Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
313 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 313 conjugacy classes of subgroups — too many to draw. Only its 37 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 | SL(2,𝔽3)×D12 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4⋊(C3⋊GL(2,𝔽3)) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3×SL(2,𝔽3)⋊C8 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #73088 | — | 3 | 1 | no | maximal |
| 5 | #68266 | — | 3 | 1 | no | maximal |
| 6 | C2×D3×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 7 | C2×C3⋊GL(2,𝔽3) | — | 2 | 1 | no | — |
| 8 | C3×C4×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#3714 | — | 4 | 1 | no | maximal |
| 10 | Q8×D12 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | D4×SL(2,𝔽3) | — | 3 | 1 | no | — |
| 12 | C4⋊GL(2,𝔽3) | — | 3 | 1 | no | — |
| 13 | C4⋊GL(2,𝔽3) | — | 6 | 1 | no | — |
| 14 | #5027 | — | 3 | 1 | no | — |
| 15 | #4843 | — | 3 | 1 | no | — |
| 16 | #4694 | — | 3 | 1 | no | — |
| 17 | SL(2,𝔽3)⋊C8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 18 | C3×Q8⋊C8 | — | 3 | 1 | no | — |
| 19 | #4486 | — | 3 | 1 | no | — |
| 20 | #4459 | — | 3 | 1 | no | — |
| 21 | C2×C4⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 22 | C2×C3×D12 | — | 4 | 1 | no | — |
| 23 | C2×C3×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 24 | D3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 25 | C3⋊GL(2,𝔽3) | — | 4 | 1 | no | — |
| 26 | C2×C3×C3⋊C8 | — | 4 | 1 | no | — |
| 27 | #3714 | — | 4 | 4 | no | — |
| 28 | #1627 | — | 9 | 1 | no | Sylow |
| 29 | C2×D3×Q8 | — | 2 | 1 | no | — |
| 30 | C22×SL(2,𝔽3) | — | 6 | 1 | no | — |
| 31 | C2×GL(2,𝔽3) | — | 6 | 2 | no | — |
| 32 | C2×GL(2,𝔽3) | — | 6 | 1 | no | — |
| 33 | C3×C4×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 34 | C2×#215 | — | 6 | 1 | no | — |
| 35 | C2×#213 | — | 12 | 1 | no | — |
| 36 | C2×D24 | — | 6 | 1 | no | — |
| 37 | #897 | — | 6 | 1 | no | — |
| 38 | #874 | — | 3 | 1 | no | — |
| 39 | C4×D12 | — | 3 | 1 | no | — |
| 40 | #869 | — | 3 | 1 | no | — |
| 41 | C4×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 42 | C4×SL(2,𝔽3) | C2×D3 | 1 | 1 | yes | characteristic |
| 43 | C3×C4⋊C8 | — | 3 | 1 | no | — |
| 44 | C3×C4×C8 | — | 3 | 1 | no | — |
| 45 | #821 | — | 6 | 2 | no | — |
| 46 | #821 | — | 6 | 1 | no | — |
| 47 | #809 | — | 6 | 1 | no | — |
| 48 | C4⋊(C3⋊C8) | — | 3 | 1 | no | — |
| 49 | C22×C3⋊D3 | — | 8 | 1 | no | — |
| 50 | C22×C3×D3 | — | 8 | 1 | no | — |
| 51 | C2×C32×C4 | — | 4 | 1 | no | — |
| 52 | C4⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 53 | C4⋊(C3⋊D3) | — | 8 | 1 | no | — |
| 54 | C3×D12 | — | 4 | 2 | no | — |
| 55 | C3×D12 | — | 8 | 1 | no | — |
| 56 | C3×SL(2,𝔽3) | D8 | 1 | 1 | yes | characteristic, complemented |
| 57 | C3×C3⋊C8 | — | 4 | 2 | no | — |
| 58 | D4×Q8 | — | 3 | 1 | no | — |
| 59 | #486 | — | 9 | 1 | no | — |
| 60 | #460 | — | 9 | 1 | no | — |
| 61 | #459 | — | 9 | 1 | no | — |
| 62 | #332 | — | 9 | 1 | no | — |
| 63 | Q8⋊C8 | — | 3 | 1 | no | — |
| 64 | D4⋊C8 | — | 9 | 1 | no | — |
| 65 | C2×C3×Q8 | #67 | 1 | 1 | yes | characteristic |
| 66 | C2×C3×D4 | — | 12 | 1 | no | — |
| 67 | D3×Q8 | — | 4 | 1 | no | — |
| 68 | D3×Q8 | — | 12 | 1 | no | — |
| 69 | C2×D12 | — | 6 | 1 | no | — |
| 70 | C2×D12 | — | 6 | 1 | no | — |
| 71 | C2×D12 | — | 6 | 1 | no | — |
| 72 | C2×D12 | — | 12 | 1 | no | — |
| 73 | C2×D12 | — | 24 | 1 | no | — |
| 74 | C2×D12 | S4 | 1 | 1 | yes | characteristic |
| 75 | C2×C4×D3 | — | 6 | 1 | no | — |
| 76 | C2×Dic6 | — | 6 | 1 | no | — |
| 77 | C2×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 78 | C2×SL(2,𝔽3) | — | 12 | 1 | no | — |
| 79 | C2×SL(2,𝔽3) | D12 | 1 | 1 | yes | characteristic |
| 80 | GL(2,𝔽3) | — | 12 | 2 | no | — |
| 81 | GL(2,𝔽3) | — | 12 | 1 | no | — |
| 82 | C2×C3×C8 | — | 6 | 1 | no | — |
| 83 | C2×C3×C8 | — | 6 | 1 | no | — |
| 84 | C3×C4⋊C4 | — | 3 | 1 | no | — |
| 85 | C3×C42 | — | 3 | 1 | no | — |
| 86 | #215 | — | 12 | 1 | no | — |
| 87 | #215 | — | 12 | 1 | no | — |
| 88 | #213 | — | 12 | 4 | no | — |
| 89 | (C2×D3)⋊C4 | — | 6 | 1 | no | — |
| 90 | C4⋊Dic3 | — | 3 | 2 | no | — |
| 91 | C4⋊Dic3 | — | 3 | 1 | no | — |
| 92 | C2×C3⋊C8 | — | 4 | 1 | no | — |
| 93 | C2×C3⋊C8 | — | 6 | 1 | no | — |
| 94 | D24 | — | 12 | 2 | no | — |
| 95 | C22×C32 | — | 4 | 1 | no | — |
| 96 | C2×C3⋊D3 | — | 8 | 2 | no | — |
| 97 | C2×C3⋊D3 | — | 16 | 1 | no | — |
| 98 | C2×C3⋊D3 | — | 16 | 1 | no | — |
| 99 | C2×C3×D3 | — | 8 | 2 | no | — |
| 100 | C2×C3×D3 | — | 16 | 1 | no | — |
| 101 | C2×C3×D3 | — | 16 | 1 | no | — |
| 102 | C32×C4 | — | 4 | 1 | no | — |
| 103 | C32×C4 | — | 4 | 1 | no | — |
| 104 | C22×Q8 | — | 6 | 1 | no | — |
| 105 | C2×SD16 | — | 18 | 1 | no | — |
| 106 | C2×D8 | — | 18 | 1 | no | — |
| 107 | C4⋊Q8 | — | 9 | 1 | no | — |
| 108 | #127 | — | 9 | 1 | no | — |
| 109 | C22⋊Q8 | — | 18 | 1 | no | — |
| 110 | C4×Q8 | D32 | 1 | 1 | yes | characteristic |
| 111 | C4×D4 | — | 9 | 1 | no | — |
| 112 | C4⋊C8 | — | 3 | 1 | no | — |
| 113 | C4⋊C8 | — | 9 | 1 | no | — |
| 114 | #102 | — | 18 | 1 | no | — |
| 115 | #102 | — | 18 | 1 | no | — |
| 116 | #102 | — | 18 | 2 | no | — |
| 117 | C4×C8 | — | 3 | 1 | no | — |
| 118 | C23×C3 | — | 24 | 1 | no | — |
| 119 | C22×D3 | — | 2 | 1 | no | — |
| 120 | C22×D3 | — | 12 | 1 | no | — |
| 121 | C22×D3 | — | 24 | 1 | no | — |
| 122 | C22×D3 | — | 24 | 2 | no | — |
| 123 | C3×Q8 | — | 3 | 1 | no | — |
| 124 | C3×Q8 | #213 | 1 | 1 | yes | characteristic, complemented |
| 125 | C3×D4 | — | 12 | 2 | no | — |
| 126 | C3×D4 | — | 24 | 1 | no | — |
| 127 | C2×C3×C4 | — | 3 | 1 | no | — |
| 128 | C2×C3×C4 | — | 3 | 1 | no | — |
| 129 | C2×C3×C4 | — | 4 | 1 | no | — |
| 130 | C2×C3×C4 | — | 8 | 1 | no | — |
| 131 | C2×C3×C4 | C2×S4 | 1 | 1 | yes | characteristic |
| 132 | C2×Dic3 | — | 6 | 1 | no | — |
| 133 | D12 | — | 2 | 1 | no | — |
| 134 | D12 | — | 12 | 1 | no | — |
| 135 | D12 | — | 12 | 1 | no | — |
| 136 | D12 | — | 12 | 2 | no | — |
| 137 | D12 | — | 12 | 2 | no | — |
| 138 | D12 | — | 12 | 1 | no | — |
| 139 | D12 | — | 12 | 1 | no | — |
| 140 | D12 | — | 24 | 1 | no | — |
| 141 | D12 | — | 24 | 2 | no | — |
| 142 | D12 | — | 24 | 2 | no | — |
| 143 | D12 | GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 144 | C4×D3 | — | 12 | 1 | no | — |
| 145 | C4×D3 | — | 12 | 1 | no | — |
| 146 | Dic6 | — | 12 | 1 | no | — |
| 147 | Dic6 | — | 12 | 1 | no | — |
| 148 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 149 | SL(2,𝔽3) | D24 | 1 | 1 | yes | characteristic, complemented |
| 150 | C3×C8 | — | 12 | 1 | no | — |
| 151 | C3×C8 | — | 12 | 1 | no | — |
| 152 | C3⋊C8 | — | 4 | 2 | no | — |
| 153 | C3⋊C8 | — | 12 | 1 | no | — |
| 154 | C2×C32 | — | 4 | 1 | no | — |
| 155 | C2×C32 | — | 4 | 1 | no | — |
| 156 | C2×C32 | — | 4 | 1 | no | — |
| 157 | C3⋊D3 | — | 16 | 2 | no | — |
| 158 | C3×D3 | — | 16 | 2 | no | — |
| 159 | C2×Q8 | — | 12 | 1 | no | — |
| 160 | C2×Q8 | — | 18 | 1 | no | — |
| 161 | C2×Q8 | — | 36 | 1 | no | — |
| 162 | C2×Q8 | #626 | 1 | 1 | yes | characteristic |
| 163 | C2×D4 | — | 3 | 1 | no | — |
| 164 | C2×D4 | — | 18 | 1 | no | — |
| 165 | C2×D4 | — | 18 | 1 | no | — |
| 166 | C2×D4 | — | 18 | 1 | no | — |
| 167 | C22×C4 | — | 18 | 1 | no | — |
| 168 | SD16 | — | 36 | 1 | no | — |
| 169 | SD16 | — | 36 | 1 | no | — |
| 170 | D8 | — | 36 | 2 | no | — |
| 171 | C2×C8 | — | 6 | 1 | no | — |
| 172 | C2×C8 | — | 6 | 1 | no | — |
| 173 | C2×C8 | — | 18 | 1 | no | — |
| 174 | C4⋊C4 | — | 3 | 1 | no | — |
| 175 | C4⋊C4 | — | 9 | 2 | no | — |
| 176 | C4⋊C4 | — | 9 | 1 | no | — |
| 177 | C22⋊C4 | — | 18 | 1 | no | — |
| 178 | C42 | — | 3 | 1 | no | — |
| 179 | C22×C3 | — | 4 | 1 | no | — |
| 180 | C22×C3 | — | 8 | 1 | no | — |
| 181 | C22×C3 | — | 24 | 2 | no | — |
| 182 | C22×C3 | — | 48 | 1 | no | — |
| 183 | C22×C3 | — | 48 | 1 | no | — |
| 184 | C22×C3 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 185 | C2×D3 | — | 2 | 2 | no | — |
| 186 | C2×D3 | — | 4 | 1 | no | — |
| 187 | C2×D3 | — | 4 | 1 | no | — |
| 188 | C2×D3 | — | 24 | 2 | no | — |
| 189 | C2×D3 | — | 24 | 4 | no | — |
| 190 | C2×D3 | — | 24 | 1 | no | — |
| 191 | C2×D3 | — | 24 | 1 | no | — |
| 192 | C2×D3 | — | 24 | 1 | no | — |
| 193 | C2×D3 | — | 48 | 1 | no | — |
| 194 | C2×D3 | — | 48 | 1 | no | — |
| 195 | C2×D3 | — | 48 | 2 | no | — |
| 196 | C2×D3 | — | 48 | 2 | no | — |
| 197 | C3×C4 | — | 3 | 1 | no | — |
| 198 | C3×C4 | — | 3 | 1 | no | — |
| 199 | C3×C4 | — | 4 | 1 | no | — |
| 200 | C3×C4 | — | 4 | 1 | no | — |
| 201 | C3×C4 | — | 6 | 1 | no | — |
| 202 | C3×C4 | — | 8 | 1 | no | — |
| 203 | C3×C4 | — | 8 | 1 | no | — |
| 204 | C3×C4 | C2×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 205 | C3×C4 | #983 | 1 | 1 | yes | characteristic |
| 206 | Dic3 | — | 12 | 1 | no | — |
| 207 | C32 | — | 4 | 1 | no | Sylow |
| 208 | C23 | — | 6 | 1 | no | — |
| 209 | C23 | — | 36 | 1 | no | — |
| 210 | Q8 | — | 3 | 1 | no | — |
| 211 | Q8 | — | 36 | 1 | no | — |
| 212 | Q8 | — | 36 | 1 | no | — |
| 213 | Q8 | #3714 | 1 | 1 | yes | characteristic, complemented |
| 214 | D4 | — | 3 | 2 | no | — |
| 215 | D4 | — | 6 | 1 | no | — |
| 216 | D4 | — | 36 | 1 | no | — |
| 217 | D4 | — | 36 | 1 | no | — |
| 218 | D4 | — | 36 | 2 | no | — |
| 219 | D4 | — | 36 | 1 | no | — |
| 220 | D4 | — | 36 | 1 | no | — |
| 221 | C2×C4 | — | 3 | 1 | no | — |
| 222 | C2×C4 | — | 3 | 1 | no | — |
| 223 | C2×C4 | — | 18 | 1 | no | — |
| 224 | C2×C4 | — | 36 | 1 | no | — |
| 225 | C2×C4 | — | 36 | 1 | no | — |
| 226 | C2×C4 | D3×S4 | 1 | 1 | yes | characteristic |
| 227 | C8 | — | 12 | 1 | no | — |
| 228 | C8 | — | 12 | 1 | no | — |
| 229 | C8 | — | 36 | 1 | no | — |
| 230 | C2×C3 | — | 4 | 1 | no | — |
| 231 | C2×C3 | — | 4 | 1 | no | — |
| 232 | C2×C3 | — | 4 | 1 | no | — |
| 233 | C2×C3 | — | 8 | 1 | no | — |
| 234 | C2×C3 | — | 8 | 1 | no | — |
| 235 | C2×C3 | — | 8 | 1 | no | — |
| 236 | C2×C3 | — | 48 | 2 | no | — |
| 237 | C2×C3 | #5415 | 1 | 1 | yes | characteristic |
| 238 | C2×C3 | #5421 | 1 | 1 | yes | characteristic |
| 239 | C2×C3 | #5431 | 1 | 1 | yes | characteristic |
| 240 | D3 | — | 4 | 2 | no | — |
| 241 | D3 | — | 48 | 2 | no | — |
| 242 | D3 | — | 48 | 4 | no | — |
| 243 | D3 | — | 48 | 1 | no | — |
| 244 | C22 | — | 6 | 2 | no | — |
| 245 | C22 | — | 12 | 1 | no | — |
| 246 | C22 | — | 12 | 1 | no | — |
| 247 | C22 | — | 72 | 1 | no | — |
| 248 | C22 | — | 72 | 1 | no | — |
| 249 | C22 | — | 72 | 1 | no | — |
| 250 | C22 | #64210 | 1 | 1 | yes | characteristic |
| 251 | C4 | — | 3 | 1 | no | — |
| 252 | C4 | — | 3 | 1 | no | — |
| 253 | C4 | — | 6 | 1 | no | — |
| 254 | C4 | — | 36 | 1 | no | — |
| 255 | C4 | #64202 | 1 | 1 | yes | characteristic |
| 256 | C4 | D3×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 257 | C3 | — | 4 | 1 | no | — |
| 258 | C3 | — | 8 | 1 | no | — |
| 259 | C3 | #73088 | 1 | 1 | yes | characteristic, complemented |
| 260 | C2 | — | 12 | 2 | no | — |
| 261 | C2 | — | 144 | 1 | no | — |
| 262 | C2 | #10593184 | 1 | 1 | yes | characteristic |
| 263 | C2 | #10593218 | 1 | 1 | yes | characteristic |
| 264 | C2 | #10593230 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. SL(2,𝔽3)×D12 | C2 | semidirect |
| 2. C4⋊(C3⋊GL(2,𝔽3)) | C2 | semidirect |
| 3. C3×SL(2,𝔽3)⋊C8 | C2 | semidirect |
| 10. Q8×D12 | D3 | semidirect |
| 17. SL(2,𝔽3)⋊C8 | D3 | semidirect |
| 56. C3×SL(2,𝔽3) | D8 | semidirect |
| 124. C3×Q8 | #213 | semidirect |
| 143. D12 | GL(2,𝔽3) | semidirect |
| 149. SL(2,𝔽3) | D24 | semidirect |
| 213. Q8 | #3714 | semidirect |
| 259. C3 | #73088 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽3)×D12 | 2 |
| C4⋊(C3⋊GL(2,𝔽3)) | 2 |
| C3×SL(2,𝔽3)⋊C8 | 2 |
| #73088 | 3 |
| #68266 | 3 |
| C2×#3714 | 4 |