Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1385 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 1385 conjugacy classes of subgroups — too many to draw. Only its 123 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×#5924 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C4⋊#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #85527 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #85407 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #85404 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | C4×#985 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #85336 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #61087 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×#994 | C22 | 1 | 2 | yes | — |
| 10 | #5924 | — | 2 | 8 | no | — |
| 11 | C2×#985 | C22 | 1 | 1 | yes | characteristic |
| 12 | C2×#985 | C22 | 1 | 4 | yes | complemented |
| 13 | C2×#984 | C22 | 1 | 2 | yes | — |
| 14 | C2×#984 | C22 | 1 | 4 | yes | complemented |
| 15 | D3×C4⋊Q8 | — | 4 | 1 | no | maximal |
| 16 | C4⋊(#41⋊C3) | C22 | 1 | 4 | yes | — |
| 17 | C4×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 18 | #5425 | C22 | 1 | 8 | yes | complemented |
| 19 | #5424 | C22 | 1 | 1 | yes | characteristic |
| 20 | C4×GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 21 | C4⋊#226 | C22 | 1 | 4 | yes | complemented |
| 22 | C4×#226 | C22 | 1 | 2 | yes | complemented |
| 23 | C2×#577 | — | 3 | 2 | no | — |
| 24 | #3499 | — | 3 | 1 | no | — |
| 25 | #3498 | D3 | 1 | 1 | yes | characteristic, complemented |
| 26 | #3284 | — | 3 | 4 | no | — |
| 27 | D4×SD16 | — | 3 | 4 | no | — |
| 28 | #3278 | — | 3 | 4 | no | — |
| 29 | #3272 | — | 3 | 4 | no | — |
| 30 | C2×#492 | — | 3 | 1 | no | — |
| 31 | #3057 | — | 3 | 2 | no | — |
| 32 | #3039 | — | 3 | 1 | no | — |
| 33 | #3038 | — | 3 | 1 | no | — |
| 34 | #3010 | — | 3 | 2 | no | — |
| 35 | #3009 | — | 3 | 2 | no | — |
| 36 | C4×#135 | — | 3 | 1 | no | — |
| 37 | #2931 | — | 3 | 1 | no | — |
| 38 | C2×D3×Q8 | — | 4 | 2 | no | — |
| 39 | #994 | — | 2 | 4 | no | — |
| 40 | C2×#41⋊C3 | C23 | 1 | 4 | yes | — |
| 41 | C2×#41⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 42 | #985 | — | 2 | 8 | no | — |
| 43 | #985 | D4 | 1 | 4 | yes | complemented |
| 44 | #984 | — | 2 | 4 | no | — |
| 45 | #984 | — | 2 | 8 | no | — |
| 46 | C2×GL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 47 | C2×#226 | C23 | 1 | 4 | yes | — |
| 48 | C2×#226 | C23 | 1 | 1 | yes | characteristic |
| 49 | C3×C4⋊Q8 | — | 4 | 1 | no | — |
| 50 | #944 | — | 4 | 2 | no | — |
| 51 | D3×C4⋊C4 | — | 4 | 4 | no | — |
| 52 | #888 | — | 4 | 4 | no | — |
| 53 | C42×D3 | — | 4 | 1 | no | — |
| 54 | #869 | — | 4 | 1 | no | — |
| 55 | C4×SL(2,𝔽3) | C23 | 1 | 2 | yes | — |
| 56 | #859 | C23 | 1 | 2 | yes | — |
| 57 | C2×2−1+4 | — | 3 | 2 | no | — |
| 58 | C2×H(5,𝔽2) | C2×D3 | 1 | 2 | yes | — |
| 59 | #577 | — | 6 | 8 | no | — |
| 60 | #577 | — | 6 | 8 | no | — |
| 61 | C2×#137 | — | 3 | 2 | no | — |
| 62 | C2×#137 | — | 3 | 4 | no | — |
| 63 | C2×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 64 | C2×AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 65 | C2×#135 | — | 3 | 1 | no | — |
| 66 | C2×#135 | — | 3 | 4 | no | — |
| 67 | C22×SD16 | — | 3 | 2 | no | — |
| 68 | C22×SD16 | — | 3 | 4 | no | — |
| 69 | C22×SD16 | — | 3 | 2 | no | — |
| 70 | C2×#131 | — | 3 | 2 | no | — |
| 71 | #550 | C2×D3 | 1 | 4 | yes | — |
| 72 | D4×Q8 | — | 3 | 4 | no | — |
| 73 | #547 | — | 3 | 4 | no | — |
| 74 | #547 | — | 6 | 4 | no | — |
| 75 | D42 | — | 3 | 4 | no | — |
| 76 | #536 | — | 3 | 2 | no | — |
| 77 | #536 | — | 6 | 2 | no | — |
| 78 | #535 | — | 3 | 2 | no | — |
| 79 | #532 | — | 3 | 1 | no | — |
| 80 | #532 | — | 6 | 1 | no | — |
| 81 | C2×C4⋊Q8 | — | 3 | 1 | no | — |
| 82 | C2×C4⋊Q8 | — | 6 | 1 | no | — |
| 83 | C2×#127 | — | 3 | 1 | no | — |
| 84 | C4×#41 | — | 3 | 1 | no | — |
| 85 | C4×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 86 | #492 | — | 3 | 4 | no | — |
| 87 | #492 | — | 3 | 4 | no | — |
| 88 | #468 | — | 3 | 8 | no | — |
| 89 | #465 | — | 3 | 8 | no | — |
| 90 | #462 | — | 3 | 4 | no | — |
| 91 | #461 | — | 3 | 4 | no | — |
| 92 | #460 | — | 3 | 4 | no | — |
| 93 | #459 | — | 3 | 4 | no | — |
| 94 | #452 | — | 3 | 8 | no | — |
| 95 | #450 | — | 3 | 8 | no | — |
| 96 | #449 | — | 3 | 8 | no | — |
| 97 | #448 | — | 3 | 8 | no | — |
| 98 | C4×Q16 | — | 3 | 2 | no | — |
| 99 | C4×SD16 | — | 3 | 2 | no | — |
| 100 | C4×SD16 | — | 3 | 2 | no | — |
| 101 | C4×D8 | — | 3 | 2 | no | — |
| 102 | #436 | — | 3 | 4 | no | — |
| 103 | C8×D4 | — | 3 | 4 | no | — |
| 104 | #427 | — | 3 | 1 | no | — |
| 105 | #423 | — | 3 | 1 | no | — |
| 106 | #419 | — | 3 | 2 | no | — |
| 107 | #418 | — | 3 | 2 | no | — |
| 108 | #416 | — | 3 | 1 | no | — |
| 109 | #416 | — | 3 | 1 | no | — |
| 110 | #408 | — | 3 | 2 | no | — |
| 111 | C2×C4×C8 | — | 3 | 1 | no | — |
| 112 | C2×C3×Q8 | — | 4 | 2 | no | — |
| 113 | D3×Q8 | — | 8 | 8 | no | — |
| 114 | C2×C4×D3 | — | 4 | 2 | no | — |
| 115 | C2×C4×D3 | — | 4 | 1 | no | — |
| 116 | C2×C4×D3 | — | 4 | 4 | no | — |
| 117 | C2×Dic6 | — | 4 | 2 | no | — |
| 118 | C2×Dic6 | — | 4 | 4 | no | — |
| 119 | #41⋊C3 | — | 2 | 4 | no | — |
| 120 | #41⋊C3 | C2×D4 | 1 | 2 | yes | — |
| 121 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 122 | GL(2,𝔽3) | C2×D4 | 1 | 2 | yes | complemented |
| 123 | #226 | — | 2 | 4 | no | — |
| 124 | #226 | C2×D4 | 1 | 2 | yes | complemented |
| 125 | C3×C4⋊C4 | — | 4 | 4 | no | — |
| 126 | C3×C42 | — | 4 | 1 | no | — |
| 127 | C4⋊Dic3 | — | 4 | 4 | no | — |
| 128 | Dic3⋊C4 | — | 4 | 8 | no | — |
| 129 | C4×Dic3 | — | 4 | 2 | no | — |
| 130 | C4×Dic3 | — | 4 | 1 | no | — |
| 131 | 2−1+4 | — | 6 | 8 | no | — |
| 132 | 2−1+4 | — | 12 | 4 | no | — |
| 133 | H(5,𝔽2) | — | 2 | 4 | no | — |
| 134 | H(5,𝔽2) | — | 6 | 4 | no | — |
| 135 | C2×#41 | — | 3 | 2 | no | — |
| 136 | C2×#41 | — | 3 | 1 | no | — |
| 137 | C2×#41 | — | 3 | 4 | no | — |
| 138 | C2×#41 | — | 6 | 2 | no | — |
| 139 | C2×#41 | — | 6 | 4 | no | — |
| 140 | C2×#41 | C22×D3 | 1 | 4 | yes | — |
| 141 | C2×#41 | C22×D3 | 1 | 1 | yes | characteristic |
| 142 | C22×Q8 | — | 3 | 2 | no | — |
| 143 | C22×Q8 | — | 3 | 4 | no | — |
| 144 | C22×Q8 | — | 6 | 2 | no | — |
| 145 | C22×D4 | — | 3 | 2 | no | — |
| 146 | C22×D4 | — | 3 | 4 | no | — |
| 147 | C22×D4 | — | 6 | 2 | no | — |
| 148 | #137 | — | 6 | 8 | no | — |
| 149 | #137 | — | 6 | 8 | no | — |
| 150 | #137 | — | 6 | 4 | no | — |
| 151 | #137 | — | 6 | 4 | no | — |
| 152 | AGL(1,ℤ/8) | — | 6 | 16 | no | — |
| 153 | AGL(1,ℤ/8) | — | 6 | 8 | no | — |
| 154 | #135 | — | 3 | 4 | no | — |
| 155 | #135 | — | 3 | 4 | no | — |
| 156 | #135 | — | 6 | 8 | no | — |
| 157 | #135 | — | 6 | 8 | no | — |
| 158 | C2×Q16 | — | 3 | 1 | no | — |
| 159 | C2×Q16 | — | 3 | 4 | no | — |
| 160 | C2×SD16 | — | 3 | 1 | no | — |
| 161 | C2×SD16 | — | 3 | 1 | no | — |
| 162 | C2×SD16 | — | 3 | 4 | no | — |
| 163 | C2×SD16 | — | 3 | 4 | no | — |
| 164 | C2×SD16 | — | 3 | 4 | no | — |
| 165 | C2×SD16 | — | 3 | 4 | no | — |
| 166 | C2×SD16 | — | 3 | 8 | no | — |
| 167 | C2×SD16 | — | 6 | 8 | no | — |
| 168 | C2×SD16 | — | 6 | 16 | no | — |
| 169 | C2×SD16 | — | 12 | 4 | no | — |
| 170 | C2×D8 | — | 3 | 1 | no | — |
| 171 | C2×D8 | — | 3 | 4 | no | — |
| 172 | #131 | — | 6 | 8 | no | — |
| 173 | C2×M16 | — | 3 | 2 | no | — |
| 174 | C2×M16 | — | 3 | 4 | no | — |
| 175 | C22×C8 | — | 3 | 1 | no | — |
| 176 | C22×C8 | — | 3 | 4 | no | — |
| 177 | C22×C8 | — | 3 | 1 | no | — |
| 178 | C4⋊Q8 | — | 3 | 4 | no | — |
| 179 | C4⋊Q8 | — | 6 | 1 | no | — |
| 180 | C4⋊Q8 | — | 6 | 2 | no | — |
| 181 | C4⋊Q8 | — | 6 | 4 | no | — |
| 182 | C4⋊Q8 | — | 6 | 1 | no | — |
| 183 | C4⋊Q8 | S4 | 1 | 1 | yes | characteristic |
| 184 | #127 | — | 3 | 4 | no | — |
| 185 | #127 | — | 3 | 1 | no | — |
| 186 | #127 | — | 6 | 1 | no | — |
| 187 | #124 | — | 3 | 2 | no | — |
| 188 | #124 | — | 6 | 2 | no | — |
| 189 | #123 | — | 6 | 8 | no | — |
| 190 | #123 | — | 6 | 4 | no | — |
| 191 | C22⋊Q8 | — | 3 | 8 | no | — |
| 192 | C22⋊Q8 | — | 6 | 8 | no | — |
| 193 | #121 | — | 3 | 4 | no | — |
| 194 | #121 | — | 3 | 8 | no | — |
| 195 | #121 | — | 6 | 4 | no | — |
| 196 | #120 | — | 6 | 4 | no | — |
| 197 | C4×Q8 | — | 3 | 2 | no | — |
| 198 | C4×Q8 | C22×D3 | 1 | 2 | yes | — |
| 199 | C4×D4 | — | 3 | 4 | no | — |
| 200 | C4×D4 | — | 3 | 2 | no | — |
| 201 | C4×D4 | — | 3 | 2 | no | — |
| 202 | C4×D4 | — | 6 | 2 | no | — |
| 203 | C4×D4 | — | 6 | 4 | no | — |
| 204 | #117 | — | 3 | 1 | no | — |
| 205 | #117 | — | 3 | 1 | no | — |
| 206 | #117 | — | 6 | 1 | no | — |
| 207 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 208 | C2×C4⋊C4 | — | 6 | 4 | no | — |
| 209 | C2×C42 | — | 3 | 1 | no | — |
| 210 | C2×C42 | — | 6 | 1 | no | — |
| 211 | #107 | — | 3 | 2 | no | — |
| 212 | #106 | — | 3 | 2 | no | — |
| 213 | C4⋊C8 | — | 3 | 4 | no | — |
| 214 | #103 | — | 3 | 2 | no | — |
| 215 | #103 | — | 3 | 8 | no | — |
| 216 | #103 | — | 3 | 2 | no | — |
| 217 | #102 | — | 3 | 2 | no | — |
| 218 | #102 | — | 3 | 2 | no | — |
| 219 | #102 | — | 3 | 8 | no | — |
| 220 | C22⋊C8 | — | 3 | 8 | no | — |
| 221 | C4×C8 | — | 3 | 2 | no | — |
| 222 | C4×C8 | — | 3 | 2 | no | — |
| 223 | C22×D3 | — | 4 | 1 | no | — |
| 224 | C3×Q8 | — | 8 | 4 | no | — |
| 225 | C2×C3×C4 | — | 4 | 2 | no | — |
| 226 | C2×C3×C4 | — | 4 | 1 | no | — |
| 227 | C2×C3×C4 | — | 4 | 4 | no | — |
| 228 | C2×Dic3 | — | 4 | 2 | no | — |
| 229 | C2×Dic3 | — | 4 | 1 | no | — |
| 230 | C2×Dic3 | — | 4 | 4 | no | — |
| 231 | C4×D3 | — | 4 | 4 | no | — |
| 232 | C4×D3 | — | 4 | 8 | no | — |
| 233 | C4×D3 | — | 8 | 8 | no | — |
| 234 | Dic6 | — | 8 | 8 | no | — |
| 235 | Dic6 | — | 8 | 4 | no | — |
| 236 | SL(2,𝔽3) | C22×D4 | 1 | 1 | yes | characteristic |
| 237 | C24 | — | 6 | 2 | no | — |
| 238 | #41 | — | 2 | 4 | no | — |
| 239 | #41 | — | 3 | 2 | no | — |
| 240 | #41 | — | 3 | 4 | no | — |
| 241 | #41 | — | 6 | 4 | no | — |
| 242 | #41 | — | 6 | 4 | no | — |
| 243 | #41 | — | 6 | 2 | no | — |
| 244 | #41 | — | 6 | 4 | no | — |
| 245 | #41 | — | 6 | 8 | no | — |
| 246 | #41 | — | 12 | 4 | no | — |
| 247 | #41 | — | 12 | 8 | no | — |
| 248 | #41 | — | 12 | 16 | no | — |
| 249 | #41 | D3×D4 | 1 | 2 | yes | — |
| 250 | C2×Q8 | — | 3 | 1 | no | — |
| 251 | C2×Q8 | — | 3 | 4 | no | — |
| 252 | C2×Q8 | — | 6 | 4 | no | — |
| 253 | C2×Q8 | — | 6 | 4 | no | — |
| 254 | C2×Q8 | — | 6 | 4 | no | — |
| 255 | C2×Q8 | — | 6 | 2 | no | — |
| 256 | C2×Q8 | — | 6 | 8 | no | — |
| 257 | C2×Q8 | — | 12 | 8 | no | — |
| 258 | C2×Q8 | — | 12 | 4 | no | — |
| 259 | C2×Q8 | C2×S4 | 1 | 2 | yes | — |
| 260 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 261 | C2×D4 | — | 3 | 1 | no | — |
| 262 | C2×D4 | — | 3 | 1 | no | — |
| 263 | C2×D4 | — | 3 | 4 | no | — |
| 264 | C2×D4 | — | 3 | 4 | no | — |
| 265 | C2×D4 | — | 3 | 4 | no | — |
| 266 | C2×D4 | — | 3 | 2 | no | — |
| 267 | C2×D4 | — | 6 | 4 | no | — |
| 268 | C2×D4 | — | 6 | 4 | no | — |
| 269 | C2×D4 | — | 6 | 4 | no | — |
| 270 | C2×D4 | — | 6 | 8 | no | — |
| 271 | C2×D4 | — | 6 | 1 | no | — |
| 272 | C2×D4 | — | 6 | 4 | no | — |
| 273 | C2×D4 | — | 12 | 4 | no | — |
| 274 | C2×D4 | — | 24 | 4 | no | — |
| 275 | C22×C4 | — | 3 | 1 | no | — |
| 276 | C22×C4 | — | 3 | 4 | no | — |
| 277 | C22×C4 | — | 3 | 2 | no | — |
| 278 | C22×C4 | — | 6 | 1 | no | — |
| 279 | C22×C4 | — | 6 | 2 | no | — |
| 280 | C22×C4 | — | 6 | 4 | no | — |
| 281 | C22×C4 | — | 6 | 2 | no | — |
| 282 | C22×C4 | — | 6 | 4 | no | — |
| 283 | Q16 | — | 3 | 2 | no | — |
| 284 | Q16 | — | 3 | 2 | no | — |
| 285 | Q16 | — | 6 | 4 | no | — |
| 286 | Q16 | — | 6 | 4 | no | — |
| 287 | SD16 | — | 3 | 4 | no | — |
| 288 | SD16 | — | 3 | 2 | no | — |
| 289 | SD16 | — | 3 | 2 | no | — |
| 290 | SD16 | — | 6 | 8 | no | — |
| 291 | SD16 | — | 6 | 8 | no | — |
| 292 | SD16 | — | 6 | 8 | no | — |
| 293 | SD16 | — | 6 | 8 | no | — |
| 294 | SD16 | — | 12 | 8 | no | — |
| 295 | D8 | — | 3 | 4 | no | — |
| 296 | D8 | — | 6 | 8 | no | — |
| 297 | M16 | — | 6 | 8 | no | — |
| 298 | M16 | — | 6 | 4 | no | — |
| 299 | C2×C8 | — | 3 | 1 | no | — |
| 300 | C2×C8 | — | 3 | 1 | no | — |
| 301 | C2×C8 | — | 3 | 4 | no | — |
| 302 | C2×C8 | — | 3 | 2 | no | — |
| 303 | C2×C8 | — | 3 | 4 | no | — |
| 304 | C2×C8 | — | 6 | 8 | no | — |
| 305 | C2×C8 | — | 6 | 2 | no | — |
| 306 | C4⋊C4 | — | 3 | 2 | no | — |
| 307 | C4⋊C4 | — | 3 | 2 | no | — |
| 308 | C4⋊C4 | — | 6 | 4 | no | — |
| 309 | C4⋊C4 | — | 6 | 2 | no | — |
| 310 | C4⋊C4 | — | 6 | 4 | no | — |
| 311 | C4⋊C4 | — | 6 | 8 | no | — |
| 312 | C4⋊C4 | C2×S4 | 1 | 4 | yes | — |
| 313 | C22⋊C4 | — | 3 | 8 | no | — |
| 314 | C22⋊C4 | — | 6 | 1 | no | — |
| 315 | C22⋊C4 | — | 6 | 2 | no | — |
| 316 | C22⋊C4 | — | 6 | 1 | no | — |
| 317 | C22⋊C4 | — | 6 | 8 | no | — |
| 318 | C42 | — | 3 | 2 | no | — |
| 319 | C42 | — | 3 | 1 | no | — |
| 320 | C42 | — | 6 | 1 | no | — |
| 321 | C42 | — | 6 | 2 | no | — |
| 322 | C42 | C2×S4 | 1 | 1 | yes | characteristic |
| 323 | C22×C3 | — | 4 | 1 | no | — |
| 324 | C2×D3 | — | 4 | 2 | no | — |
| 325 | C2×D3 | — | 4 | 4 | no | — |
| 326 | C3×C4 | — | 4 | 2 | no | — |
| 327 | C3×C4 | — | 4 | 4 | no | — |
| 328 | C3×C4 | — | 8 | 4 | no | — |
| 329 | Dic3 | — | 4 | 2 | no | — |
| 330 | Dic3 | — | 4 | 4 | no | — |
| 331 | Dic3 | — | 8 | 4 | no | — |
| 332 | C23 | — | 3 | 1 | no | — |
| 333 | C23 | — | 3 | 4 | no | — |
| 334 | C23 | — | 6 | 1 | no | — |
| 335 | C23 | — | 12 | 4 | no | — |
| 336 | C23 | — | 24 | 4 | no | — |
| 337 | Q8 | — | 2 | 4 | no | — |
| 338 | Q8 | — | 3 | 1 | no | — |
| 339 | Q8 | — | 3 | 2 | no | — |
| 340 | Q8 | — | 6 | 1 | no | — |
| 341 | Q8 | — | 6 | 4 | no | — |
| 342 | Q8 | — | 6 | 4 | no | — |
| 343 | Q8 | — | 12 | 8 | no | — |
| 344 | Q8 | — | 12 | 4 | no | — |
| 345 | Q8 | C2×D3×D4 | 1 | 1 | yes | characteristic |
| 346 | D4 | — | 3 | 2 | no | — |
| 347 | D4 | — | 3 | 2 | no | — |
| 348 | D4 | — | 6 | 8 | no | — |
| 349 | D4 | — | 6 | 1 | no | — |
| 350 | D4 | — | 6 | 4 | no | — |
| 351 | D4 | — | 6 | 4 | no | — |
| 352 | D4 | — | 6 | 1 | no | — |
| 353 | D4 | — | 6 | 4 | no | — |
| 354 | D4 | — | 6 | 4 | no | — |
| 355 | D4 | — | 6 | 8 | no | — |
| 356 | D4 | — | 12 | 8 | no | — |
| 357 | D4 | — | 24 | 4 | no | — |
| 358 | C2×C4 | — | 3 | 1 | no | — |
| 359 | C2×C4 | — | 3 | 2 | no | — |
| 360 | C2×C4 | — | 3 | 2 | no | — |
| 361 | C2×C4 | — | 3 | 2 | no | — |
| 362 | C2×C4 | — | 6 | 4 | no | — |
| 363 | C2×C4 | — | 6 | 4 | no | — |
| 364 | C2×C4 | — | 6 | 1 | no | — |
| 365 | C2×C4 | — | 6 | 4 | no | — |
| 366 | C2×C4 | — | 6 | 4 | no | — |
| 367 | C2×C4 | — | 6 | 2 | no | — |
| 368 | C2×C4 | — | 6 | 4 | no | — |
| 369 | C2×C4 | — | 12 | 8 | no | — |
| 370 | C2×C4 | — | 12 | 8 | no | — |
| 371 | C2×C4 | — | 12 | 4 | no | — |
| 372 | C2×C4 | — | 12 | 4 | no | — |
| 373 | C2×C4 | C22×S4 | 1 | 4 | yes | — |
| 374 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 375 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 376 | C8 | — | 3 | 2 | no | — |
| 377 | C8 | — | 3 | 2 | no | — |
| 378 | C8 | — | 6 | 4 | no | — |
| 379 | C8 | — | 6 | 2 | no | — |
| 380 | C2×C3 | — | 4 | 1 | no | — |
| 381 | C2×C3 | — | 4 | 2 | no | — |
| 382 | D3 | — | 4 | 4 | no | — |
| 383 | C22 | — | 3 | 2 | no | — |
| 384 | C22 | — | 6 | 2 | no | — |
| 385 | C22 | — | 6 | 2 | no | — |
| 386 | C22 | — | 6 | 4 | no | — |
| 387 | C22 | — | 6 | 8 | no | — |
| 388 | C22 | — | 12 | 2 | no | — |
| 389 | C22 | — | 48 | 4 | no | — |
| 390 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 391 | C4 | — | 2 | 4 | no | — |
| 392 | C4 | — | 3 | 1 | no | — |
| 393 | C4 | — | 3 | 1 | no | — |
| 394 | C4 | — | 6 | 2 | no | — |
| 395 | C4 | — | 6 | 2 | no | — |
| 396 | C4 | — | 12 | 4 | no | — |
| 397 | C4 | — | 12 | 2 | no | — |
| 398 | C4 | D4×S4 | 1 | 2 | yes | — |
| 399 | C4 | #5924 | 1 | 4 | yes | complemented |
| 400 | C3 | — | 4 | 1 | no | Sylow |
| 401 | C2 | — | 6 | 2 | no | — |
| 402 | C2 | — | 12 | 4 | no | — |
| 403 | C2 | — | 12 | 2 | no | — |
| 404 | C2 | C2×D4×S4 | 1 | 1 | yes | characteristic |
| 405 | C2 | C2×#5924 | 1 | 2 | yes | — |
| 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×#5924 | C2 | semidirect |
| 2. C4⋊#994 | C2 | semidirect |
| 3. #85527 | C2 | semidirect |
| 4. #85407 | C2 | semidirect |
| 5. #85404 | C2 | semidirect |
| 6. C4×#985 | C2 | semidirect |
| 7. #85336 | C2 | semidirect |
| 12. C2×#985 | C22 | semidirect |
| 14. C2×#984 | C22 | semidirect |
| 18. #5425 | C22 | semidirect |
| 20. C4×GL(2,𝔽3) | C22 | semidirect |
| 21. C4⋊#226 | C22 | semidirect |
| 22. C4×#226 | C22 | semidirect |
| 25. #3498 | D3 | semidirect |
| 43. #985 | D4 | semidirect |
| 122. GL(2,𝔽3) | C2×D4 | semidirect |
| 124. #226 | C2×D4 | semidirect |
| 399. C4 | #5924 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#5924 | 2 |
| C4⋊#994 | 2 |
| #85527 | 2 |
| #85407 | 2 |
| #85404 | 2 |
| C4×#985 | 2 |
| #85336 | 2 |
| #61087 | 3 |
| D3×C4⋊Q8 | 4 |