Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1441 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 1441 conjugacy classes of subgroups — too many to draw. Only its 157 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 | #41×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #85430 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | Q8×GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #85397 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | D4×GL(2,𝔽3) | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #85325 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | C2×C4×GL(2,𝔽3) | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | SD16×#41 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×D3×#41 | — | 4 | 1 | no | maximal |
| 10 | C22×GL(2,𝔽3) | C22 | 1 | 3 | yes | — |
| 11 | Q8×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | D4×SL(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 13 | C2×C4×SL(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 14 | #5421 | C22 | 1 | 6 | yes | complemented |
| 15 | C2×#859 | C22 | 1 | 3 | yes | complemented |
| 16 | C4⋊GL(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 17 | C4×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 18 | C4×GL(2,𝔽3) | C22 | 1 | 3 | yes | — |
| 19 | C4×GL(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 20 | C4×GL(2,𝔽3) | C22 | 1 | 3 | yes | complemented |
| 21 | #5386 | C22 | 1 | 3 | yes | complemented |
| 22 | Q8×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | D4×#41 | — | 3 | 1 | no | — |
| 24 | #3392 | — | 3 | 1 | no | — |
| 25 | Q8×SD16 | — | 3 | 1 | no | — |
| 26 | #3362 | — | 3 | 1 | no | — |
| 27 | #3359 | — | 3 | 1 | no | — |
| 28 | #3335 | — | 3 | 3 | no | — |
| 29 | #3298 | — | 3 | 3 | no | — |
| 30 | #3295 | — | 3 | 3 | no | — |
| 31 | D4×SD16 | — | 3 | 3 | no | — |
| 32 | #3167 | — | 3 | 3 | no | — |
| 33 | #3103 | — | 3 | 3 | no | — |
| 34 | #3101 | — | 3 | 3 | no | — |
| 35 | C8×#41 | — | 3 | 1 | no | — |
| 36 | C2×C4×SD16 | — | 3 | 3 | no | — |
| 37 | C2×C3×#41 | — | 4 | 1 | no | — |
| 38 | D3×#41 | — | 4 | 16 | no | — |
| 39 | C2×#239 | — | 4 | 1 | no | — |
| 40 | C2×D3×Q8 | — | 4 | 1 | no | — |
| 41 | C2×#237 | — | 4 | 3 | no | — |
| 42 | C2×D3×D4 | — | 4 | 3 | no | — |
| 43 | C2×#235 | — | 4 | 3 | no | — |
| 44 | C22×C4×D3 | — | 4 | 3 | no | — |
| 45 | C22×SL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 46 | C2×GL(2,𝔽3) | — | 2 | 6 | no | — |
| 47 | C2×GL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 48 | C2×GL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 49 | C4×SL(2,𝔽3) | C23 | 1 | 3 | yes | — |
| 50 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 51 | #859 | C23 | 1 | 3 | yes | — |
| 52 | #859 | C23 | 1 | 1 | yes | characteristic |
| 53 | C22×#41 | — | 6 | 1 | no | — |
| 54 | C22×SD16 | — | 3 | 3 | no | — |
| 55 | Q82 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 56 | #557 | — | 3 | 1 | no | — |
| 57 | #554 | — | 3 | 3 | no | — |
| 58 | #550 | — | 3 | 1 | no | — |
| 59 | D4×Q8 | — | 3 | 1 | no | — |
| 60 | D4×Q8 | C2×D3 | 1 | 3 | yes | complemented |
| 61 | #548 | — | 6 | 1 | no | — |
| 62 | #547 | — | 3 | 3 | no | — |
| 63 | #546 | — | 6 | 3 | no | — |
| 64 | D42 | — | 3 | 3 | no | — |
| 65 | #533 | — | 3 | 3 | no | — |
| 66 | #532 | — | 3 | 3 | no | — |
| 67 | #525 | — | 6 | 3 | no | — |
| 68 | C4×#41 | — | 3 | 1 | no | — |
| 69 | C2×C4×Q8 | C2×D3 | 1 | 3 | yes | complemented |
| 70 | C2×C4×D4 | — | 3 | 3 | no | — |
| 71 | #498 | — | 3 | 3 | no | — |
| 72 | #492 | — | 3 | 3 | no | — |
| 73 | #487 | — | 3 | 3 | no | — |
| 74 | #486 | — | 3 | 3 | no | — |
| 75 | #483 | — | 3 | 6 | no | — |
| 76 | #481 | — | 3 | 6 | no | — |
| 77 | #476 | — | 3 | 3 | no | — |
| 78 | #476 | — | 3 | 3 | no | — |
| 79 | #475 | — | 3 | 3 | no | — |
| 80 | #475 | — | 3 | 3 | no | — |
| 81 | #465 | — | 3 | 6 | no | — |
| 82 | #465 | — | 3 | 6 | no | — |
| 83 | #461 | — | 3 | 3 | no | — |
| 84 | #461 | — | 3 | 3 | no | — |
| 85 | #460 | — | 3 | 3 | no | — |
| 86 | #460 | — | 3 | 3 | no | — |
| 87 | #450 | — | 3 | 6 | no | — |
| 88 | #448 | — | 3 | 6 | no | — |
| 89 | C8×Q8 | — | 3 | 1 | no | — |
| 90 | C8×Q8 | — | 3 | 1 | no | — |
| 91 | C4×SD16 | — | 3 | 3 | no | — |
| 92 | C4×SD16 | — | 3 | 3 | no | — |
| 93 | C4×SD16 | — | 3 | 6 | no | — |
| 94 | C4×SD16 | — | 3 | 3 | no | — |
| 95 | C4×SD16 | — | 3 | 3 | no | — |
| 96 | C4×SD16 | — | 3 | 3 | no | — |
| 97 | C4×SD16 | — | 3 | 1 | no | — |
| 98 | C4×SD16 | — | 3 | 3 | no | — |
| 99 | C4×SD16 | — | 3 | 3 | no | — |
| 100 | C8×D4 | — | 3 | 3 | no | — |
| 101 | C8×D4 | — | 3 | 3 | no | — |
| 102 | #431 | — | 3 | 3 | no | — |
| 103 | C2×#106 | — | 3 | 3 | no | — |
| 104 | C2×#103 | — | 3 | 3 | no | — |
| 105 | C2×#102 | — | 3 | 3 | no | — |
| 106 | C2×C4×C8 | — | 3 | 3 | no | — |
| 107 | C23×D3 | — | 4 | 3 | no | — |
| 108 | C3×#41 | — | 4 | 8 | no | — |
| 109 | C2×C3×Q8 | — | 4 | 1 | no | — |
| 110 | C2×C3×D4 | — | 4 | 3 | no | — |
| 111 | C22×C3×C4 | — | 4 | 3 | no | — |
| 112 | C2×#67 | — | 4 | 6 | no | — |
| 113 | C22×Dic3 | — | 4 | 3 | no | — |
| 114 | #239 | — | 4 | 8 | no | — |
| 115 | D3×Q8 | — | 4 | 8 | no | — |
| 116 | #237 | — | 4 | 24 | no | — |
| 117 | D3×D4 | — | 4 | 24 | no | — |
| 118 | #235 | — | 4 | 24 | no | — |
| 119 | C2×D12 | — | 4 | 3 | no | — |
| 120 | C2×C4×D3 | — | 4 | 1 | no | — |
| 121 | C2×C4×D3 | — | 4 | 3 | no | — |
| 122 | C2×C4×D3 | — | 4 | 3 | no | — |
| 123 | C2×C4×D3 | — | 4 | 3 | no | — |
| 124 | C2×C4×D3 | — | 4 | 24 | no | — |
| 125 | C2×Dic6 | — | 4 | 3 | no | — |
| 126 | C2×SL(2,𝔽3) | — | 2 | 3 | no | — |
| 127 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 128 | GL(2,𝔽3) | — | 2 | 3 | no | — |
| 129 | GL(2,𝔽3) | #41 | 1 | 2 | yes | direct product factor |
| 130 | C2×#41 | — | 6 | 1 | no | — |
| 131 | C2×#41 | — | 6 | 3 | no | — |
| 132 | C2×#41 | — | 6 | 3 | no | — |
| 133 | C2×#41 | — | 12 | 8 | no | — |
| 134 | C2×#41 | S4 | 1 | 1 | yes | characteristic |
| 135 | C22×Q8 | — | 6 | 1 | no | — |
| 136 | C22×Q8 | C22×D3 | 1 | 3 | yes | — |
| 137 | C22×D4 | — | 3 | 3 | no | — |
| 138 | C22×D4 | — | 6 | 3 | no | — |
| 139 | C23×C4 | — | 6 | 3 | no | — |
| 140 | C2×SD16 | — | 3 | 1 | no | — |
| 141 | C2×SD16 | — | 3 | 3 | no | — |
| 142 | C2×SD16 | — | 3 | 3 | no | — |
| 143 | C2×SD16 | — | 3 | 3 | no | — |
| 144 | C2×SD16 | — | 3 | 6 | no | — |
| 145 | C2×SD16 | — | 6 | 6 | no | — |
| 146 | C2×SD16 | — | 6 | 6 | no | — |
| 147 | C22×C8 | — | 3 | 3 | no | — |
| 148 | C22×C8 | — | 3 | 3 | no | — |
| 149 | C4⋊Q8 | — | 3 | 3 | no | — |
| 150 | C4⋊Q8 | — | 3 | 3 | no | — |
| 151 | C4⋊Q8 | — | 3 | 3 | no | — |
| 152 | #127 | — | 3 | 3 | no | — |
| 153 | #125 | — | 6 | 3 | no | — |
| 154 | #124 | — | 6 | 3 | no | — |
| 155 | #123 | — | 6 | 6 | no | — |
| 156 | C22⋊Q8 | — | 3 | 6 | no | — |
| 157 | C22⋊Q8 | — | 6 | 3 | no | — |
| 158 | C22⋊Q8 | — | 6 | 3 | no | — |
| 159 | C22⋊Q8 | — | 6 | 3 | no | — |
| 160 | #121 | — | 3 | 6 | no | — |
| 161 | #121 | — | 6 | 3 | no | — |
| 162 | #121 | — | 6 | 3 | no | — |
| 163 | #121 | — | 6 | 3 | no | — |
| 164 | #120 | — | 6 | 6 | no | — |
| 165 | C4×Q8 | — | 3 | 1 | no | — |
| 166 | C4×Q8 | — | 3 | 1 | no | — |
| 167 | C4×Q8 | — | 3 | 3 | no | — |
| 168 | C4×Q8 | — | 3 | 3 | no | — |
| 169 | C4×Q8 | — | 6 | 1 | no | — |
| 170 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 171 | C4×Q8 | C22×D3 | 1 | 3 | yes | — |
| 172 | C4×D4 | — | 3 | 3 | no | — |
| 173 | C4×D4 | — | 3 | 3 | no | — |
| 174 | C4×D4 | — | 3 | 1 | no | — |
| 175 | C4×D4 | — | 3 | 3 | no | — |
| 176 | C4×D4 | — | 3 | 3 | no | — |
| 177 | C4×D4 | — | 3 | 3 | no | — |
| 178 | C4×D4 | — | 6 | 3 | no | — |
| 179 | C4×D4 | — | 6 | 3 | no | — |
| 180 | C4×D4 | — | 6 | 3 | no | — |
| 181 | #117 | — | 3 | 3 | no | — |
| 182 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 183 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 184 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 185 | C2×C42 | — | 3 | 3 | no | — |
| 186 | #106 | — | 3 | 1 | no | — |
| 187 | #106 | — | 3 | 3 | no | — |
| 188 | #106 | — | 3 | 3 | no | — |
| 189 | #106 | — | 3 | 3 | no | — |
| 190 | #106 | — | 3 | 6 | no | — |
| 191 | C4⋊C8 | — | 3 | 3 | no | — |
| 192 | C4⋊C8 | — | 3 | 3 | no | — |
| 193 | #103 | — | 3 | 3 | no | — |
| 194 | #103 | — | 3 | 3 | no | — |
| 195 | #103 | — | 3 | 6 | no | — |
| 196 | #103 | — | 3 | 1 | no | — |
| 197 | #103 | — | 3 | 3 | no | — |
| 198 | #102 | — | 3 | 1 | no | — |
| 199 | #102 | — | 3 | 3 | no | — |
| 200 | #102 | — | 3 | 3 | no | — |
| 201 | #102 | — | 3 | 3 | no | — |
| 202 | #102 | — | 3 | 6 | no | — |
| 203 | C22⋊C8 | — | 3 | 6 | no | — |
| 204 | C4×C8 | — | 3 | 1 | no | — |
| 205 | C4×C8 | — | 3 | 3 | no | — |
| 206 | C4×C8 | — | 3 | 3 | no | — |
| 207 | C4×C8 | — | 3 | 3 | no | — |
| 208 | C23×C3 | — | 4 | 3 | no | — |
| 209 | C22×D3 | — | 4 | 1 | no | — |
| 210 | C22×D3 | — | 4 | 3 | no | — |
| 211 | C22×D3 | — | 4 | 12 | no | — |
| 212 | C22×D3 | — | 8 | 6 | no | — |
| 213 | C22×D3 | — | 8 | 6 | no | — |
| 214 | C3×Q8 | — | 4 | 4 | no | — |
| 215 | C3×D4 | — | 4 | 12 | no | — |
| 216 | C2×C3×C4 | — | 4 | 12 | no | — |
| 217 | C2×C3×C4 | — | 4 | 3 | no | — |
| 218 | C2×C3×C4 | — | 4 | 1 | no | — |
| 219 | #67 | — | 4 | 24 | no | — |
| 220 | C2×Dic3 | — | 4 | 3 | no | — |
| 221 | C2×Dic3 | — | 4 | 1 | no | — |
| 222 | C2×Dic3 | — | 4 | 12 | no | — |
| 223 | D12 | — | 4 | 12 | no | — |
| 224 | C4×D3 | — | 4 | 4 | no | — |
| 225 | C4×D3 | — | 4 | 12 | no | — |
| 226 | C4×D3 | — | 4 | 12 | no | — |
| 227 | C4×D3 | — | 4 | 12 | no | — |
| 228 | Dic6 | — | 4 | 12 | no | — |
| 229 | SL(2,𝔽3) | C2×#41 | 1 | 1 | yes | characteristic, complemented |
| 230 | C24 | — | 6 | 3 | no | — |
| 231 | #41 | — | 12 | 4 | no | — |
| 232 | #41 | — | 12 | 12 | no | — |
| 233 | #41 | — | 12 | 12 | no | — |
| 234 | #41 | GL(2,𝔽3) | 1 | 8 | yes | direct product factor |
| 235 | C2×Q8 | — | 2 | 3 | no | — |
| 236 | C2×Q8 | — | 3 | 3 | no | — |
| 237 | C2×Q8 | — | 6 | 1 | no | — |
| 238 | C2×Q8 | — | 6 | 3 | no | — |
| 239 | C2×Q8 | — | 6 | 3 | no | — |
| 240 | C2×Q8 | — | 12 | 4 | no | — |
| 241 | C2×Q8 | C2×S4 | 1 | 1 | yes | characteristic |
| 242 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 243 | C2×D4 | — | 3 | 1 | no | — |
| 244 | C2×D4 | — | 3 | 3 | no | — |
| 245 | C2×D4 | — | 6 | 3 | no | — |
| 246 | C2×D4 | — | 6 | 3 | no | — |
| 247 | C2×D4 | — | 6 | 6 | no | — |
| 248 | C2×D4 | — | 6 | 6 | no | — |
| 249 | C2×D4 | — | 6 | 3 | no | — |
| 250 | C2×D4 | — | 12 | 3 | no | — |
| 251 | C2×D4 | — | 12 | 12 | no | — |
| 252 | C2×D4 | C2×S4 | 1 | 3 | yes | — |
| 253 | C22×C4 | — | 3 | 3 | no | — |
| 254 | C22×C4 | — | 3 | 3 | no | — |
| 255 | C22×C4 | — | 6 | 3 | no | — |
| 256 | C22×C4 | — | 6 | 3 | no | — |
| 257 | C22×C4 | — | 6 | 3 | no | — |
| 258 | C22×C4 | — | 6 | 1 | no | — |
| 259 | C22×C4 | — | 6 | 3 | no | — |
| 260 | C22×C4 | — | 12 | 12 | no | — |
| 261 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 262 | SD16 | — | 3 | 2 | no | — |
| 263 | SD16 | — | 3 | 2 | no | — |
| 264 | SD16 | — | 6 | 6 | no | — |
| 265 | SD16 | — | 6 | 6 | no | — |
| 266 | SD16 | — | 6 | 3 | no | — |
| 267 | SD16 | — | 6 | 3 | no | — |
| 268 | SD16 | — | 12 | 6 | no | — |
| 269 | C2×C8 | — | 3 | 1 | no | — |
| 270 | C2×C8 | — | 3 | 1 | no | — |
| 271 | C2×C8 | — | 3 | 3 | no | — |
| 272 | C2×C8 | — | 3 | 3 | no | — |
| 273 | C2×C8 | — | 6 | 6 | no | — |
| 274 | C2×C8 | — | 12 | 3 | no | — |
| 275 | C4⋊C4 | — | 3 | 1 | no | — |
| 276 | C4⋊C4 | — | 3 | 3 | no | — |
| 277 | C4⋊C4 | — | 3 | 3 | no | — |
| 278 | C4⋊C4 | — | 3 | 3 | no | — |
| 279 | C4⋊C4 | — | 3 | 1 | no | — |
| 280 | C4⋊C4 | — | 3 | 3 | no | — |
| 281 | C4⋊C4 | — | 6 | 3 | no | — |
| 282 | C4⋊C4 | — | 6 | 3 | no | — |
| 283 | C4⋊C4 | — | 6 | 3 | no | — |
| 284 | C4⋊C4 | — | 6 | 3 | no | — |
| 285 | C22⋊C4 | — | 3 | 6 | no | — |
| 286 | C22⋊C4 | — | 6 | 1 | no | — |
| 287 | C22⋊C4 | — | 6 | 3 | no | — |
| 288 | C22⋊C4 | — | 6 | 3 | no | — |
| 289 | C22⋊C4 | — | 6 | 3 | no | — |
| 290 | C22⋊C4 | — | 6 | 6 | no | — |
| 291 | C42 | — | 3 | 3 | no | — |
| 292 | C42 | — | 3 | 3 | no | — |
| 293 | C42 | — | 3 | 1 | no | — |
| 294 | C42 | — | 3 | 3 | no | — |
| 295 | C22×C3 | — | 4 | 6 | no | — |
| 296 | C22×C3 | — | 4 | 1 | no | — |
| 297 | C22×C3 | — | 8 | 3 | no | — |
| 298 | C22×C3 | — | 8 | 3 | no | — |
| 299 | C2×D3 | — | 4 | 2 | no | — |
| 300 | C2×D3 | — | 4 | 2 | no | — |
| 301 | C2×D3 | — | 4 | 2 | no | — |
| 302 | C2×D3 | — | 4 | 6 | no | — |
| 303 | C2×D3 | — | 8 | 3 | no | — |
| 304 | C2×D3 | — | 8 | 3 | no | — |
| 305 | C2×D3 | — | 8 | 24 | no | — |
| 306 | C3×C4 | — | 4 | 6 | no | — |
| 307 | C3×C4 | — | 4 | 2 | no | — |
| 308 | Dic3 | — | 4 | 6 | no | — |
| 309 | Dic3 | — | 4 | 2 | no | — |
| 310 | C23 | — | 6 | 1 | no | — |
| 311 | C23 | — | 6 | 3 | no | — |
| 312 | C23 | — | 12 | 6 | no | — |
| 313 | C23 | — | 12 | 6 | no | — |
| 314 | C23 | — | 24 | 3 | no | — |
| 315 | C23 | C22×S4 | 1 | 3 | yes | — |
| 316 | Q8 | — | 3 | 1 | no | — |
| 317 | Q8 | — | 6 | 3 | no | — |
| 318 | Q8 | — | 6 | 3 | no | — |
| 319 | Q8 | — | 12 | 6 | no | — |
| 320 | Q8 | — | 24 | 1 | no | — |
| 321 | Q8 | C2×GL(2,𝔽3) | 1 | 4 | yes | complemented |
| 322 | Q8 | D3×#41 | 1 | 1 | yes | characteristic, complemented |
| 323 | D4 | — | 3 | 2 | no | — |
| 324 | D4 | — | 6 | 1 | no | — |
| 325 | D4 | — | 6 | 3 | no | — |
| 326 | D4 | — | 6 | 3 | no | — |
| 327 | D4 | — | 12 | 6 | no | — |
| 328 | D4 | — | 12 | 12 | no | — |
| 329 | D4 | — | 12 | 6 | no | — |
| 330 | D4 | — | 24 | 3 | no | — |
| 331 | D4 | C2×GL(2,𝔽3) | 1 | 12 | yes | complemented |
| 332 | C2×C4 | — | 3 | 1 | no | — |
| 333 | C2×C4 | — | 3 | 1 | no | — |
| 334 | C2×C4 | — | 3 | 3 | no | — |
| 335 | C2×C4 | — | 3 | 3 | no | — |
| 336 | C2×C4 | — | 6 | 3 | no | — |
| 337 | C2×C4 | — | 6 | 3 | no | — |
| 338 | C2×C4 | — | 6 | 1 | no | — |
| 339 | C2×C4 | — | 6 | 3 | no | — |
| 340 | C2×C4 | — | 12 | 3 | no | — |
| 341 | C2×C4 | — | 12 | 6 | no | — |
| 342 | C2×C4 | — | 12 | 2 | no | — |
| 343 | C2×C4 | — | 12 | 6 | no | — |
| 344 | C2×C4 | — | 12 | 6 | no | — |
| 345 | C2×C4 | — | 12 | 6 | no | — |
| 346 | C2×C4 | C2×GL(2,𝔽3) | 1 | 12 | yes | complemented |
| 347 | C2×C4 | C22×S4 | 1 | 3 | yes | — |
| 348 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 349 | C8 | — | 3 | 2 | no | — |
| 350 | C8 | — | 6 | 3 | no | — |
| 351 | C8 | — | 6 | 1 | no | — |
| 352 | C8 | — | 6 | 3 | no | — |
| 353 | C2×C3 | — | 4 | 1 | no | — |
| 354 | C2×C3 | — | 4 | 1 | no | — |
| 355 | C2×C3 | — | 4 | 1 | no | — |
| 356 | C2×C3 | — | 8 | 6 | no | — |
| 357 | D3 | — | 4 | 4 | no | — |
| 358 | D3 | — | 8 | 6 | no | — |
| 359 | C22 | — | 2 | 3 | no | — |
| 360 | C22 | — | 2 | 3 | no | — |
| 361 | C22 | — | 6 | 2 | no | — |
| 362 | C22 | — | 12 | 1 | no | — |
| 363 | C22 | — | 12 | 1 | no | — |
| 364 | C22 | — | 12 | 3 | no | — |
| 365 | C22 | — | 12 | 3 | no | — |
| 366 | C22 | — | 12 | 3 | no | — |
| 367 | C22 | — | 24 | 12 | no | — |
| 368 | C22 | C22×GL(2,𝔽3) | 1 | 6 | yes | — |
| 369 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 370 | C4 | — | 3 | 1 | no | — |
| 371 | C4 | — | 3 | 1 | no | — |
| 372 | C4 | — | 6 | 3 | no | — |
| 373 | C4 | — | 6 | 1 | no | — |
| 374 | C4 | — | 6 | 3 | no | — |
| 375 | C4 | — | 12 | 1 | no | — |
| 376 | C4 | — | 12 | 3 | no | — |
| 377 | C4 | C22×GL(2,𝔽3) | 1 | 2 | yes | — |
| 378 | C4 | C22×GL(2,𝔽3) | 1 | 6 | yes | — |
| 379 | C3 | — | 4 | 1 | no | Sylow |
| 380 | C2 | — | 2 | 6 | no | — |
| 381 | C2 | — | 12 | 2 | no | — |
| 382 | C2 | — | 24 | 3 | no | — |
| 383 | C2 | C23×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 384 | C2 | #41×S4 | 1 | 1 | yes | characteristic |
| 385 | C2 | #87497 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C24 |
| Frattini, socle | C22 | C23×S4 |
| Fitting | Q8×#41 | D3 |
| hypercenter | C2×#41 | S4 |
| hypocenter | SL(2,𝔽3) | C2×#41 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C4 ◁ C2×#41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #41×SL(2,𝔽3) | C2 | semidirect |
| 2. #85430 | C2 | semidirect |
| 3. Q8×GL(2,𝔽3) | C2 | semidirect |
| 4. #85397 | C2 | semidirect |
| 5. D4×GL(2,𝔽3) | C2 | semidirect |
| 6. #85325 | C2 | semidirect |
| 7. C2×C4×GL(2,𝔽3) | C2 | semidirect |
| 11. Q8×SL(2,𝔽3) | C22 | semidirect |
| 12. D4×SL(2,𝔽3) | C22 | semidirect |
| 13. C2×C4×SL(2,𝔽3) | C22 | semidirect |
| 14. #5421 | C22 | semidirect |
| 15. C2×#859 | C22 | semidirect |
| 16. C4⋊GL(2,𝔽3) | C22 | semidirect |
| 19. C4×GL(2,𝔽3) | C22 | semidirect |
| 20. C4×GL(2,𝔽3) | C22 | semidirect |
| 21. #5386 | C22 | semidirect |
| 22. Q8×#41 | D3 | semidirect |
| 55. Q82 | C2×D3 | semidirect |
| 60. D4×Q8 | C2×D3 | semidirect |
| 69. C2×C4×Q8 | C2×D3 | semidirect |
| 129. GL(2,𝔽3) | #41 | direct (N × H) |
| 229. SL(2,𝔽3) | C2×#41 | semidirect |
| 234. #41 | GL(2,𝔽3) | direct (N × H) |
| 321. Q8 | C2×GL(2,𝔽3) | semidirect |
| 322. Q8 | D3×#41 | semidirect |
| 331. D4 | C2×GL(2,𝔽3) | semidirect |
| 346. C2×C4 | C2×GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #41×SL(2,𝔽3) | 2 |
| #85430 | 2 |
| Q8×GL(2,𝔽3) | 2 |
| #85397 | 2 |
| D4×GL(2,𝔽3) | 2 |
| #85325 | 2 |
| C2×C4×GL(2,𝔽3) | 2 |
| SD16×#41 | 3 |
| C2×D3×#41 | 4 |