no | — |
| 78 | C22×D8 | — | 3 | 4 | no | — |
| 79 | C2×#131 | — | 3 | 2 | no | — |
| 80 | #550 | C2×D3 | 1 | 2 | yes | — |
| 81 | #550 | C2×D3 | 1 | 2 | yes | — |
| 82 | #547 | — | 3 | 2 | no | — |
| 83 | #547 | — | 6 | 1 | no | — |
| 84 | #546 | — | 3 | 4 | no | — |
| 85 | #546 | — | 6 | 2 | no | — |
| 86 | D42 | — | 3 | 2 | no | — |
| 87 | D42 | — | 3 | 2 | no | — |
| 88 | D42 | — | 3 | 2 | no | — |
| 89 | D42 | — | 6 | 1 | no | — |
| 90 | #537 | — | 3 | 2 | no | — |
| 91 | #535 | — | 3 | 2 | no | — |
| 92 | #534 | — | 6 | 2 | no | — |
| 93 | #532 | — | 3 | 1 | no | — |
| 94 | #532 | — | 3 | 1 | no | — |
| 95 | C2×#127 | — | 3 | 1 | no | — |
| 96 | C2×#121 | — | 6 | 2 | no | — |
| 97 | C4×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 98 | C2×C4×D4 | — | 3 | 1 | no | — |
| 99 | #496 | — | 3 | 4 | no | — |
| 100 | #493 | — | 3 | 2 | no | — |
| 101 | #492 | — | 3 | 2 | no | — |
| 102 | #469 | — | 3 | 4 | no | — |
| 103 | #468 | — | 3 | 4 | no | — |
| 104 | #466 | — | 3 | 4 | no | — |
| 105 | #465 | — | 3 | 4 | no | — |
| 106 | #464 | — | 3 | 2 | no | — |
| 107 | #463 | — | 3 | 2 | no | — |
| 108 | #462 | — | 3 | 1 | no | — |
| 109 | #461 | — | 3 | 1 | no | — |
| 110 | #460 | — | 3 | 1 | no | — |
| 111 | #459 | — | 3 | 4 | no | — |
| 112 | #459 | — | 3 | 2 | no | — |
| 113 | #459 | — | 3 | 1 | no | — |
| 114 | #459 | — | 3 | 2 | no | — |
| 115 | #452 | — | 3 | 4 | no | — |
| 116 | #450 | — | 3 | 4 | no | — |
| 117 | #449 | — | 3 | 4 | no | — |
| 118 | #449 | — | 3 | 4 | no | — |
| 119 | #449 | — | 3 | 4 | no | — |
| 120 | #447 | — | 3 | 4 | no | — |
| 121 | #447 | — | 3 | 4 | no | — |
| 122 | #447 | — | 3 | 4 | no | — |
| 123 | #442 | — | 3 | 2 | no | — |
| 124 | #440 | — | 3 | 2 | no | — |
| 125 | C4×SD16 | — | 3 | 2 | no | — |
| 126 | C4×D8 | — | 3 | 2 | no | — |
| 127 | #436 | — | 3 | 2 | no | — |
| 128 | #435 | — | 3 | 4 | no | — |
| 129 | C8×D4 | — | 3 | 2 | no | — |
| 130 | #428 | — | 3 | 1 | no | — |
| 131 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 132 | #418 | — | 3 | 1 | no | — |
| 133 | #417 | — | 3 | 1 | no | — |
| 134 | #416 | — | 3 | 2 | no | — |
| 135 | C2×#102 | — | 3 | 2 | no | — |
| 136 | #408 | — | 3 | 2 | no | — |
| 137 | C4×M16 | — | 3 | 1 | no | — |
| 138 | C2×C3×Q8 | — | 4 | 2 | no | — |
| 139 | #239 | — | 8 | 8 | no | — |
| 140 | C2×D12 | — | 4 | 1 | no | — |
| 141 | C2×D12 | — | 4 | 1 | no | — |
| 142 | C2×D12 | — | 4 | 4 | no | — |
| 143 | C2×C4×D3 | — | 4 | 2 | no | — |
| 144 | C2×C4×D3 | — | 4 | 2 | no | — |
| 145 | C2×C4×D3 | — | 4 | 4 | no | — |
| 146 | #41⋊C3 | — | 2 | 4 | no | — |
| 147 | #41⋊C3 | C2×D4 | 1 | 2 | yes | complemented |
| 148 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 149 | GL(2,𝔽3) | — | 2 | 2 | no | — |
| 150 | GL(2,𝔽3) | C2×D4 | 1 | 4 | yes | complemented |
| 151 | #226 | — | 2 | 2 | no | — |
| 152 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 153 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 154 | C3×C42 | — | 4 | 1 | no | — |
| 155 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 156 | (C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 157 | (C2×D3)⋊C4 | — | 4 | 4 | no | — |
| 158 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 159 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 160 | Dic3⋊C4 | — | 4 | 4 | no | — |
| 161 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 162 | H(5,𝔽2) | — | 6 | 8 | no | — |
| 163 | H(5,𝔽2) | — | 6 | 4 | no | — |
| 164 | H(5,𝔽2) | — | 12 | 4 | no | — |
| 165 | C2×#41 | — | 3 | 2 | no | — |
| 166 | C2×#41 | — | 3 | 2 | no | — |
| 167 | C2×#41 | — | 3 | 4 | no | — |
| 168 | C2×#41 | — | 6 | 2 | no | — |
| 169 | C2×#41 | C22×D3 | 1 | 4 | yes | — |
| 170 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 171 | C22×D4 | — | 3 | 1 | no | — |
| 172 | C22×D4 | — | 3 | 2 | no | — |
| 173 | C22×D4 | — | 3 | 4 | no | — |
| 174 | C22×D4 | — | 3 | 4 | no | — |
| 175 | C22×D4 | — | 6 | 1 | no | — |
| 176 | C22×D4 | — | 6 | 1 | no | — |
| 177 | C22×D4 | — | 6 | 4 | no | — |
| 178 | C22×D4 | — | 6 | 2 | no | — |
| 179 | C22×D4 | — | 6 | 2 | no | — |
| 180 | C23×C4 | — | 6 | 1 | no | — |
| 181 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 182 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 183 | AGL(1,ℤ/8) | — | 6 | 8 | no | — |
| 184 | AGL(1,ℤ/8) | — | 6 | 16 | no | — |
| 185 | AGL(1,ℤ/8) | — | 6 | 8 | no | — |
| 186 | AGL(1,ℤ/8) | — | 6 | 8 | no | — |
| 187 | AGL(1,ℤ/8) | — | 12 | 4 | no | — |
| 188 | #135 | — | 6 | 8 | no | — |
| 189 | #135 | — | 6 | 8 | no | — |
| 190 | #135 | — | 6 | 4 | no | — |
| 191 | #135 | — | 6 | 4 | no | — |
| 192 | C2×Q16 | — | 3 | 2 | no | — |
| 193 | C2×SD16 | — | 3 | 2 | no | — |
| 194 | C2×SD16 | — | 3 | 2 | no | — |
| 195 | C2×SD16 | — | 3 | 4 | no | — |
| 196 | C2×SD16 | — | 3 | 2 | no | — |
| 197 | C2×SD16 | — | 3 | 4 | no | — |
| 198 | C2×D8 | — | 3 | 2 | no | — |
| 199 | C2×D8 | — | 3 | 2 | no | — |
| 200 | C2×D8 | — | 3 | 4 | no | — |
| 201 | C2×D8 | — | 3 | 4 | no | — |
| 202 | C2×D8 | — | 3 | 4 | no | — |
| 203 | C2×D8 | — | 3 | 4 | no | — |
| 204 | C2×D8 | — | 6 | 16 | no | — |
| 205 | C2×D8 | — | 6 | 8 | no | — |
| 206 | #131 | — | 6 | 8 | no | — |
| 207 | C2×M16 | — | 3 | 1 | no | — |
| 208 | C2×M16 | — | 3 | 4 | no | — |
| 209 | C2×M16 | — | 3 | 1 | no | — |
| 210 | C22×C8 | — | 3 | 2 | no | — |
| 211 | C22×C8 | — | 3 | 4 | no | — |
| 212 | C4⋊Q8 | — | 3 | 1 | no | — |
| 213 | C4⋊Q8 | S4 | 1 | 1 | yes | characteristic |
| 214 | #127 | — | 3 | 2 | no | — |
| 215 | #127 | — | 3 | 2 | no | — |
| 216 | #127 | — | 3 | 1 | no | — |
| 217 | #127 | — | 3 | 1 | no | — |
| 218 | #127 | — | 6 | 1 | no | — |
| 219 | #124 | — | 3 | 2 | no | — |
| 220 | #124 | — | 3 | 2 | no | — |
| 221 | #123 | — | 6 | 4 | no | — |
| 222 | #123 | — | 6 | 4 | no | — |
| 223 | C22⋊Q8 | — | 3 | 4 | no | — |
| 224 | C22⋊Q8 | — | 6 | 2 | no | — |
| 225 | #121 | — | 3 | 2 | no | — |
| 226 | #121 | — | 3 | 2 | no | — |
| 227 | #121 | — | 3 | 4 | no | — |
| 228 | #121 | — | 3 | 4 | no | — |
| 229 | #121 | — | 3 | 4 | no | — |
| 230 | #121 | — | 6 | 4 | no | — |
| 231 | #121 | — | 6 | 1 | no | — |
| 232 | #121 | — | 6 | 1 | no | — |
| 233 | #121 | — | 6 | 4 | no | — |
| 234 | #121 | — | 6 | 2 | no | — |
| 235 | #121 | — | 6 | 2 | no | — |
| 236 | #121 | — | 6 | 2 | no | — |
| 237 | #120 | — | 6 | 4 | no | — |
| 238 | #120 | — | 6 | 4 | no | — |
| 239 | #120 | — | 6 | 4 | no | — |
| 240 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 241 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 242 | C4×D4 | — | 3 | 2 | no | — |
| 243 | C4×D4 | — | 3 | 2 | no | — |
| 244 | C4×D4 | — | 3 | 1 | no | — |
| 245 | C4×D4 | — | 3 | 1 | no | — |
| 246 | C4×D4 | — | 3 | 2 | no | — |
| 247 | C4×D4 | — | 3 | 2 | no | — |
| 248 | C4×D4 | — | 6 | 1 | no | — |
| 249 | C4×D4 | — | 6 | 1 | no | — |
| 250 | C4×D4 | — | 6 | 1 | no | — |
| 251 | C4×D4 | — | 6 | 1 | no | — |
| 252 | #117 | — | 3 | 1 | no | — |
| 253 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 254 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 255 | C2×C22⋊C4 | — | 6 | 4 | no | — |
| 256 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 257 | C2×C42 | — | 3 | 1 | no | — |
| 258 | #107 | — | 3 | 2 | no | — |
| 259 | #106 | — | 3 | 2 | no | — |
| 260 | C4⋊C8 | — | 3 | 1 | no | — |
| 261 | C4⋊C8 | — | 3 | 2 | no | — |
| 262 | C4⋊C8 | — | 3 | 1 | no | — |
| 263 | #103 | — | 3 | 2 | no | — |
| 264 | #103 | — | 3 | 2 | no | — |
| 265 | #103 | — | 3 | 2 | no | — |
| 266 | #102 | — | 3 | 4 | no | — |
| 267 | #102 | — | 3 | 4 | no | — |
| 268 | #102 | — | 3 | 2 | no | — |
| 269 | #102 | — | 3 | 2 | no | — |
| 270 | #102 | — | 3 | 2 | no | — |
| 271 | #102 | — | 3 | 2 | no | — |
| 272 | #102 | — | 3 | 2 | no | — |
| 273 | C22⋊C8 | — | 3 | 4 | no | — |
| 274 | C22⋊C8 | — | 3 | 4 | no | — |
| 275 | #97 | — | 3 | 2 | no | — |
| 276 | C4×C8 | — | 3 | 2 | no | — |
| 277 | C22×D3 | — | 4 | 2 | no | — |
| 278 | C22×D3 | — | 4 | 2 | no | — |
| 279 | C3×Q8 | — | 8 | 4 | no | — |
| 280 | C2×C3×C4 | — | 4 | 1 | no | — |
| 281 | C2×C3×C4 | — | 4 | 1 | no | — |
| 282 | C2×C3×C4 | — | 4 | 1 | no | — |
| 283 | C2×C3×C4 | — | 4 | 4 | no | — |
| 284 | C2×Dic3 | — | 4 | 2 | no | — |
| 285 | C2×Dic3 | — | 4 | 2 | no | — |
| 286 | D12 | — | 4 | 4 | no | — |
| 287 | D12 | — | 8 | 2 | no | — |
| 288 | D12 | — | 8 | 8 | no | — |
| 289 | C4×D3 | — | 8 | 4 | no | — |
| 290 | C4×D3 | — | 8 | 4 | no | — |
| 291 | C4×D3 | — | 8 | 8 | no | — |
| 292 | SL(2,𝔽3) | C22×D4 | 1 | 1 | yes | characteristic |
| 293 | C24 | — | 6 | 2 | no | — |
| 294 | C24 | — | 6 | 1 | no | — |
| 295 | C24 | — | 6 | 4 | no | — |
| 296 | #41 | — | 2 | 4 | no | — |
| 297 | #41 | — | 3 | 2 | no | — |
| 298 | #41 | — | 6 | 4 | no | — |
| 299 | #41 | — | 6 | 4 | no | — |
| 300 | #41 | — | 6 | 4 | no | — |
| 301 | #41 | — | 6 | 4 | no | — |
| 302 | #41 | — | 6 | 4 | no | — |
| 303 | #41 | — | 6 | 8 | no | — |
| 304 | #41 | — | 12 | 4 | no | — |
| 305 | #41 | — | 12 | 8 | no | — |
| 306 | #41 | D3×D4 | 1 | 2 | yes | — |
| 307 | C2×Q8 | — | 3 | 2 | no | — |
| 308 | C2×Q8 | C2×S4 | 1 | 2 | yes | — |
| 309 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 310 | C2×D4 | — | 3 | 1 | no | — |
| 311 | C2×D4 | — | 3 | 2 | no | — |
| 312 | C2×D4 | — | 3 | 4 | no | — |
| 313 | C2×D4 | — | 3 | 4 | no | — |
| 314 | C2×D4 | — | 3 | 2 | no | — |
| 315 | C2×D4 | — | 3 | 2 | no | — |
| 316 | C2×D4 | — | 3 | 4 | no | — |
| 317 | C2×D4 | — | 3 | 2 | no | — |
| 318 | C2×D4 | — | 3 | 2 | no | — |
| 319 | C2×D4 | — | 6 | 2 | no | — |
| 320 | C2×D4 | — | 6 | 4 | no | — |
| 321 | C2×D4 | — | 6 | 4 | no | — |
| 322 | C2×D4 | — | 6 | 2 | no | — |
| 323 | C2×D4 | — | 6 | 2 | no | — |
| 324 | C2×D4 | — | 6 | 1 | no | — |
| 325 | C2×D4 | — | 6 | 8 | no | — |
| 326 | C2×D4 | — | 6 | 8 | no | — |
| 327 | C2×D4 | — | 6 | 4 | no | — |
| 328 | C2×D4 | — | 6 | 2 | no | — |
| 329 | C2×D4 | — | 6 | 4 | no | — |
| 330 | C2×D4 | — | 6 | 1 | no | — |
| 331 | C2×D4 | — | 6 | 1 | no | — |
| 332 | C2×D4 | — | 6 | 4 | no | — |
| 333 | C2×D4 | — | 12 | 4 | no | — |
| 334 | C2×D4 | — | 12 | 4 | no | — |
| 335 | C2×D4 | — | 12 | 8 | no | — |
| 336 | C2×D4 | — | 12 | 16 | no | — |
| 337 | C2×D4 | — | 24 | 2 | no | — |
| 338 | C2×D4 | — | 24 | 4 | no | — |
| 339 | C2×D4 | — | 24 | 2 | no | — |
| 340 | C22×C4 | — | 3 | 1 | no | — |
| 341 | C22×C4 | — | 3 | 4 | no | — |
| 342 | C22×C4 | — | 3 | 1 | no | — |
| 343 | C22×C4 | — | 3 | 1 | no | — |
| 344 | C22×C4 | — | 6 | 2 | no | — |
| 345 | C22×C4 | — | 6 | 1 | no | — |
| 346 | C22×C4 | — | 6 | 4 | no | — |
| 347 | C22×C4 | — | 6 | 2 | no | — |
| 348 | C22×C4 | — | 6 | 2 | no | — |
| 349 | C22×C4 | — | 24 | 2 | no | — |
| 350 | Q16 | — | 6 | 2 | no | — |
| 351 | Q16 | — | 6 | 2 | no | — |
| 352 | SD16 | — | 3 | 4 | no | — |
| 353 | SD16 | — | 3 | 4 | no | — |
| 354 | SD16 | — | 6 | 8 | no | — |
| 355 | SD16 | — | 6 | 4 | no | — |
| 356 | SD16 | — | 6 | 2 | no | — |
| 357 | SD16 | — | 6 | 2 | no | — |
| 358 | SD16 | — | 12 | 4 | no | — |
| 359 | D8 | — | 3 | 8 | no | — |
| 360 | D8 | — | 6 | 8 | no | — |
| 361 | D8 | — | 6 | 4 | no | — |
| 362 | D8 | — | 6 | 8 | no | — |
| 363 | D8 | — | 6 | 8 | no | — |
| 364 | D8 | — | 12 | 4 | no | — |
| 365 | M16 | — | 3 | 4 | no | — |
| 366 | M16 | — | 6 | 8 | no | — |
| 367 | M16 | — | 6 | 2 | no | — |
| 368 | C2×C8 | — | 3 | 2 | no | — |
| 369 | C2×C8 | — | 3 | 2 | no | — |
| 370 | C2×C8 | — | 3 | 2 | no | — |
| 371 | C2×C8 | — | 3 | 2 | no | — |
| 372 | C2×C8 | — | 6 | 8 | no | — |
| 373 | C2×C8 | — | 6 | 4 | no | — |
| 374 | C4⋊C4 | — | 3 | 1 | no | — |
| 375 | C4⋊C4 | — | 3 | 1 | no | — |
| 376 | C4⋊C4 | — | 3 | 2 | no | — |
| 377 | C4⋊C4 | — | 6 | 1 | no | — |
| 378 | C4⋊C4 | — | 6 | 1 | no | — |
| 379 | C4⋊C4 | — | 6 | 1 | no | — |
| 380 | C4⋊C4 | — | 6 | 4 | no | — |
| 381 | C4⋊C4 | C2×S4 | 1 | 2 | yes | — |
| 382 | C4⋊C4 | C2×S4 | 1 | 2 | yes | — |
| 383 | C22⋊C4 | — | 3 | 4 | no | — |
| 384 | C22⋊C4 | — | 3 | 4 | no | — |
| 385 | C22⋊C4 | — | 6 | 1 | no | — |
| 386 | C22⋊C4 | — | 6 | 2 | no | — |
| 387 | C22⋊C4 | — | 6 | 2 | no | — |
| 388 | C22⋊C4 | — | 6 | 4 | no | — |
| 389 | C22⋊C4 | — | 6 | 2 | no | — |
| 390 | C22⋊C4 | — | 6 | 4 | no | — |
| 391 | C22⋊C4 | — | 6 | 2 | no | — |
| 392 | C42 | — | 3 | 1 | no | — |
| 393 | C42 | — | 3 | 1 | no | — |
| 394 | C42 | — | 3 | 1 | no | — |
| 395 | C42 | C2×S4 | 1 | 1 | yes | characteristic |
| 396 | C22×C3 | — | 4 | 1 | no | — |
| 397 | C2×D3 | — | 4 | 4 | no | — |
| 398 | C2×D3 | — | 8 | 2 | no | — |
| 399 | C2×D3 | — | 8 | 2 | no | — |
| 400 | C2×D3 | — | 8 | 2 | no | — |
| 401 | C2×D3 | — | 8 | 2 | no | — |
| 402 | C2×D3 | — | 8 | 2 | no | — |
| 403 | C3×C4 | — | 4 | 2 | no | — |
| 404 | C3×C4 | — | 4 | 2 | no | — |
| 405 | C3×C4 | — | 8 | 4 | no | — |
| 406 | C3×C4 | — | 8 | 1 | no | — |
| 407 | Dic3 | — | 8 | 2 | no | — |
| 408 | Dic3 | — | 8 | 2 | no | — |
| 409 | C23 | — | 3 | 1 | no | — |
| 410 | C23 | — | 3 | 4 | no | — |
| 411 | C23 | — | 6 | 4 | no | — |
| 412 | C23 | — | 6 | 2 | no | — |
| 413 | C23 | — | 6 | 2 | no | — |
| 414 | C23 | — | 12 | 4 | no | — |
| 415 | C23 | — | 12 | 8 | no | — |
| 416 | C23 | — | 24 | 2 | no | — |
| 417 | C23 | — | 24 | 2 | no | — |
| 418 | C23 | — | 24 | 1 | no | — |
| 419 | C23 | — | 24 | 1 | no | — |
| 420 | C23 | — | 24 | 4 | no | — |
| 421 | C23 | — | 24 | 4 | no | — |
| 422 | Q8 | — | 2 | 4 | no | — |
| 423 | Q8 | — | 3 | 1 | no | — |
| 424 | Q8 | — | 6 | 2 | no | — |
| 425 | Q8 | — | 6 | 2 | no | — |
| 426 | Q8 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 427 | D4 | — | 3 | 2 | no | — |
| 428 | D4 | — | 3 | 4 | no | — |
| 429 | D4 | — | 6 | 8 | no | — |
| 430 | D4 | — | 6 | 1 | no | — |
| 431 | D4 | — | 6 | 4 | no | — |
| 432 | D4 | — | 6 | 4 | no | — |
| 433 | D4 | — | 6 | 2 | no | — |
| 434 | D4 | — | 6 | 2 | no | — |
| 435 | D4 | — | 6 | 2 | no | — |
| 436 | D4 | — | 6 | 4 | no | — |
| 437 | D4 | — | 6 | 4 | no | — |
| 438 | D4 | — | 6 | 4 | no | — |
| 439 | D4 | — | 12 | 2 | no | — |
| 440 | D4 | — | 12 | 8 | no | — |
| 441 | D4 | — | 12 | 4 | no | — |
| 442 | D4 | — | 12 | 8 | no | — |
| 443 | D4 | — | 12 | 2 | no | — |
| 444 | D4 | — | 24 | 2 | no | — |
| 445 | D4 | — | 24 | 2 | no | — |
| 446 | D4 | — | 24 | 1 | no | — |
| 447 | D4 | — | 24 | 4 | no | — |
| 448 | C2×C4 | — | 3 | 1 | no | — |
| 449 | C2×C4 | — | 3 | 2 | no | — |
| 450 | C2×C4 | — | 3 | 2 | no | — |
| 451 | C2×C4 | — | 3 | 1 | no | — |
| 452 | C2×C4 | — | 3 | 1 | no | — |
| 453 | C2×C4 | — | 6 | 4 | no | — |
| 454 | C2×C4 | — | 6 | 4 | no | — |
| 455 | C2×C4 | — | 6 | 2 | no | — |
| 456 | C2×C4 | — | 6 | 2 | no | — |
| 457 | C2×C4 | — | 6 | 2 | no | — |
| 458 | C2×C4 | — | 12 | 1 | no | — |
| 459 | C2×C4 | — | 12 | 8 | no | — |
| 460 | C2×C4 | — | 12 | 4 | no | — |
| 461 | C2×C4 | — | 12 | 4 | no | — |
| 462 | C2×C4 | — | 24 | 2 | no | — |
| 463 | C2×C4 | — | 24 | 1 | no | — |
| 464 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 465 | C2×C4 | C22×S4 | 1 | 4 | yes | — |
| 466 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 467 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 468 | C8 | — | 3 | 4 | no | — |
| 469 | C8 | — | 6 | 2 | no | — |
| 470 | C8 | — | 6 | 2 | no | — |
| 471 | C8 | — | 6 | 2 | no | — |
| 472 | C2×C3 | — | 4 | 1 | no | — |
| 473 | C2×C3 | — | 4 | 1 | no | — |
| 474 | C2×C3 | — | 4 | 1 | no | — |
| 475 | D3 | — | 8 | 4 | no | — |
| 476 | D3 | — | 16 | 2 | no | — |
| 477 | C22 | — | 3 | 2 | no | — |
| 478 | C22 | — | 6 | 1 | no | — |
| 479 | C22 | — | 6 | 1 | no | — |
| 480 | C22 | — | 6 | 4 | no | — |
| 481 | C22 | — | 6 | 4 | no | — |
| 482 | C22 | — | 6 | 4 | no | — |
| 483 | C22 | — | 6 | 4 | no | — |
| 484 | C22 | — | 12 | 2 | no | — |
| 485 | C22 | — | 12 | 2 | no | — |
| 486 | C22 | — | 12 | 2 | no | — |
| 487 | C22 | — | 12 | 2 | no | — |
| 488 | C22 | — | 12 | 2 | no | — |
| 489 | C22 | — | 24 | 4 | no | — |
| 490 | C22 | — | 48 | 4 | no | — |
| 491 | C22 | — | 48 | 8 | no | — |
| 492 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 493 | C4 | — | 2 | 4 | no | — |
| 494 | C4 | — | 2 | 1 | no | — |
| 495 | C4 | — | 3 | 1 | no | — |
| 496 | C4 | — | 3 | 1 | no | — |
| 497 | C4 | — | 6 | 1 | no | — |
| 498 | C4 | — | 6 | 1 | no | — |
| 499 | C4 | — | 12 | 2 | no | — |
| 500 | C4 | — | 12 | 2 | no | — |
| 501 | C4 | D4×S4 | 1 | 2 | yes | — |
| 502 | C4 | #5925 | 1 | 2 | yes | complemented |
| 503 | C3 | — | 4 | 1 | no | Sylow |
| 504 | C2 | — | 6 | 2 | no | — |
| 505 | C2 | — | 12 | 4 | no | — |
| 506 | C2 | — | 12 | 4 | no | — |
| 507 | C2 | — | 24 | 2 | no | — |
| 508 | C2 | C2×D4×S4 | 1 | 1 | yes | characteristic |
| 509 | C2 | C2×#5925 | 1 | 1 | yes | characteristic |
| 510 | C2 | #87493 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C23×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C24 |
| Fitting | #3498 | D3 |
| hypercenter | C4⋊Q8 | S4 |
| hypocenter | SL(2,𝔽3) | C22×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C22 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#5925 | C2 | semidirect |
| 2. C4⋊#994 | C2 | semidirect |
| 3. #85529 | C2 | semidirect |
| 4. #85410 | C2 | semidirect |
| 5. #85409 | C2 | semidirect |
| 6. #85407 | C2 | semidirect |
| 7. #85344 | C2 | semidirect |
| 8. C4×#986 | C2 | semidirect |
| 10. C2×#994 | C22 | semidirect |
| 13. C2×#986 | C22 | semidirect |
| 14. C2×#985 | C22 | semidirect |
| 15. C2×#985 | C22 | semidirect |
| 17. C4⋊(#41⋊C3) | C22 | semidirect |
| 18. C4⋊(#41⋊C3) | C22 | semidirect |
| 19. C4×#41⋊C3 | C22 | semidirect |
| 20. #5427 | C22 | semidirect |
| 21. #5426 | C22 | semidirect |
| 22. #5425 | C22 | semidirect |
| 23. #5395 | C22 | semidirect |
| 24. #5394 | C22 | semidirect |
| 25. C4⋊GL(2,𝔽3) | C22 | semidirect |
| 26. C4×GL(2,𝔽3) | C22 | semidirect |
| 27. C4⋊#226 | C22 | semidirect |
| 29. #3498 | D3 | semidirect |
| 50. #986 | D4 | semidirect |
| 147. #41⋊C3 | C2×D4 | semidirect |
| 150. GL(2,𝔽3) | C2×D4 | semidirect |
| 502. C4 | #5925 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5925 | 2 |
| C4⋊#994 | 2 |
| #85529 | 2 |
| #85410 | 2 |
| #85409 | 2 |
| #85407 | 2 |
| #85344 | 2 |
| C4×#986 | 2 |
| #61093 | 3 |
| #5726 | 4 |