Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
2017 conjugacy classes of subgroups; only the 151 normal subgroups are shown.
This group has 2017 conjugacy classes of subgroups — too many to draw. Only its 151 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 | C23×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #85523 | C2 | 1 | 8 | yes | maximal, direct product factor |
| 4 | C2×#5427 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C2×#5426 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C2×#5421 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×#3006 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C22×#41⋊C3 | — | 2 | 4 | no | — |
| 9 | C22×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×#41⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22×#41⋊C3 | C22 | 1 | 4 | yes | complemented |
| 12 | C23×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×#986 | — | 2 | 4 | no | — |
| 14 | C2×#986 | C22 | 1 | 4 | yes | complemented |
| 15 | C22×GL(2,𝔽3) | C22 | 1 | 2 | yes | complemented |
| 16 | C2×#930 | — | 4 | 1 | no | maximal |
| 17 | #5427 | C22 | 1 | 8 | yes | complemented |
| 18 | #5426 | C22 | 1 | 4 | yes | complemented |
| 19 | #5421 | C22 | 1 | 8 | yes | complemented |
| 20 | C2×#859 | C22 | 1 | 2 | yes | complemented |
| 21 | C23×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 22 | C22×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 23 | C22×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 24 | C22×#121 | — | 3 | 1 | no | — |
| 25 | #3006 | — | 3 | 8 | no | — |
| 26 | #3006 | — | 3 | 8 | no | — |
| 27 | C2×#449 | — | 3 | 2 | no | — |
| 28 | C2×#449 | — | 3 | 2 | no | — |
| 29 | C2×#448 | — | 3 | 2 | no | — |
| 30 | C2×#447 | — | 3 | 2 | no | — |
| 31 | C2×#417 | — | 3 | 1 | no | — |
| 32 | C2×#417 | — | 3 | 1 | no | — |
| 33 | C2×#407 | — | 3 | 1 | no | — |
| 34 | C23×C3×C4 | — | 4 | 1 | no | — |
| 35 | C22×#67 | — | 4 | 2 | no | — |
| 36 | C22×D12 | — | 4 | 1 | no | — |
| 37 | C2×#41⋊C3 | — | 2 | 8 | no | — |
| 38 | C2×#41⋊C3 | — | 2 | 2 | no | — |
| 39 | C2×#41⋊C3 | D4 | 1 | 4 | yes | complemented |
| 40 | C2×#41⋊C3 | C23 | 1 | 2 | yes | — |
| 41 | C2×#41⋊C3 | C23 | 1 | 2 | yes | complemented |
| 42 | C22×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 43 | C22×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 44 | C22×SL(2,𝔽3) | C23 | 1 | 2 | yes | complemented |
| 45 | #986 | — | 2 | 8 | no | — |
| 46 | C2×GL(2,𝔽3) | — | 2 | 4 | no | — |
| 47 | C2×GL(2,𝔽3) | C23 | 1 | 4 | yes | complemented |
| 48 | #930 | — | 4 | 8 | no | — |
| 49 | C2×(C2×D3)⋊C4 | — | 4 | 2 | no | — |
| 50 | C2×C4⋊Dic3 | — | 4 | 1 | no | — |
| 51 | #859 | C23 | 1 | 4 | yes | complemented |
| 52 | C22×#41 | — | 2 | 4 | no | — |
| 53 | C22×#41 | — | 3 | 1 | no | — |
| 54 | C22×#41 | — | 3 | 1 | no | — |
| 55 | C22×#41 | — | 3 | 4 | no | — |
| 56 | C22×#41 | — | 6 | 4 | no | — |
| 57 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 58 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 59 | C22×#41 | C2×D3 | 1 | 4 | yes | complemented |
| 60 | C23×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | C23×D4 | — | 3 | 1 | no | — |
| 62 | C23×D4 | — | 3 | 1 | no | — |
| 63 | C23×D4 | — | 6 | 1 | no | — |
| 64 | C24×C4 | — | 3 | 1 | no | — |
| 65 | C2×AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 66 | C2×AGL(1,ℤ/8) | — | 3 | 8 | no | — |
| 67 | C2×AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 68 | C2×AGL(1,ℤ/8) | — | 6 | 4 | no | — |
| 69 | C2×AGL(1,ℤ/8) | — | 6 | 8 | no | — |
| 70 | C2×AGL(1,ℤ/8) | — | 6 | 4 | no | — |
| 71 | C22×SD16 | — | 3 | 2 | no | — |
| 72 | C22×SD16 | — | 3 | 2 | no | — |
| 73 | C22×D8 | — | 3 | 2 | no | — |
| 74 | C22×D8 | — | 3 | 2 | no | — |
| 75 | C22×M16 | — | 3 | 1 | no | — |
| 76 | C22×M16 | — | 3 | 1 | no | — |
| 77 | C2×#121 | — | 3 | 2 | no | — |
| 78 | C2×#121 | — | 3 | 2 | no | — |
| 79 | C2×#121 | — | 3 | 8 | no | — |
| 80 | C2×#121 | — | 6 | 4 | no | — |
| 81 | C2×#121 | — | 6 | 1 | no | — |
| 82 | C2×#121 | — | 6 | 1 | no | — |
| 83 | C22×C4⋊C4 | — | 3 | 1 | no | — |
| 84 | C22×C22⋊C4 | — | 6 | 1 | no | — |
| 85 | #449 | — | 3 | 8 | no | — |
| 86 | #449 | — | 3 | 16 | no | — |
| 87 | #449 | — | 3 | 8 | no | — |
| 88 | #448 | — | 3 | 8 | no | — |
| 89 | #448 | — | 3 | 8 | no | — |
| 90 | #447 | — | 3 | 16 | no | — |
| 91 | #417 | — | 3 | 4 | no | — |
| 92 | #417 | — | 3 | 8 | no | — |
| 93 | #417 | — | 3 | 4 | no | — |
| 94 | C2×#103 | — | 3 | 2 | no | — |
| 95 | C2×#103 | — | 3 | 2 | no | — |
| 96 | C2×#102 | — | 3 | 2 | no | — |
| 97 | C2×#102 | — | 3 | 2 | no | — |
| 98 | #407 | — | 3 | 8 | no | — |
| 99 | C2×C22⋊C8 | — | 3 | 2 | no | — |
| 100 | C2×C22⋊C8 | — | 3 | 2 | no | — |
| 101 | C24×C3 | — | 4 | 1 | no | — |
| 102 | C23×D3 | — | 4 | 2 | no | — |
| 103 | C22×C3×C4 | — | 4 | 1 | no | — |
| 104 | C22×C3×C4 | — | 4 | 1 | no | — |
| 105 | C22×C3×C4 | — | 4 | 4 | no | — |
| 106 | C22×C3×C4 | — | 8 | 4 | no | — |
| 107 | C2×#67 | — | 4 | 8 | no | — |
| 108 | C2×#67 | — | 8 | 8 | no | — |
| 109 | C22×Dic3 | — | 4 | 2 | no | — |
| 110 | C2×D12 | — | 4 | 4 | no | — |
| 111 | C2×D12 | — | 8 | 4 | no | — |
| 112 | #41⋊C3 | — | 2 | 2 | no | — |
| 113 | #41⋊C3 | C2×D4 | 1 | 4 | yes | complemented |
| 114 | C2×SL(2,𝔽3) | — | 2 | 2 | no | — |
| 115 | C2×SL(2,𝔽3) | C2×D4 | 1 | 2 | yes | complemented |
| 116 | C2×SL(2,𝔽3) | C24 | 1 | 1 | yes | characteristic |
| 117 | GL(2,𝔽3) | — | 2 | 4 | no | — |
| 118 | (C2×D3)⋊C4 | — | 4 | 8 | no | — |
| 119 | C4⋊Dic3 | — | 4 | 4 | no | — |
| 120 | C25 | — | 3 | 1 | no | — |
| 121 | C25 | — | 6 | 1 | no | — |
| 122 | C2×#41 | — | 2 | 8 | no | — |
| 123 | C2×#41 | — | 2 | 2 | no | — |
| 124 | C2×#41 | — | 3 | 2 | no | — |
| 125 | C2×#41 | — | 3 | 4 | no | — |
| 126 | C2×#41 | — | 3 | 4 | no | — |
| 127 | C2×#41 | — | 3 | 2 | no | — |
| 128 | C2×#41 | — | 3 | 4 | no | — |
| 129 | C2×#41 | — | 3 | 8 | no | — |
| 130 | C2×#41 | — | 6 | 2 | no | — |
| 131 | C2×#41 | — | 6 | 2 | no | — |
| 132 | C2×#41 | — | 6 | 4 | no | — |
| 133 | C2×#41 | — | 6 | 4 | no | — |
| 134 | C2×#41 | — | 6 | 16 | no | — |
| 135 | C2×#41 | — | 6 | 2 | no | — |
| 136 | C2×#41 | — | 6 | 4 | no | — |
| 137 | C2×#41 | — | 6 | 8 | no | — |
| 138 | C2×#41 | — | 6 | 16 | no | — |
| 139 | C2×#41 | #67 | 1 | 4 | yes | complemented |
| 140 | C2×#41 | C22×D3 | 1 | 2 | yes | — |
| 141 | C2×#41 | C22×D3 | 1 | 2 | yes | — |
| 142 | C22×Q8 | — | 2 | 2 | no | — |
| 143 | C22×Q8 | — | 3 | 1 | no | — |
| 144 | C22×Q8 | — | 3 | 2 | no | — |
| 145 | C22×Q8 | — | 6 | 2 | no | — |
| 146 | C22×Q8 | C22×D3 | 1 | 2 | yes | — |
| 147 | C22×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 148 | C22×D4 | — | 3 | 1 | no | — |
| 149 | C22×D4 | — | 3 | 1 | no | — |
| 150 | C22×D4 | — | 3 | 4 | no | — |
| 151 | C22×D4 | — | 3 | 2 | no | — |
| 152 | C22×D4 | — | 3 | 4 | no | — |
| 153 | C22×D4 | — | 6 | 2 | no | — |
| 154 | C22×D4 | — | 6 | 1 | no | — |
| 155 | C22×D4 | — | 6 | 4 | no | — |
| 156 | C22×D4 | — | 6 | 4 | no | — |
| 157 | C22×D4 | — | 6 | 2 | no | — |
| 158 | C22×D4 | — | 6 | 4 | no | — |
| 159 | C22×D4 | — | 6 | 4 | no | — |
| 160 | C22×D4 | — | 6 | 8 | no | — |
| 161 | C22×D4 | — | 6 | 2 | no | — |
| 162 | C22×D4 | — | 6 | 2 | no | — |
| 163 | C22×D4 | — | 6 | 1 | no | — |
| 164 | C22×D4 | — | 12 | 8 | no | — |
| 165 | C23×C4 | — | 3 | 1 | no | — |
| 166 | C23×C4 | — | 3 | 1 | no | — |
| 167 | C23×C4 | — | 3 | 1 | no | — |
| 168 | C23×C4 | — | 3 | 1 | no | — |
| 169 | C23×C4 | — | 3 | 4 | no | — |
| 170 | C23×C4 | — | 3 | 4 | no | — |
| 171 | C23×C4 | — | 3 | 1 | no | — |
| 172 | C23×C4 | — | 6 | 4 | no | — |
| 173 | C23×C4 | — | 6 | 4 | no | — |
| 174 | C23×C4 | — | 6 | 1 | no | — |
| 175 | C23×C4 | S4 | 1 | 1 | yes | characteristic |
| 176 | AGL(1,ℤ/8) | — | 6 | 16 | no | — |
| 177 | AGL(1,ℤ/8) | — | 6 | 8 | no | — |
| 178 | AGL(1,ℤ/8) | — | 6 | 32 | no | — |
| 179 | AGL(1,ℤ/8) | — | 6 | 8 | no | — |
| 180 | C2×SD16 | — | 3 | 8 | no | — |
| 181 | C2×SD16 | — | 3 | 4 | no | — |
| 182 | C2×SD16 | — | 3 | 4 | no | — |
| 183 | C2×SD16 | — | 6 | 4 | no | — |
| 184 | C2×SD16 | — | 6 | 8 | no | — |
| 185 | C2×SD16 | — | 6 | 4 | no | — |
| 186 | C2×D8 | — | 3 | 8 | no | — |
| 187 | C2×D8 | — | 3 | 8 | no | — |
| 188 | C2×D8 | — | 6 | 8 | no | — |
| 189 | C2×D8 | — | 6 | 8 | no | — |
| 190 | C2×M16 | — | 3 | 4 | no | — |
| 191 | C2×M16 | — | 3 | 4 | no | — |
| 192 | C2×M16 | — | 6 | 4 | no | — |
| 193 | C2×M16 | — | 6 | 4 | no | — |
| 194 | C22×C8 | — | 3 | 2 | no | — |
| 195 | C22×C8 | — | 3 | 2 | no | — |
| 196 | #121 | — | 3 | 8 | no | — |
| 197 | #121 | — | 3 | 8 | no | — |
| 198 | #121 | — | 6 | 4 | no | — |
| 199 | #121 | — | 6 | 8 | no | — |
| 200 | #121 | — | 6 | 4 | no | — |
| 201 | #121 | — | 6 | 8 | no | — |
| 202 | C2×C4⋊C4 | — | 3 | 2 | no | — |
| 203 | C2×C4⋊C4 | — | 3 | 4 | no | — |
| 204 | C2×C4⋊C4 | — | 6 | 2 | no | — |
| 205 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 206 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 207 | C2×C22⋊C4 | — | 6 | 8 | no | — |
| 208 | C2×C22⋊C4 | — | 6 | 2 | no | — |
| 209 | #103 | — | 3 | 8 | no | — |
| 210 | #103 | — | 3 | 4 | no | — |
| 211 | #103 | — | 3 | 4 | no | — |
| 212 | #102 | — | 3 | 8 | no | — |
| 213 | #102 | — | 3 | 8 | no | — |
| 214 | C22⋊C8 | — | 3 | 8 | no | — |
| 215 | C22⋊C8 | — | 3 | 8 | no | — |
| 216 | C23×C3 | — | 4 | 4 | no | — |
| 217 | C23×C3 | — | 4 | 2 | no | — |
| 218 | C23×C3 | — | 4 | 1 | no | — |
| 219 | C23×C3 | — | 8 | 2 | no | — |
| 220 | C23×C3 | — | 8 | 2 | no | — |
| 221 | C22×D3 | — | 4 | 4 | no | — |
| 222 | C22×D3 | — | 8 | 4 | no | — |
| 223 | C22×D3 | — | 8 | 4 | no | — |
| 224 | C22×D3 | — | 8 | 4 | no | — |
| 225 | C2×C3×C4 | — | 4 | 4 | no | — |
| 226 | C2×C3×C4 | — | 4 | 2 | no | — |
| 227 | C2×C3×C4 | — | 4 | 2 | no | — |
| 228 | C2×C3×C4 | — | 8 | 8 | no | — |
| 229 | C2×C3×C4 | — | 8 | 2 | no | — |
| 230 | #67 | — | 8 | 16 | no | — |
| 231 | C2×Dic3 | — | 4 | 4 | no | — |
| 232 | C2×Dic3 | — | 8 | 4 | no | — |
| 233 | D12 | — | 8 | 8 | no | — |
| 234 | SL(2,𝔽3) | C22×D4 | 1 | 1 | yes | characteristic, complemented |
| 235 | C24 | — | 3 | 1 | no | — |
| 236 | C24 | — | 3 | 1 | no | — |
| 237 | C24 | — | 3 | 4 | no | — |
| 238 | C24 | — | 6 | 4 | no | — |
| 239 | C24 | — | 6 | 4 | no | — |
| 240 | C24 | — | 6 | 4 | no | — |
| 241 | C24 | — | 6 | 2 | no | — |
| 242 | C24 | — | 12 | 2 | no | — |
| 243 | C24 | — | 12 | 4 | no | — |
| 244 | C24 | — | 12 | 4 | no | — |
| 245 | C24 | — | 12 | 4 | no | — |
| 246 | C24 | C2×S4 | 1 | 1 | yes | characteristic |
| 247 | #41 | — | 2 | 2 | no | — |
| 248 | #41 | — | 3 | 8 | no | — |
| 249 | #41 | — | 3 | 4 | no | — |
| 250 | #41 | — | 3 | 16 | no | — |
| 251 | #41 | — | 6 | 4 | no | — |
| 252 | #41 | — | 6 | 16 | no | — |
| 253 | #41 | — | 6 | 8 | no | — |
| 254 | #41 | — | 6 | 4 | no | — |
| 255 | #41 | — | 6 | 8 | no | — |
| 256 | #41 | — | 6 | 16 | no | — |
| 257 | #41 | — | 6 | 2 | no | — |
| 258 | #41 | — | 6 | 8 | no | — |
| 259 | #41 | — | 6 | 4 | no | — |
| 260 | #41 | C2×#67 | 1 | 4 | yes | complemented |
| 261 | C2×Q8 | — | 2 | 2 | no | — |
| 262 | C2×Q8 | — | 3 | 1 | no | — |
| 263 | C2×Q8 | — | 3 | 2 | no | — |
| 264 | C2×Q8 | — | 3 | 2 | no | — |
| 265 | C2×Q8 | — | 3 | 4 | no | — |
| 266 | C2×Q8 | — | 6 | 1 | no | — |
| 267 | C2×Q8 | — | 6 | 2 | no | — |
| 268 | C2×Q8 | — | 6 | 4 | no | — |
| 269 | C2×Q8 | — | 6 | 2 | no | — |
| 270 | C2×Q8 | — | 6 | 4 | no | — |
| 271 | C2×Q8 | C2×#67 | 1 | 2 | yes | — |
| 272 | C2×Q8 | C23×D3 | 1 | 1 | yes | characteristic |
| 273 | C2×D4 | — | 3 | 2 | no | — |
| 274 | C2×D4 | — | 3 | 2 | no | — |
| 275 | C2×D4 | — | 3 | 4 | no | — |
| 276 | C2×D4 | — | 3 | 4 | no | — |
| 277 | C2×D4 | — | 3 | 4 | no | — |
| 278 | C2×D4 | — | 6 | 2 | no | — |
| 279 | C2×D4 | — | 6 | 1 | no | — |
| 280 | C2×D4 | — | 6 | 1 | no | — |
| 281 | C2×D4 | — | 6 | 4 | no | — |
| 282 | C2×D4 | — | 6 | 2 | no | — |
| 283 | C2×D4 | — | 6 | 4 | no | — |
| 284 | C2×D4 | — | 6 | 4 | no | — |
| 285 | C2×D4 | — | 6 | 8 | no | — |
| 286 | C2×D4 | — | 6 | 8 | no | — |
| 287 | C2×D4 | — | 6 | 2 | no | — |
| 288 | C2×D4 | — | 6 | 8 | no | — |
| 289 | C2×D4 | — | 6 | 8 | no | — |
| 290 | C2×D4 | — | 6 | 8 | no | — |
| 291 | C2×D4 | — | 6 | 16 | no | — |
| 292 | C2×D4 | — | 6 | 8 | no | — |
| 293 | C2×D4 | — | 6 | 8 | no | — |
| 294 | C2×D4 | — | 6 | 4 | no | — |
| 295 | C2×D4 | — | 12 | 8 | no | — |
| 296 | C2×D4 | — | 12 | 32 | no | — |
| 297 | C2×D4 | — | 12 | 8 | no | — |
| 298 | C2×D4 | — | 12 | 8 | no | — |
| 299 | C2×D4 | — | 12 | 4 | no | — |
| 300 | C22×C4 | — | 2 | 4 | no | — |
| 301 | C22×C4 | — | 3 | 1 | no | — |
| 302 | C22×C4 | — | 3 | 4 | no | — |
| 303 | C22×C4 | — | 3 | 4 | no | — |
| 304 | C22×C4 | — | 3 | 2 | no | — |
| 305 | C22×C4 | — | 3 | 2 | no | — |
| 306 | C22×C4 | — | 3 | 2 | no | — |
| 307 | C22×C4 | — | 3 | 2 | no | — |
| 308 | C22×C4 | — | 3 | 1 | no | — |
| 309 | C22×C4 | — | 3 | 2 | no | — |
| 310 | C22×C4 | — | 3 | 4 | no | — |
| 311 | C22×C4 | — | 3 | 2 | no | — |
| 312 | C22×C4 | — | 3 | 4 | no | — |
| 313 | C22×C4 | — | 6 | 2 | no | — |
| 314 | C22×C4 | — | 6 | 8 | no | — |
| 315 | C22×C4 | — | 6 | 2 | no | — |
| 316 | C22×C4 | — | 6 | 8 | no | — |
| 317 | C22×C4 | — | 6 | 16 | no | — |
| 318 | C22×C4 | — | 6 | 4 | no | — |
| 319 | C22×C4 | — | 6 | 2 | no | — |
| 320 | C22×C4 | — | 6 | 2 | no | — |
| 321 | C22×C4 | — | 6 | 4 | no | — |
| 322 | C22×C4 | — | 6 | 4 | no | — |
| 323 | C22×C4 | — | 6 | 2 | no | — |
| 324 | C22×C4 | — | 12 | 4 | no | — |
| 325 | C22×C4 | C2×S4 | 1 | 4 | yes | — |
| 326 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 327 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 328 | SD16 | — | 6 | 8 | no | — |
| 329 | SD16 | — | 6 | 4 | no | — |
| 330 | SD16 | — | 6 | 16 | no | — |
| 331 | SD16 | — | 6 | 4 | no | — |
| 332 | D8 | — | 6 | 16 | no | — |
| 333 | D8 | — | 6 | 16 | no | — |
| 334 | M16 | — | 6 | 8 | no | — |
| 335 | M16 | — | 6 | 8 | no | — |
| 336 | C2×C8 | — | 3 | 4 | no | — |
| 337 | C2×C8 | — | 3 | 4 | no | — |
| 338 | C2×C8 | — | 6 | 4 | no | — |
| 339 | C2×C8 | — | 6 | 4 | no | — |
| 340 | C4⋊C4 | — | 3 | 4 | no | — |
| 341 | C4⋊C4 | — | 6 | 2 | no | — |
| 342 | C4⋊C4 | — | 6 | 4 | no | — |
| 343 | C22⋊C4 | — | 6 | 8 | no | — |
| 344 | C22⋊C4 | — | 6 | 8 | no | — |
| 345 | C22×C3 | — | 4 | 2 | no | — |
| 346 | C22×C3 | — | 4 | 4 | no | — |
| 347 | C22×C3 | — | 4 | 2 | no | — |
| 348 | C22×C3 | — | 4 | 2 | no | — |
| 349 | C22×C3 | — | 4 | 1 | no | — |
| 350 | C22×C3 | — | 8 | 2 | no | — |
| 351 | C22×C3 | — | 8 | 8 | no | — |
| 352 | C22×C3 | — | 8 | 2 | no | — |
| 353 | C2×D3 | — | 8 | 4 | no | — |
| 354 | C2×D3 | — | 8 | 4 | no | — |
| 355 | C2×D3 | — | 8 | 4 | no | — |
| 356 | C2×D3 | — | 16 | 8 | no | — |
| 357 | C3×C4 | — | 4 | 4 | no | — |
| 358 | C3×C4 | — | 8 | 2 | no | — |
| 359 | Dic3 | — | 8 | 4 | no | — |
| 360 | C23 | — | 2 | 2 | no | — |
| 361 | C23 | — | 2 | 2 | no | — |
| 362 | C23 | — | 3 | 4 | no | — |
| 363 | C23 | — | 3 | 2 | no | — |
| 364 | C23 | — | 3 | 2 | no | — |
| 365 | C23 | — | 6 | 8 | no | — |
| 366 | C23 | — | 6 | 2 | no | — |
| 367 | C23 | — | 6 | 2 | no | — |
| 368 | C23 | — | 6 | 2 | no | — |
| 369 | C23 | — | 6 | 8 | no | — |
| 370 | C23 | — | 6 | 2 | no | — |
| 371 | C23 | — | 6 | 2 | no | — |
| 372 | C23 | — | 6 | 4 | no | — |
| 373 | C23 | — | 12 | 4 | no | — |
| 374 | C23 | — | 12 | 16 | no | — |
| 375 | C23 | — | 12 | 8 | no | — |
| 376 | C23 | — | 12 | 8 | no | — |
| 377 | C23 | — | 12 | 8 | no | — |
| 378 | C23 | — | 12 | 4 | no | — |
| 379 | C23 | — | 12 | 4 | no | — |
| 380 | C23 | — | 12 | 4 | no | — |
| 381 | C23 | — | 24 | 16 | no | — |
| 382 | C23 | #986 | 1 | 4 | yes | complemented |
| 383 | C23 | C22×S4 | 1 | 2 | yes | — |
| 384 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 385 | Q8 | — | 3 | 2 | no | — |
| 386 | Q8 | — | 3 | 1 | no | — |
| 387 | Q8 | — | 3 | 4 | no | — |
| 388 | Q8 | — | 6 | 1 | no | — |
| 389 | Q8 | — | 6 | 2 | no | — |
| 390 | Q8 | — | 6 | 4 | no | — |
| 391 | Q8 | C22×#67 | 1 | 1 | yes | characteristic |
| 392 | D4 | — | 3 | 4 | no | — |
| 393 | D4 | — | 3 | 4 | no | — |
| 394 | D4 | — | 6 | 2 | no | — |
| 395 | D4 | — | 6 | 4 | no | — |
| 396 | D4 | — | 6 | 2 | no | — |
| 397 | D4 | — | 6 | 4 | no | — |
| 398 | D4 | — | 6 | 4 | no | — |
| 399 | D4 | — | 6 | 4 | no | — |
| 400 | D4 | — | 6 | 16 | no | — |
| 401 | D4 | — | 12 | 4 | no | — |
| 402 | D4 | — | 12 | 16 | no | — |
| 403 | D4 | — | 12 | 16 | no | — |
| 404 | D4 | — | 12 | 8 | no | — |
| 405 | C2×C4 | — | 2 | 2 | no | — |
| 406 | C2×C4 | — | 2 | 8 | no | — |
| 407 | C2×C4 | — | 3 | 2 | no | — |
| 408 | C2×C4 | — | 3 | 2 | no | — |
| 409 | C2×C4 | — | 3 | 1 | no | — |
| 410 | C2×C4 | — | 3 | 1 | no | — |
| 411 | C2×C4 | — | 3 | 4 | no | — |
| 412 | C2×C4 | — | 3 | 4 | no | — |
| 413 | C2×C4 | — | 3 | 8 | no | — |
| 414 | C2×C4 | — | 3 | 2 | no | — |
| 415 | C2×C4 | — | 6 | 2 | no | — |
| 416 | C2×C4 | — | 6 | 2 | no | — |
| 417 | C2×C4 | — | 6 | 4 | no | — |
| 418 | C2×C4 | — | 6 | 8 | no | — |
| 419 | C2×C4 | — | 6 | 2 | no | — |
| 420 | C2×C4 | — | 6 | 2 | no | — |
| 421 | C2×C4 | — | 6 | 4 | no | — |
| 422 | C2×C4 | — | 6 | 2 | no | — |
| 423 | C2×C4 | — | 6 | 8 | no | — |
| 424 | C2×C4 | — | 6 | 4 | no | — |
| 425 | C2×C4 | — | 12 | 8 | no | — |
| 426 | C2×C4 | — | 12 | 4 | no | — |
| 427 | C2×C4 | GL(2,ℤ/4) | 1 | 4 | yes | — |
| 428 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 429 | C2×C4 | C22×S4 | 1 | 2 | yes | — |
| 430 | C8 | — | 6 | 4 | no | — |
| 431 | C8 | — | 6 | 4 | no | — |
| 432 | C2×C3 | — | 4 | 4 | no | — |
| 433 | C2×C3 | — | 4 | 1 | no | — |
| 434 | C2×C3 | — | 4 | 1 | no | — |
| 435 | C2×C3 | — | 4 | 1 | no | — |
| 436 | C2×C3 | — | 8 | 4 | no | — |
| 437 | D3 | — | 16 | 4 | no | — |
| 438 | C22 | — | 2 | 8 | no | — |
| 439 | C22 | — | 2 | 2 | no | — |
| 440 | C22 | — | 2 | 2 | no | — |
| 441 | C22 | — | 3 | 4 | no | — |
| 442 | C22 | — | 6 | 8 | no | — |
| 443 | C22 | — | 6 | 2 | no | — |
| 444 | C22 | — | 6 | 2 | no | — |
| 445 | C22 | — | 6 | 2 | no | — |
| 446 | C22 | — | 6 | 2 | no | — |
| 447 | C22 | — | 6 | 2 | no | — |
| 448 | C22 | — | 12 | 16 | no | — |
| 449 | C22 | — | 12 | 4 | no | — |
| 450 | C22 | — | 12 | 4 | no | — |
| 451 | C22 | — | 12 | 4 | no | — |
| 452 | C22 | — | 12 | 4 | no | — |
| 453 | C22 | — | 24 | 16 | no | — |
| 454 | C22 | — | 24 | 8 | no | — |
| 455 | C22 | C2×#986 | 1 | 2 | yes | — |
| 456 | C22 | C2×#986 | 1 | 4 | yes | complemented |
| 457 | C22 | #5923 | 1 | 2 | yes | — |
| 458 | C22 | C2×GL(2,ℤ/4) | 1 | 2 | yes | — |
| 459 | C22 | C23×S4 | 1 | 1 | yes | characteristic |
| 460 | C4 | — | 2 | 2 | no | — |
| 461 | C4 | — | 3 | 1 | no | — |
| 462 | C4 | — | 3 | 1 | no | — |
| 463 | C4 | — | 3 | 2 | no | — |
| 464 | C4 | — | 6 | 2 | no | — |
| 465 | C4 | — | 12 | 4 | no | — |
| 466 | C4 | C2×GL(2,ℤ/4) | 1 | 4 | yes | — |
| 467 | C3 | — | 4 | 1 | no | Sylow |
| 468 | C2 | — | 2 | 4 | no | — |
| 469 | C2 | — | 6 | 4 | no | — |
| 470 | C2 | — | 12 | 2 | no | — |
| 471 | C2 | — | 24 | 4 | no | — |
| 472 | C2 | #85523 | 1 | 4 | yes | direct product factor |
| 473 | C2 | C22×#986 | 1 | 1 | yes | characteristic |
| 474 | C2 | C2×#5923 | 1 | 1 | yes | characteristic |
| 475 | C2 | C22×GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×S4 |
| commutator, cosocle | C2×SL(2,𝔽3) | C24 |
| Frattini | C22 | C23×S4 |
| Fitting | C23×#41 | D3 |
| hypercenter | C23×C4 | S4 |
| hypocenter | SL(2,𝔽3) | C22×D4 |
| LCS | G ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#41⋊C3 | C2 | semidirect |
| 2. C22×#986 | C2 | semidirect |
| 3. #85523 | C2 | direct (N × H) |
| 4. C2×#5427 | C2 | semidirect |
| 5. C2×#5426 | C2 | semidirect |
| 6. C2×#5421 | C2 | semidirect |
| 10. C22×#41⋊C3 | C22 | semidirect |
| 11. C22×#41⋊C3 | C22 | semidirect |
| 12. C23×SL(2,𝔽3) | C22 | semidirect |
| 14. C2×#986 | C22 | semidirect |
| 15. C22×GL(2,𝔽3) | C22 | semidirect |
| 17. #5427 | C22 | semidirect |
| 18. #5426 | C22 | semidirect |
| 19. #5421 | C22 | semidirect |
| 20. C2×#859 | C22 | semidirect |
| 21. C23×#41 | D3 | semidirect |
| 39. C2×#41⋊C3 | D4 | semidirect |
| 41. C2×#41⋊C3 | C23 | semidirect |
| 44. C22×SL(2,𝔽3) | C23 | semidirect |
| 47. C2×GL(2,𝔽3) | C23 | semidirect |
| 51. #859 | C23 | semidirect |
| 59. C22×#41 | C2×D3 | semidirect |
| 113. #41⋊C3 | C2×D4 | semidirect |
| 115. C2×SL(2,𝔽3) | C2×D4 | semidirect |
| 139. C2×#41 | #67 | semidirect |
| 234. SL(2,𝔽3) | C22×D4 | semidirect |
| 260. #41 | C2×#67 | semidirect |
| 382. C23 | #986 | semidirect |
| 456. C22 | C2×#986 | semidirect |
| 472. C2 | #85523 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41⋊C3 | 2 |
| C22×#986 | 2 |
| #85523 | 2 |
| C2×#5427 | 2 |
| C2×#5426 | 2 |
| C2×#5421 | 2 |
| C2×#3006 | 3 |
| C2×#930 | 4 |