Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5289 conjugacy classes of subgroups; only the 519 normal subgroups are shown.
This group has 5289 conjugacy classes of subgroups — too many to draw. Only its 519 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 | C22×2−1+4⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#5927 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 3 | C22×#985 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C22×#984 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C22×#983 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C23×#226 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C22×#578 | — | 3 | 1 | no | maximal, Sylow |
| 8 | C22×#237 | — | 4 | 1 | no | maximal |
| 9 | C2×2−1+4⋊C3 | C22 | 1 | 12 | yes | complemented |
| 10 | C22×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 11 | C23×SL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 12 | #5927 | C22 | 1 | 64 | yes | direct product factor |
| 13 | C2×#985 | C22 | 1 | 12 | yes | complemented |
| 14 | C2×#984 | C22 | 1 | 12 | yes | complemented |
| 15 | C2×#983 | C22 | 1 | 24 | yes | complemented |
| 16 | C22×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 17 | C22×#226 | C22 | 1 | 1 | yes | characteristic |
| 18 | C22×#226 | C22 | 1 | 2 | yes | — |
| 19 | C22×#226 | C22 | 1 | 24 | yes | complemented |
| 20 | C22×2−1+4 | — | 3 | 1 | no | — |
| 21 | C22×2−1+4 | D3 | 1 | 1 | yes | characteristic, complemented |
| 22 | C2×#578 | — | 3 | 24 | no | — |
| 23 | C2×#578 | — | 3 | 24 | no | — |
| 24 | C22×#137 | — | 3 | 1 | no | — |
| 25 | C22×#137 | — | 3 | 2 | no | — |
| 26 | C22×#137 | — | 3 | 1 | no | — |
| 27 | C22×#137 | — | 3 | 2 | no | — |
| 28 | C22×#135 | — | 3 | 2 | no | — |
| 29 | C22×#135 | — | 3 | 1 | no | — |
| 30 | C23×Q16 | — | 3 | 1 | no | — |
| 31 | C23×Q16 | — | 3 | 2 | no | — |
| 32 | C22×#131 | — | 3 | 1 | no | — |
| 33 | C22×C3×D4 | — | 4 | 1 | no | — |
| 34 | C22×#67 | — | 4 | 2 | no | — |
| 35 | C23×Dic3 | — | 4 | 2 | no | — |
| 36 | C2×#237 | — | 4 | 24 | no | — |
| 37 | C22×C4×D3 | — | 4 | 1 | no | — |
| 38 | C22×Dic6 | — | 4 | 1 | no | — |
| 39 | 2−1+4⋊C3 | C23 | 1 | 16 | yes | complemented |
| 40 | C2×#41⋊C3 | C23 | 1 | 6 | yes | — |
| 41 | C22×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 42 | C22×SL(2,𝔽3) | C23 | 1 | 12 | yes | — |
| 43 | #985 | C23 | 1 | 16 | yes | complemented |
| 44 | #984 | C23 | 1 | 16 | yes | complemented |
| 45 | #983 | C23 | 1 | 32 | yes | complemented |
| 46 | C2×GL(2,𝔽3) | C23 | 1 | 6 | yes | — |
| 47 | C2×#226 | C23 | 1 | 6 | yes | — |
| 48 | C2×#226 | C23 | 1 | 12 | yes | — |
| 49 | C2×#226 | C23 | 1 | 32 | yes | complemented |
| 50 | C2×2−1+4 | — | 3 | 12 | no | — |
| 51 | C2×2−1+4 | — | 3 | 24 | no | — |
| 52 | C2×2−1+4 | — | 6 | 12 | no | — |
| 53 | C2×2−1+4 | C2×D3 | 1 | 12 | yes | complemented |
| 54 | C22×#41 | — | 3 | 1 | no | — |
| 55 | C22×#41 | — | 3 | 2 | no | — |
| 56 | C22×#41 | — | 3 | 2 | no | — |
| 57 | C22×#41 | — | 3 | 1 | no | — |
| 58 | C22×#41 | — | 6 | 2 | no | — |
| 59 | C22×#41 | — | 6 | 1 | no | — |
| 60 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 61 | C23×Q8 | — | 3 | 1 | no | — |
| 62 | C23×Q8 | — | 3 | 2 | no | — |
| 63 | C23×Q8 | — | 3 | 1 | no | — |
| 64 | C23×Q8 | — | 6 | 1 | no | — |
| 65 | C23×Q8 | C2×D3 | 1 | 2 | yes | — |
| 66 | #578 | — | 3 | 192 | no | — |
| 67 | #578 | — | 3 | 64 | no | — |
| 68 | C2×#137 | — | 3 | 24 | no | — |
| 69 | C2×#137 | — | 3 | 24 | no | — |
| 70 | C2×#137 | — | 3 | 12 | no | — |
| 71 | C2×#137 | — | 3 | 48 | no | — |
| 72 | C2×#137 | — | 3 | 12 | no | — |
| 73 | C2×#137 | — | 3 | 24 | no | — |
| 74 | C2×#135 | — | 3 | 12 | no | — |
| 75 | C2×#135 | — | 3 | 48 | no | — |
| 76 | C2×#135 | — | 3 | 12 | no | — |
| 77 | C22×Q16 | — | 3 | 2 | no | — |
| 78 | C22×Q16 | — | 3 | 2 | no | — |
| 79 | C22×Q16 | — | 3 | 2 | no | — |
| 80 | C22×Q16 | — | 3 | 24 | no | — |
| 81 | C22×Q16 | — | 3 | 24 | no | — |
| 82 | C22×Q16 | — | 3 | 1 | no | — |
| 83 | C22×Q16 | — | 3 | 2 | no | — |
| 84 | C22×Q16 | — | 3 | 24 | no | — |
| 85 | C22×SD16 | — | 3 | 2 | no | — |
| 86 | C22×SD16 | — | 3 | 1 | no | — |
| 87 | C22×SD16 | — | 3 | 2 | no | — |
| 88 | C22×SD16 | — | 3 | 1 | no | — |
| 89 | C22×D8 | — | 3 | 1 | no | — |
| 90 | C2×#131 | — | 3 | 24 | no | — |
| 91 | C22×M16 | — | 3 | 2 | no | — |
| 92 | C22×M16 | — | 3 | 1 | no | — |
| 93 | C23×C8 | — | 3 | 2 | no | — |
| 94 | C23×C8 | — | 3 | 1 | no | — |
| 95 | C24×C3 | — | 4 | 2 | no | — |
| 96 | C23×D3 | — | 4 | 1 | no | — |
| 97 | C2×C3×D4 | — | 4 | 12 | no | — |
| 98 | C22×C3×C4 | — | 4 | 1 | no | — |
| 99 | C2×#67 | — | 4 | 24 | no | — |
| 100 | C22×Dic3 | — | 4 | 2 | no | — |
| 101 | C22×Dic3 | — | 4 | 1 | no | — |
| 102 | C22×Dic3 | — | 4 | 24 | no | — |
| 103 | #237 | — | 4 | 64 | no | — |
| 104 | C2×C4×D3 | — | 4 | 12 | no | — |
| 105 | C2×Dic6 | — | 4 | 12 | no | — |
| 106 | #41⋊C3 | C24 | 1 | 4 | yes | — |
| 107 | C2×SL(2,𝔽3) | C24 | 1 | 3 | yes | — |
| 108 | C2×SL(2,𝔽3) | C24 | 1 | 8 | yes | — |
| 109 | GL(2,𝔽3) | C24 | 1 | 4 | yes | — |
| 110 | #226 | C24 | 1 | 4 | yes | — |
| 111 | #226 | C24 | 1 | 8 | yes | — |
| 112 | 2−1+4 | — | 3 | 48 | no | — |
| 113 | 2−1+4 | — | 3 | 64 | no | — |
| 114 | 2−1+4 | — | 6 | 16 | no | — |
| 115 | 2−1+4 | — | 6 | 96 | no | — |
| 116 | 2−1+4 | C22×D3 | 1 | 16 | yes | complemented |
| 117 | C2×#41 | — | 3 | 12 | no | — |
| 118 | C2×#41 | — | 3 | 12 | no | — |
| 119 | C2×#41 | — | 3 | 6 | no | — |
| 120 | C2×#41 | — | 3 | 24 | no | — |
| 121 | C2×#41 | — | 3 | 24 | no | — |
| 122 | C2×#41 | — | 3 | 12 | no | — |
| 123 | C2×#41 | — | 6 | 12 | no | — |
| 124 | C2×#41 | — | 6 | 12 | no | — |
| 125 | C2×#41 | — | 6 | 6 | no | — |
| 126 | C2×#41 | — | 6 | 48 | no | — |
| 127 | C2×#41 | — | 6 | 24 | no | — |
| 128 | C2×#41 | C22×D3 | 1 | 6 | yes | — |
| 129 | C22×Q8 | — | 3 | 1 | no | — |
| 130 | C22×Q8 | — | 3 | 2 | no | — |
| 131 | C22×Q8 | — | 3 | 12 | no | — |
| 132 | C22×Q8 | — | 3 | 12 | no | — |
| 133 | C22×Q8 | — | 3 | 2 | no | — |
| 134 | C22×Q8 | — | 3 | 1 | no | — |
| 135 | C22×Q8 | — | 3 | 24 | no | — |
| 136 | C22×Q8 | — | 3 | 12 | no | — |
| 137 | C22×Q8 | — | 6 | 6 | no | — |
| 138 | C22×Q8 | — | 6 | 2 | no | — |
| 139 | C22×Q8 | — | 6 | 12 | no | — |
| 140 | C22×Q8 | — | 6 | 6 | no | — |
| 141 | C22×Q8 | — | 6 | 24 | no | — |
| 142 | C22×Q8 | — | 6 | 1 | no | — |
| 143 | C22×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 144 | C22×Q8 | C22×D3 | 1 | 12 | yes | — |
| 145 | C22×D4 | — | 3 | 2 | no | — |
| 146 | C22×D4 | — | 3 | 1 | no | — |
| 147 | C22×D4 | — | 3 | 1 | no | — |
| 148 | C22×D4 | — | 6 | 1 | no | — |
| 149 | C22×D4 | — | 6 | 2 | no | — |
| 150 | C22×D4 | S4 | 1 | 1 | yes | characteristic |
| 151 | C23×C4 | — | 3 | 2 | no | — |
| 152 | C23×C4 | — | 3 | 1 | no | — |
| 153 | C23×C4 | — | 6 | 1 | no | — |
| 154 | C23×C4 | — | 6 | 2 | no | — |
| 155 | C23×C4 | — | 6 | 1 | no | — |
| 156 | C23×C4 | — | 6 | 2 | no | — |
| 157 | C23×C4 | — | 6 | 1 | no | — |
| 158 | #137 | — | 3 | 96 | no | — |
| 159 | #137 | — | 3 | 64 | no | — |
| 160 | #137 | — | 3 | 48 | no | — |
| 161 | #137 | — | 3 | 128 | no | — |
| 162 | #137 | — | 3 | 16 | no | — |
| 163 | #137 | — | 3 | 32 | no | — |
| 164 | #135 | — | 3 | 48 | no | — |
| 165 | #135 | — | 3 | 128 | no | — |
| 166 | #135 | — | 3 | 16 | no | — |
| 167 | C2×Q16 | — | 3 | 24 | no | — |
| 168 | C2×Q16 | — | 3 | 6 | no | — |
| 169 | C2×Q16 | — | 3 | 24 | no | — |
| 170 | C2×Q16 | — | 3 | 24 | no | — |
| 171 | C2×Q16 | — | 3 | 12 | no | — |
| 172 | C2×Q16 | — | 3 | 96 | no | — |
| 173 | C2×Q16 | — | 3 | 64 | no | — |
| 174 | C2×Q16 | — | 3 | 6 | no | — |
| 175 | C2×Q16 | — | 3 | 12 | no | — |
| 176 | C2×Q16 | — | 3 | 32 | no | — |
| 177 | C2×SD16 | — | 3 | 24 | no | — |
| 178 | C2×SD16 | — | 3 | 12 | no | — |
| 179 | C2×SD16 | — | 3 | 24 | no | — |
| 180 | C2×SD16 | — | 3 | 6 | no | — |
| 181 | C2×SD16 | — | 3 | 6 | no | — |
| 182 | C2×D8 | — | 3 | 12 | no | — |
| 183 | #131 | — | 3 | 64 | no | — |
| 184 | C2×M16 | — | 3 | 24 | no | — |
| 185 | C2×M16 | — | 3 | 12 | no | — |
| 186 | C22×C8 | — | 3 | 1 | no | — |
| 187 | C22×C8 | — | 3 | 2 | no | — |
| 188 | C22×C8 | — | 3 | 1 | no | — |
| 189 | C22×C8 | — | 3 | 24 | no | — |
| 190 | C22×C8 | — | 3 | 12 | no | — |
| 191 | C23×C3 | — | 4 | 12 | no | — |
| 192 | C23×C3 | — | 4 | 1 | no | — |
| 193 | C23×C3 | — | 8 | 8 | no | — |
| 194 | C22×D3 | — | 4 | 6 | no | — |
| 195 | C22×D3 | — | 8 | 4 | no | — |
| 196 | C3×D4 | — | 4 | 16 | no | — |
| 197 | C2×C3×C4 | — | 4 | 6 | no | — |
| 198 | #67 | — | 4 | 32 | no | — |
| 199 | C2×Dic3 | — | 4 | 12 | no | — |
| 200 | C2×Dic3 | — | 4 | 6 | no | — |
| 201 | C2×Dic3 | — | 4 | 32 | no | — |
| 202 | C4×D3 | — | 4 | 16 | no | — |
| 203 | Dic6 | — | 4 | 16 | no | — |
| 204 | SL(2,𝔽3) | C25 | 1 | 1 | yes | characteristic |
| 205 | C24 | — | 3 | 1 | no | — |
| 206 | C24 | — | 6 | 1 | no | — |
| 207 | C24 | C2×S4 | 1 | 2 | yes | — |
| 208 | #41 | — | 3 | 16 | no | — |
| 209 | #41 | — | 3 | 48 | no | — |
| 210 | #41 | — | 3 | 12 | no | — |
| 211 | #41 | — | 3 | 32 | no | — |
| 212 | #41 | — | 3 | 32 | no | — |
| 213 | #41 | — | 3 | 16 | no | — |
| 214 | #41 | — | 6 | 4 | no | — |
| 215 | #41 | — | 6 | 48 | no | — |
| 216 | #41 | — | 6 | 48 | no | — |
| 217 | #41 | — | 6 | 24 | no | — |
| 218 | #41 | — | 6 | 128 | no | — |
| 219 | #41 | — | 6 | 64 | no | — |
| 220 | #41 | C23×D3 | 1 | 4 | yes | — |
| 221 | C2×Q8 | — | 3 | 12 | no | — |
| 222 | C2×Q8 | — | 3 | 3 | no | — |
| 223 | C2×Q8 | — | 3 | 12 | no | — |
| 224 | C2×Q8 | — | 3 | 24 | no | — |
| 225 | C2×Q8 | — | 3 | 16 | no | — |
| 226 | C2×Q8 | — | 3 | 12 | no | — |
| 227 | C2×Q8 | — | 3 | 6 | no | — |
| 228 | C2×Q8 | — | 3 | 32 | no | — |
| 229 | C2×Q8 | — | 3 | 16 | no | — |
| 230 | C2×Q8 | — | 6 | 6 | no | — |
| 231 | C2×Q8 | — | 6 | 8 | no | — |
| 232 | C2×Q8 | — | 6 | 24 | no | — |
| 233 | C2×Q8 | — | 6 | 6 | no | — |
| 234 | C2×Q8 | — | 6 | 3 | no | — |
| 235 | C2×Q8 | — | 6 | 24 | no | — |
| 236 | C2×Q8 | — | 6 | 48 | no | — |
| 237 | C2×Q8 | — | 6 | 24 | no | — |
| 238 | C2×Q8 | — | 6 | 64 | no | — |
| 239 | C2×Q8 | — | 6 | 12 | no | — |
| 240 | C2×Q8 | C23×D3 | 1 | 3 | yes | — |
| 241 | C2×Q8 | C23×D3 | 1 | 8 | yes | — |
| 242 | C2×D4 | — | 3 | 24 | no | — |
| 243 | C2×D4 | — | 3 | 6 | no | — |
| 244 | C2×D4 | — | 3 | 6 | no | — |
| 245 | C2×D4 | — | 6 | 3 | no | — |
| 246 | C2×D4 | — | 6 | 3 | no | — |
| 247 | C2×D4 | — | 6 | 12 | no | — |
| 248 | C2×D4 | — | 6 | 24 | no | — |
| 249 | C2×D4 | C2×S4 | 1 | 12 | yes | — |
| 250 | C22×C4 | — | 3 | 12 | no | — |
| 251 | C22×C4 | — | 3 | 12 | no | — |
| 252 | C22×C4 | — | 3 | 1 | no | — |
| 253 | C22×C4 | — | 3 | 2 | no | — |
| 254 | C22×C4 | — | 3 | 6 | no | — |
| 255 | C22×C4 | — | 3 | 6 | no | — |
| 256 | C22×C4 | — | 6 | 12 | no | — |
| 257 | C22×C4 | — | 6 | 24 | no | — |
| 258 | C22×C4 | — | 6 | 12 | no | — |
| 259 | C22×C4 | — | 6 | 24 | no | — |
| 260 | C22×C4 | — | 6 | 2 | no | — |
| 261 | C22×C4 | — | 6 | 1 | no | — |
| 262 | C22×C4 | — | 6 | 12 | no | — |
| 263 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 264 | Q16 | — | 3 | 32 | no | — |
| 265 | Q16 | — | 3 | 12 | no | — |
| 266 | Q16 | — | 3 | 32 | no | — |
| 267 | Q16 | — | 3 | 32 | no | — |
| 268 | Q16 | — | 3 | 24 | no | — |
| 269 | Q16 | — | 3 | 4 | no | — |
| 270 | Q16 | — | 3 | 8 | no | — |
| 271 | SD16 | — | 3 | 32 | no | — |
| 272 | SD16 | — | 3 | 16 | no | — |
| 273 | SD16 | — | 3 | 32 | no | — |
| 274 | SD16 | — | 3 | 12 | no | — |
| 275 | SD16 | — | 3 | 4 | no | — |
| 276 | D8 | — | 3 | 16 | no | — |
| 277 | M16 | — | 3 | 32 | no | — |
| 278 | M16 | — | 3 | 16 | no | — |
| 279 | C2×C8 | — | 3 | 6 | no | — |
| 280 | C2×C8 | — | 3 | 12 | no | — |
| 281 | C2×C8 | — | 3 | 6 | no | — |
| 282 | C2×C8 | — | 3 | 32 | no | — |
| 283 | C2×C8 | — | 3 | 16 | no | — |
| 284 | C22×C3 | — | 4 | 4 | no | — |
| 285 | C22×C3 | — | 4 | 3 | no | — |
| 286 | C22×C3 | — | 4 | 8 | no | — |
| 287 | C22×C3 | — | 8 | 24 | no | — |
| 288 | C2×D3 | — | 4 | 4 | no | — |
| 289 | C2×D3 | — | 8 | 12 | no | — |
| 290 | C3×C4 | — | 4 | 4 | no | — |
| 291 | Dic3 | — | 4 | 8 | no | — |
| 292 | Dic3 | — | 4 | 4 | no | — |
| 293 | C23 | — | 2 | 8 | no | — |
| 294 | C23 | — | 3 | 6 | no | — |
| 295 | C23 | — | 6 | 4 | no | — |
| 296 | C23 | — | 6 | 6 | no | — |
| 297 | C23 | — | 12 | 4 | no | — |
| 298 | C23 | C22×S4 | 1 | 12 | yes | — |
| 299 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 300 | Q8 | — | 3 | 3 | no | — |
| 301 | Q8 | — | 3 | 8 | no | — |
| 302 | Q8 | — | 3 | 16 | no | — |
| 303 | Q8 | — | 3 | 8 | no | — |
| 304 | Q8 | — | 3 | 4 | no | — |
| 305 | Q8 | — | 6 | 1 | no | — |
| 306 | Q8 | — | 6 | 12 | no | — |
| 307 | Q8 | — | 6 | 12 | no | — |
| 308 | Q8 | — | 6 | 6 | no | — |
| 309 | Q8 | — | 6 | 32 | no | — |
| 310 | Q8 | — | 6 | 16 | no | — |
| 311 | Q8 | C24×D3 | 1 | 1 | yes | characteristic |
| 312 | D4 | — | 3 | 4 | no | — |
| 313 | D4 | — | 3 | 32 | no | — |
| 314 | D4 | — | 3 | 4 | no | — |
| 315 | D4 | — | 6 | 6 | no | — |
| 316 | D4 | — | 6 | 6 | no | — |
| 317 | D4 | — | 6 | 16 | no | — |
| 318 | D4 | — | 6 | 32 | no | — |
| 319 | D4 | C22×S4 | 1 | 16 | yes | — |
| 320 | C2×C4 | — | 3 | 3 | no | — |
| 321 | C2×C4 | — | 3 | 8 | no | — |
| 322 | C2×C4 | — | 3 | 3 | no | — |
| 323 | C2×C4 | — | 3 | 12 | no | — |
| 324 | C2×C4 | — | 3 | 24 | no | — |
| 325 | C2×C4 | — | 3 | 4 | no | — |
| 326 | C2×C4 | — | 3 | 12 | no | — |
| 327 | C2×C4 | — | 6 | 16 | no | — |
| 328 | C2×C4 | — | 6 | 32 | no | — |
| 329 | C2×C4 | — | 6 | 16 | no | — |
| 330 | C2×C4 | — | 6 | 12 | no | — |
| 331 | C2×C4 | — | 6 | 6 | no | — |
| 332 | C2×C4 | — | 6 | 32 | no | — |
| 333 | C2×C4 | — | 6 | 16 | no | — |
| 334 | C2×C4 | C22×S4 | 1 | 6 | yes | — |
| 335 | C8 | — | 3 | 4 | no | — |
| 336 | C8 | — | 3 | 8 | no | — |
| 337 | C8 | — | 3 | 4 | no | — |
| 338 | C2×C3 | — | 4 | 1 | no | — |
| 339 | C2×C3 | — | 4 | 6 | no | — |
| 340 | C2×C3 | — | 8 | 8 | no | — |
| 341 | D3 | — | 8 | 4 | no | — |
| 342 | C22 | — | 2 | 24 | no | — |
| 343 | C22 | — | 3 | 4 | no | — |
| 344 | C22 | — | 6 | 12 | no | — |
| 345 | C22 | — | 6 | 4 | no | — |
| 346 | C22 | — | 12 | 12 | no | — |
| 347 | C22 | #5927 | 1 | 4 | yes | direct product factor |
| 348 | C22 | C23×S4 | 1 | 8 | yes | — |
| 349 | C22 | C23×S4 | 1 | 3 | yes | — |
| 350 | C4 | — | 3 | 1 | no | — |
| 351 | C4 | — | 3 | 3 | no | — |
| 352 | C4 | — | 3 | 8 | no | — |
| 353 | C4 | — | 6 | 8 | no | — |
| 354 | C4 | — | 6 | 4 | no | — |
| 355 | C4 | C23×S4 | 1 | 4 | yes | — |
| 356 | C3 | — | 4 | 1 | no | Sylow |
| 357 | C2 | — | 2 | 8 | no | — |
| 358 | C2 | — | 6 | 4 | no | — |
| 359 | C2 | — | 12 | 4 | no | — |
| 360 | C2 | C2×#5927 | 1 | 6 | yes | direct product factor |
| 361 | C2 | C24×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C22×2−1+4 | D3 |
| hypercenter | C22×D4 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×2−1+4⋊C3 | C2 | semidirect |
| 2. C2×#5927 | C2 | direct (N × H) |
| 3. C22×#985 | C2 | semidirect |
| 4. C22×#984 | C2 | semidirect |
| 5. C22×#983 | C2 | semidirect |
| 6. C23×#226 | C2 | semidirect |
| 9. C2×2−1+4⋊C3 | C22 | semidirect |
| 12. #5927 | C22 | direct (N × H) |
| 13. C2×#985 | C22 | semidirect |
| 14. C2×#984 | C22 | semidirect |
| 15. C2×#983 | C22 | semidirect |
| 19. C22×#226 | C22 | semidirect |
| 21. C22×2−1+4 | D3 | semidirect |
| 39. 2−1+4⋊C3 | C23 | semidirect |
| 43. #985 | C23 | semidirect |
| 44. #984 | C23 | semidirect |
| 45. #983 | C23 | semidirect |
| 49. C2×#226 | C23 | semidirect |
| 53. C2×2−1+4 | C2×D3 | semidirect |
| 116. 2−1+4 | C22×D3 | semidirect |
| 347. C22 | #5927 | direct (N × H) |
| 360. C2 | C2×#5927 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×2−1+4⋊C3 | 2 |
| C2×#5927 | 2 |
| C22×#985 | 2 |
| C22×#984 | 2 |
| C22×#983 | 2 |
| C23×#226 | 2 |
| C22×#578 | 3 |
| C22×#237 | 4 |