Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
1393 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 1393 conjugacy classes of subgroups — too many to draw. Only its 183 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×#985 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C22×#131⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C22×GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#5405 | C2 | 1 | 12 | yes | maximal, direct product factor |
| 5 | C22×#444 | — | 3 | 1 | no | maximal, Sylow |
| 6 | C22×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#985 | C4 | 1 | 6 | yes | complemented |
| 8 | C2×#985 | C22 | 1 | 6 | yes | — |
| 9 | C22×GL(2,𝔽3) | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×#226 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C22×ΓL(1,𝔽25) | — | 4 | 1 | no | maximal |
| 12 | C2×#131⋊C3 | C22 | 1 | 6 | yes | complemented |
| 13 | C2×GU(2,𝔽3) | C22 | 1 | 6 | yes | complemented |
| 14 | #5405 | C22 | 1 | 16 | yes | direct product factor |
| 15 | C22×#135 | — | 3 | 1 | no | — |
| 16 | C22×#131 | — | 3 | 1 | no | — |
| 17 | C22×#131 | D3 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×#444 | — | 3 | 12 | no | — |
| 19 | C2×#444 | — | 3 | 12 | no | — |
| 20 | C22×#108 | — | 3 | 1 | no | — |
| 21 | C22×#104 | — | 3 | 1 | no | — |
| 22 | C22×#104 | — | 3 | 1 | no | — |
| 23 | C22×#97 | — | 3 | 1 | no | — |
| 24 | C22×C4×D3 | — | 4 | 1 | no | — |
| 25 | C2×#41⋊C3 | C2×C4 | 1 | 3 | yes | — |
| 26 | C2×#41⋊C3 | C23 | 1 | 3 | yes | — |
| 27 | C22×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | #985 | C2×C4 | 1 | 12 | yes | complemented |
| 29 | #985 | C23 | 1 | 4 | yes | — |
| 30 | C2×GL(2,𝔽3) | C2×C4 | 1 | 6 | yes | complemented |
| 31 | C2×#226 | C2×C4 | 1 | 6 | yes | complemented |
| 32 | C22×C3×C8 | — | 4 | 1 | no | — |
| 33 | C22×C3⋊C8 | — | 4 | 1 | no | — |
| 34 | C2×ΓL(1,𝔽25) | — | 4 | 12 | no | — |
| 35 | #131⋊C3 | C23 | 1 | 4 | yes | complemented |
| 36 | GU(2,𝔽3) | C23 | 1 | 4 | yes | complemented |
| 37 | C22×#41 | — | 3 | 1 | no | — |
| 38 | C22×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 39 | C2×#135 | — | 3 | 6 | no | — |
| 40 | C2×#135 | — | 3 | 6 | no | — |
| 41 | C2×#135 | — | 3 | 6 | no | — |
| 42 | C2×#135 | — | 3 | 6 | no | — |
| 43 | C22×Q16 | — | 3 | 1 | no | — |
| 44 | C22×SD16 | — | 3 | 1 | no | — |
| 45 | C22×SD16 | — | 3 | 1 | no | — |
| 46 | C22×D8 | — | 3 | 1 | no | — |
| 47 | C2×#131 | — | 3 | 6 | no | — |
| 48 | C2×#131 | — | 3 | 12 | no | — |
| 49 | C2×#131 | — | 6 | 6 | no | — |
| 50 | C2×#131 | C2×D3 | 1 | 6 | yes | complemented |
| 51 | C22×M16 | — | 3 | 1 | no | — |
| 52 | C22×M16 | — | 3 | 1 | no | — |
| 53 | C22×M16 | — | 6 | 1 | no | — |
| 54 | C23×C8 | — | 3 | 1 | no | — |
| 55 | C23×C8 | — | 3 | 1 | no | — |
| 56 | C23×C8 | — | 6 | 1 | no | — |
| 57 | C22×C42 | — | 3 | 1 | no | — |
| 58 | #444 | — | 3 | 48 | no | — |
| 59 | #444 | — | 3 | 16 | no | — |
| 60 | C2×#108 | — | 3 | 12 | no | — |
| 61 | C2×#104 | — | 3 | 12 | no | — |
| 62 | C2×#104 | — | 3 | 6 | no | — |
| 63 | C2×#104 | — | 3 | 6 | no | — |
| 64 | C2×#97 | — | 3 | 12 | no | — |
| 65 | C23×D3 | — | 4 | 1 | no | — |
| 66 | C22×C3×C4 | — | 4 | 1 | no | — |
| 67 | C22×Dic3 | — | 4 | 1 | no | — |
| 68 | C2×C4×D3 | — | 4 | 6 | no | — |
| 69 | C2×C4×D3 | — | 4 | 6 | no | — |
| 70 | #41⋊C3 | C22×C4 | 1 | 3 | yes | — |
| 71 | #41⋊C3 | C24 | 1 | 1 | yes | characteristic |
| 72 | C2×SL(2,𝔽3) | C22×C4 | 1 | 3 | yes | — |
| 73 | GL(2,𝔽3) | C22×C4 | 1 | 4 | yes | complemented |
| 74 | #226 | C22×C4 | 1 | 4 | yes | complemented |
| 75 | C2×C3×C8 | — | 4 | 6 | no | — |
| 76 | C2×C3⋊C8 | — | 4 | 6 | no | — |
| 77 | ΓL(1,𝔽25) | — | 4 | 16 | no | — |
| 78 | C2×#41 | — | 3 | 3 | no | — |
| 79 | C2×#41 | — | 3 | 3 | no | — |
| 80 | C2×#41 | — | 3 | 6 | no | — |
| 81 | C2×#41 | — | 3 | 6 | no | — |
| 82 | C2×#41 | — | 6 | 3 | no | — |
| 83 | C2×#41 | — | 6 | 3 | no | — |
| 84 | C2×#41 | C4×D3 | 1 | 3 | yes | — |
| 85 | C2×#41 | C22×D3 | 1 | 3 | yes | — |
| 86 | C22×Q8 | — | 3 | 1 | no | — |
| 87 | C22×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 88 | C22×D4 | — | 3 | 1 | no | — |
| 89 | C22×D4 | — | 3 | 1 | no | — |
| 90 | C22×D4 | — | 6 | 1 | no | — |
| 91 | C23×C4 | — | 3 | 1 | no | — |
| 92 | C23×C4 | — | 6 | 1 | no | — |
| 93 | C23×C4 | — | 6 | 1 | no | — |
| 94 | #135 | — | 3 | 24 | no | — |
| 95 | #135 | — | 3 | 12 | no | — |
| 96 | #135 | — | 3 | 12 | no | — |
| 97 | #135 | — | 3 | 12 | no | — |
| 98 | #135 | — | 3 | 4 | no | — |
| 99 | C2×Q16 | — | 3 | 6 | no | — |
| 100 | C2×Q16 | — | 3 | 6 | no | — |
| 101 | C2×SD16 | — | 3 | 6 | no | — |
| 102 | C2×SD16 | — | 3 | 6 | no | — |
| 103 | C2×SD16 | — | 3 | 6 | no | — |
| 104 | C2×SD16 | — | 3 | 6 | no | — |
| 105 | C2×D8 | — | 3 | 6 | no | — |
| 106 | C2×D8 | — | 3 | 6 | no | — |
| 107 | #131 | — | 3 | 12 | no | — |
| 108 | #131 | — | 3 | 16 | no | — |
| 109 | #131 | — | 6 | 4 | no | — |
| 110 | #131 | — | 6 | 24 | no | — |
| 111 | #131 | C22×D3 | 1 | 4 | yes | complemented |
| 112 | C2×M16 | — | 3 | 6 | no | — |
| 113 | C2×M16 | — | 3 | 6 | no | — |
| 114 | C2×M16 | — | 6 | 3 | no | — |
| 115 | C2×M16 | — | 6 | 3 | no | — |
| 116 | C2×M16 | — | 6 | 12 | no | — |
| 117 | C22×C8 | — | 3 | 1 | no | — |
| 118 | C22×C8 | — | 3 | 1 | no | — |
| 119 | C22×C8 | — | 3 | 6 | no | — |
| 120 | C22×C8 | — | 3 | 6 | no | — |
| 121 | C22×C8 | — | 3 | 1 | no | — |
| 122 | C22×C8 | — | 3 | 6 | no | — |
| 123 | C22×C8 | — | 3 | 6 | no | — |
| 124 | C22×C8 | — | 6 | 12 | no | — |
| 125 | C22×C8 | — | 6 | 1 | no | — |
| 126 | C22×C8 | S4 | 1 | 1 | yes | characteristic |
| 127 | C2×C42 | — | 3 | 6 | no | — |
| 128 | C2×C42 | — | 6 | 3 | no | — |
| 129 | #108 | — | 3 | 16 | no | — |
| 130 | #104 | — | 3 | 16 | no | — |
| 131 | #104 | — | 3 | 12 | no | — |
| 132 | #104 | — | 3 | 4 | no | — |
| 133 | #97 | — | 3 | 16 | no | — |
| 134 | C23×C3 | — | 4 | 1 | no | — |
| 135 | C22×D3 | — | 4 | 6 | no | — |
| 136 | C22×D3 | — | 8 | 4 | no | — |
| 137 | C2×C3×C4 | — | 4 | 3 | no | — |
| 138 | C2×C3×C4 | — | 4 | 3 | no | — |
| 139 | C2×Dic3 | — | 4 | 6 | no | — |
| 140 | C4×D3 | — | 4 | 4 | no | — |
| 141 | C4×D3 | — | 4 | 12 | no | — |
| 142 | SL(2,𝔽3) | C23×C4 | 1 | 1 | yes | characteristic |
| 143 | C3×C8 | — | 4 | 4 | no | — |
| 144 | C3⋊C8 | — | 4 | 4 | no | — |
| 145 | C24 | — | 3 | 1 | no | — |
| 146 | C24 | — | 6 | 1 | no | — |
| 147 | #41 | — | 3 | 3 | no | — |
| 148 | #41 | — | 3 | 6 | no | — |
| 149 | #41 | — | 3 | 3 | no | — |
| 150 | #41 | — | 3 | 4 | no | — |
| 151 | #41 | — | 3 | 12 | no | — |
| 152 | #41 | — | 6 | 1 | no | — |
| 153 | #41 | — | 6 | 3 | no | — |
| 154 | #41 | — | 6 | 6 | no | — |
| 155 | #41 | — | 6 | 6 | no | — |
| 156 | #41 | — | 6 | 12 | no | — |
| 157 | #41 | C2×C4×D3 | 1 | 3 | yes | — |
| 158 | #41 | C23×D3 | 1 | 1 | yes | characteristic |
| 159 | C2×Q8 | — | 3 | 3 | no | — |
| 160 | C2×Q8 | — | 3 | 6 | no | — |
| 161 | C2×Q8 | — | 6 | 3 | no | — |
| 162 | C2×Q8 | C2×C4×D3 | 1 | 3 | yes | — |
| 163 | C2×D4 | — | 3 | 3 | no | — |
| 164 | C2×D4 | — | 3 | 3 | no | — |
| 165 | C2×D4 | — | 3 | 6 | no | — |
| 166 | C2×D4 | — | 6 | 3 | no | — |
| 167 | C2×D4 | — | 6 | 3 | no | — |
| 168 | C2×D4 | — | 6 | 6 | no | — |
| 169 | C2×D4 | — | 6 | 6 | no | — |
| 170 | C22×C4 | — | 3 | 1 | no | — |
| 171 | C22×C4 | — | 3 | 3 | no | — |
| 172 | C22×C4 | — | 3 | 3 | no | — |
| 173 | C22×C4 | — | 3 | 3 | no | — |
| 174 | C22×C4 | — | 3 | 3 | no | — |
| 175 | C22×C4 | — | 6 | 1 | no | — |
| 176 | C22×C4 | — | 6 | 6 | no | — |
| 177 | C22×C4 | — | 6 | 6 | no | — |
| 178 | C22×C4 | — | 6 | 6 | no | — |
| 179 | C22×C4 | — | 12 | 4 | no | — |
| 180 | C22×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 181 | Q16 | — | 3 | 12 | no | — |
| 182 | Q16 | — | 3 | 4 | no | — |
| 183 | SD16 | — | 3 | 12 | no | — |
| 184 | SD16 | — | 3 | 4 | no | — |
| 185 | SD16 | — | 3 | 12 | no | — |
| 186 | SD16 | — | 3 | 4 | no | — |
| 187 | D8 | — | 3 | 12 | no | — |
| 188 | D8 | — | 3 | 4 | no | — |
| 189 | M16 | — | 3 | 4 | no | — |
| 190 | M16 | — | 3 | 4 | no | — |
| 191 | M16 | — | 6 | 6 | no | — |
| 192 | M16 | — | 6 | 6 | no | — |
| 193 | M16 | — | 6 | 16 | no | — |
| 194 | C2×C8 | — | 3 | 4 | no | — |
| 195 | C2×C8 | — | 3 | 6 | no | — |
| 196 | C2×C8 | — | 3 | 6 | no | — |
| 197 | C2×C8 | — | 3 | 12 | no | — |
| 198 | C2×C8 | — | 3 | 6 | no | — |
| 199 | C2×C8 | — | 3 | 4 | no | — |
| 200 | C2×C8 | — | 3 | 12 | no | — |
| 201 | C2×C8 | — | 6 | 16 | no | — |
| 202 | C2×C8 | — | 6 | 6 | no | — |
| 203 | C2×C8 | C2×S4 | 1 | 6 | yes | — |
| 204 | C42 | — | 3 | 4 | no | — |
| 205 | C42 | — | 6 | 6 | no | — |
| 206 | C22×C3 | — | 4 | 3 | no | — |
| 207 | C22×C3 | — | 4 | 4 | no | — |
| 208 | C2×D3 | — | 4 | 4 | no | — |
| 209 | C2×D3 | — | 8 | 12 | no | — |
| 210 | C3×C4 | — | 4 | 3 | no | — |
| 211 | C3×C4 | — | 4 | 1 | no | — |
| 212 | Dic3 | — | 4 | 4 | no | — |
| 213 | C23 | — | 3 | 3 | no | — |
| 214 | C23 | — | 3 | 3 | no | — |
| 215 | C23 | — | 6 | 4 | no | — |
| 216 | C23 | — | 6 | 6 | no | — |
| 217 | C23 | — | 12 | 4 | no | — |
| 218 | C23 | C4×S4 | 1 | 1 | yes | characteristic |
| 219 | Q8 | — | 3 | 3 | no | — |
| 220 | Q8 | — | 3 | 4 | no | — |
| 221 | Q8 | — | 6 | 1 | no | — |
| 222 | Q8 | — | 6 | 6 | no | — |
| 223 | Q8 | C22×C4×D3 | 1 | 1 | yes | characteristic |
| 224 | D4 | — | 3 | 3 | no | — |
| 225 | D4 | — | 3 | 1 | no | — |
| 226 | D4 | — | 3 | 4 | no | — |
| 227 | D4 | — | 6 | 3 | no | — |
| 228 | D4 | — | 6 | 3 | no | — |
| 229 | D4 | — | 6 | 6 | no | — |
| 230 | D4 | — | 6 | 4 | no | — |
| 231 | D4 | — | 6 | 12 | no | — |
| 232 | C2×C4 | — | 3 | 3 | no | — |
| 233 | C2×C4 | — | 3 | 3 | no | — |
| 234 | C2×C4 | — | 3 | 3 | no | — |
| 235 | C2×C4 | — | 3 | 3 | no | — |
| 236 | C2×C4 | — | 3 | 6 | no | — |
| 237 | C2×C4 | — | 3 | 1 | no | — |
| 238 | C2×C4 | — | 3 | 3 | no | — |
| 239 | C2×C4 | — | 6 | 6 | no | — |
| 240 | C2×C4 | — | 6 | 12 | no | — |
| 241 | C2×C4 | — | 6 | 4 | no | — |
| 242 | C2×C4 | — | 6 | 4 | no | — |
| 243 | C2×C4 | — | 12 | 12 | no | — |
| 244 | C2×C4 | C4×S4 | 1 | 3 | yes | — |
| 245 | C2×C4 | C22×S4 | 1 | 3 | yes | — |
| 246 | C8 | — | 3 | 4 | no | — |
| 247 | C8 | — | 3 | 4 | no | — |
| 248 | C8 | — | 3 | 4 | no | — |
| 249 | C8 | — | 6 | 4 | no | — |
| 250 | C8 | C22×S4 | 1 | 4 | yes | — |
| 251 | C2×C3 | — | 4 | 1 | no | — |
| 252 | C2×C3 | — | 4 | 6 | no | — |
| 253 | D3 | — | 8 | 4 | no | — |
| 254 | C22 | — | 3 | 1 | no | — |
| 255 | C22 | — | 3 | 3 | no | — |
| 256 | C22 | — | 6 | 6 | no | — |
| 257 | C22 | — | 6 | 6 | no | — |
| 258 | C22 | — | 6 | 4 | no | — |
| 259 | C22 | — | 12 | 12 | no | — |
| 260 | C22 | #5405 | 1 | 4 | yes | direct product factor |
| 261 | C22 | C2×C4×S4 | 1 | 3 | yes | — |
| 262 | C4 | — | 3 | 1 | no | — |
| 263 | C4 | — | 3 | 3 | no | — |
| 264 | C4 | — | 6 | 4 | no | — |
| 265 | C4 | — | 12 | 4 | no | — |
| 266 | C4 | C2×C4×S4 | 1 | 3 | yes | — |
| 267 | C4 | C23×S4 | 1 | 1 | yes | characteristic |
| 268 | C3 | — | 4 | 1 | no | Sylow |
| 269 | C2 | — | 6 | 1 | no | — |
| 270 | C2 | — | 6 | 3 | no | — |
| 271 | C2 | — | 12 | 4 | no | — |
| 272 | C2 | C2×#5405 | 1 | 6 | yes | direct product factor |
| 273 | C2 | C22×C4×S4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×#131⋊C3 | C2 | semidirect |
| 3. C22×GU(2,𝔽3) | C2 | semidirect |
| 4. C2×#5405 | C2 | direct (N × H) |
| 7. C2×#985 | C4 | semidirect |
| 9. C22×GL(2,𝔽3) | C4 | semidirect |
| 10. C22×#226 | C4 | semidirect |
| 12. C2×#131⋊C3 | C22 | semidirect |
| 13. C2×GU(2,𝔽3) | C22 | semidirect |
| 14. #5405 | C22 | direct (N × H) |
| 17. C22×#131 | D3 | semidirect |
| 28. #985 | C2×C4 | semidirect |
| 30. C2×GL(2,𝔽3) | C2×C4 | semidirect |
| 31. C2×#226 | C2×C4 | semidirect |
| 35. #131⋊C3 | C23 | semidirect |
| 36. GU(2,𝔽3) | C23 | semidirect |
| 50. C2×#131 | C2×D3 | semidirect |
| 73. GL(2,𝔽3) | C22×C4 | semidirect |
| 74. #226 | C22×C4 | semidirect |
| 111. #131 | C22×D3 | semidirect |
| 260. C22 | #5405 | direct (N × H) |
| 272. C2 | C2×#5405 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#985 | 2 |
| C22×#131⋊C3 | 2 |
| C22×GU(2,𝔽3) | 2 |
| C2×#5405 | 2 |
| C22×#444 | 3 |
| C22×ΓL(1,𝔽25) | 4 |