Finite Group DatabasebetaFinite Group DatabasebetaOrder 3456.
327 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 26 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)×PSU(3,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #157⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | SL(2,𝔽3)⋊AGL(1,𝔽9) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11779068 | — | 3 | 1 | no | maximal |
| 5 | C2×#89092 | — | 4 | 1 | no | maximal |
| 6 | #11724211 | — | 2 | 1 | no | — |
| 7 | SL(2,𝔽3)×#157 | — | 2 | 1 | no | — |
| 8 | SL(2,𝔽3)×#157 | C22 | 1 | 1 | yes | characteristic |
| 9 | Q8×PSU(3,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | #10596410 | — | 3 | 1 | no | — |
| 11 | #10596408 | — | 3 | 1 | no | — |
| 12 | #10593476 | — | 3 | 1 | no | — |
| 13 | Q8⋊AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 14 | #10593158 | — | 3 | 1 | no | — |
| 15 | PSU(3,𝔽2)⋊C8 | — | 3 | 1 | no | — |
| 16 | C2×C3×PSU(3,𝔽2) | — | 4 | 1 | no | — |
| 17 | C2×#157⋊D3 | — | 4 | 1 | no | — |
| 18 | C2×C3⋊AGL(1,𝔽9) | — | 4 | 1 | no | — |
| 19 | #89092 | — | 4 | 4 | no | — |
| 20 | C3⋊D3×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 21 | C3×C3⋊GL(2,𝔽3) | — | 4 | 1 | no | — |
| 22 | #73100 | — | 9 | 1 | no | maximal |
| 23 | C2×AΓL(1,𝔽9) | — | 6 | 2 | no | — |
| 24 | C2×AΓL(1,𝔽9) | — | 6 | 1 | no | — |
| 25 | Q8×#157 | — | 2 | 1 | no | — |
| 26 | Q8×#157 | C2×D3 | 1 | 1 | yes | characteristic |
| 27 | C4⋊PSU(3,𝔽2) | — | 3 | 1 | no | — |
| 28 | C4⋊PSU(3,𝔽2) | — | 6 | 1 | no | — |
| 29 | C4×PSU(3,𝔽2) | — | 3 | 1 | no | — |
| 30 | #64231 | — | 3 | 1 | no | — |
| 31 | C4⋊AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 32 | C4×AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 33 | D3×GL(2,𝔽3) | — | 12 | 1 | no | — |
| 34 | GO+(4,𝔽2)⋊C4 | — | 6 | 1 | no | — |
| 35 | #63940 | — | 6 | 1 | no | — |
| 36 | #63732 | — | 3 | 1 | no | — |
| 37 | #63729 | — | 6 | 1 | no | — |
| 38 | C2×#1113 | — | 8 | 1 | no | — |
| 39 | C2×C3×#157 | — | 4 | 1 | no | — |
| 40 | C2×C3×#157 | — | 8 | 1 | no | — |
| 41 | C3×PSU(3,𝔽2) | — | 4 | 2 | no | — |
| 42 | C3×PSU(3,𝔽2) | — | 8 | 1 | no | — |
| 43 | #157⋊D3 | — | 4 | 2 | no | — |
| 44 | #157⋊D3 | — | 8 | 1 | no | — |
| 45 | C3⋊AGL(1,𝔽9) | — | 4 | 2 | no | — |
| 46 | C32×SL(2,𝔽3) | SD16 | 1 | 1 | yes | characteristic, complemented |
| 47 | Q8×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 48 | C4⋊GL(2,𝔽3) | — | 9 | 1 | no | — |
| 49 | SL(2,𝔽3)⋊C8 | — | 9 | 1 | no | — |
| 50 | C2×PSU(3,𝔽2) | — | 3 | 2 | no | — |
| 51 | C2×PSU(3,𝔽2) | S4 | 1 | 1 | yes | characteristic |
| 52 | C2×GO+(4,𝔽2) | — | 6 | 1 | no | — |
| 53 | C2×GO+(4,𝔽2) | — | 6 | 1 | no | — |
| 54 | C2×AGL(1,𝔽9) | — | 6 | 1 | no | — |
| 55 | C2×AGL(1,𝔽9) | — | 6 | 1 | no | — |
| 56 | AΓL(1,𝔽9) | — | 12 | 4 | no | — |
| 57 | AΓL(1,𝔽9) | — | 12 | 2 | no | — |
| 58 | Q8×C3⋊D3 | #67 | 1 | 1 | yes | characteristic |
| 59 | #3802 | — | 6 | 1 | no | — |
| 60 | C4⋊#157 | — | 3 | 1 | no | — |
| 61 | C4⋊#157 | — | 6 | 1 | no | — |
| 62 | C4×#157 | — | 3 | 1 | no | — |
| 63 | C4×#157 | — | 6 | 1 | no | — |
| 64 | D3×SL(2,𝔽3) | — | 12 | 1 | no | — |
| 65 | C3⋊GL(2,𝔽3) | — | 4 | 1 | no | — |
| 66 | C3×GL(2,𝔽3) | — | 12 | 1 | no | — |
| 67 | C3×GL(2,𝔽3) | — | 24 | 1 | no | — |
| 68 | #3777 | — | 3 | 1 | no | — |
| 69 | #3777 | — | 6 | 1 | no | — |
| 70 | C3×#215 | — | 12 | 1 | no | — |
| 71 | #3716 | — | 12 | 1 | no | — |
| 72 | #3710 | — | 6 | 1 | no | — |
| 73 | #1623 | — | 27 | 1 | no | Sylow |
| 74 | C2×C3×C3⋊D3 | — | 4 | 1 | no | — |
| 75 | C2×C3×C3⋊D3 | — | 16 | 1 | no | — |
| 76 | #1113 | — | 8 | 2 | no | — |
| 77 | #1113 | — | 16 | 1 | no | — |
| 78 | C3×#157 | — | 4 | 1 | no | — |
| 79 | C3×#157 | — | 4 | 1 | no | — |
| 80 | C3×#157 | — | 8 | 2 | no | — |
| 81 | C2×GL(2,𝔽3) | — | 18 | 1 | no | — |
| 82 | C2×#215 | — | 36 | 1 | no | — |
| 83 | D3×SD16 | — | 36 | 1 | no | — |
| 84 | C4×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 85 | C4×SL(2,𝔽3) | — | 18 | 1 | no | — |
| 86 | C2×D32 | — | 12 | 1 | no | — |
| 87 | C2×D32 | — | 48 | 1 | no | — |
| 88 | C2×#157 | — | 2 | 1 | no | — |
| 89 | C2×#157 | — | 3 | 1 | no | — |
| 90 | C2×#157 | — | 6 | 1 | no | — |
| 91 | C2×#157 | C2×S4 | 1 | 1 | yes | characteristic |
| 92 | PSU(3,𝔽2) | — | 2 | 1 | no | — |
| 93 | PSU(3,𝔽2) | — | 12 | 2 | no | — |
| 94 | PSU(3,𝔽2) | GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 95 | GO+(4,𝔽2) | — | 12 | 2 | no | — |
| 96 | GO+(4,𝔽2) | — | 12 | 1 | no | — |
| 97 | GO+(4,𝔽2) | — | 12 | 1 | no | — |
| 98 | AGL(1,𝔽9) | — | 6 | 2 | no | — |
| 99 | AGL(1,𝔽9) | — | 12 | 1 | no | — |
| 100 | C32×Q8 | #215 | 1 | 1 | yes | characteristic, complemented |
| 101 | C4×C3⋊D3 | — | 3 | 1 | no | — |
| 102 | #634 | — | 3 | 1 | no | — |
| 103 | C3×D12 | — | 12 | 1 | no | — |
| 104 | C3×SL(2,𝔽3) | — | 4 | 1 | no | — |
| 105 | C3×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 106 | #625 | — | 12 | 1 | no | — |
| 107 | C3×C3⋊C8 | — | 12 | 1 | no | — |
| 108 | Q82 | — | 9 | 1 | no | — |
| 109 | #492 | — | 27 | 1 | no | — |
| 110 | #460 | — | 27 | 1 | no | — |
| 111 | #460 | — | 27 | 1 | no | — |
| 112 | #333 | — | 27 | 1 | no | — |
| 113 | Q8⋊C8 | — | 27 | 1 | no | — |
| 114 | Q8⋊C8 | — | 27 | 1 | no | — |
| 115 | C2×C33 | — | 4 | 1 | no | — |
| 116 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 117 | C3×C3⋊D3 | — | 4 | 1 | no | — |
| 118 | C3×C3⋊D3 | — | 16 | 2 | no | — |
| 119 | C2×C3×Q8 | — | 36 | 1 | no | — |
| 120 | D3×Q8 | — | 12 | 1 | no | — |
| 121 | D3×D4 | — | 36 | 1 | no | — |
| 122 | C2×D12 | — | 36 | 1 | no | — |
| 123 | C2×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 124 | GL(2,𝔽3) | — | 12 | 1 | no | — |
| 125 | GL(2,𝔽3) | — | 24 | 1 | no | — |
| 126 | C3×SD16 | — | 36 | 1 | no | — |
| 127 | #215 | — | 12 | 1 | no | — |
| 128 | #215 | — | 36 | 4 | no | — |
| 129 | #214 | — | 36 | 1 | no | — |
| 130 | C2×C3⋊C8 | — | 36 | 1 | no | — |
| 131 | #204 | — | 36 | 1 | no | — |
| 132 | D3×C8 | — | 36 | 1 | no | — |
| 133 | C2×C3⋊D3 | — | 16 | 1 | no | — |
| 134 | C2×C3⋊D3 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 135 | C2×C3×D3 | — | 24 | 1 | no | — |
| 136 | C2×C3×D3 | — | 48 | 1 | no | — |
| 137 | C2×C3×D3 | — | 48 | 1 | no | — |
| 138 | C2×C3×D3 | — | 96 | 1 | no | — |
| 139 | D32 | — | 24 | 1 | no | — |
| 140 | D32 | — | 24 | 1 | no | — |
| 141 | D32 | — | 48 | 2 | no | — |
| 142 | D32 | — | 48 | 2 | no | — |
| 143 | #157 | — | 2 | 2 | no | — |
| 144 | #157 | — | 6 | 1 | no | — |
| 145 | #157 | — | 12 | 1 | no | — |
| 146 | #157 | C2×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 147 | #157 | #983 | 1 | 1 | yes | characteristic |
| 148 | C32×C4 | — | 3 | 1 | no | — |
| 149 | C3⋊Dic3 | — | 3 | 1 | no | — |
| 150 | C2×SD16 | — | 54 | 1 | no | — |
| 151 | C2×SD16 | — | 54 | 1 | no | — |
| 152 | C2×SD16 | — | 54 | 2 | no | — |
| 153 | C4⋊Q8 | — | 27 | 1 | no | — |
| 154 | C4⋊Q8 | — | 54 | 1 | no | — |
| 155 | #127 | — | 27 | 1 | no | — |
| 156 | C4×Q8 | — | 9 | 1 | no | — |
| 157 | C4×Q8 | — | 18 | 1 | no | — |
| 158 | C4×Q8 | — | 27 | 1 | no | — |
| 159 | C4⋊C8 | — | 27 | 1 | no | — |
| 160 | C4⋊C8 | — | 27 | 1 | no | — |
| 161 | #102 | — | 54 | 1 | no | — |
| 162 | #102 | — | 54 | 1 | no | — |
| 163 | C4×C8 | — | 27 | 1 | no | — |
| 164 | C33 | — | 4 | 1 | no | Sylow |
| 165 | C22×D3 | — | 72 | 1 | no | — |
| 166 | C22×D3 | — | 72 | 1 | no | — |
| 167 | C3×Q8 | — | 4 | 1 | no | — |
| 168 | C3×Q8 | — | 36 | 2 | no | — |
| 169 | C3×Q8 | — | 72 | 1 | no | — |
| 170 | C3×D4 | — | 36 | 1 | no | — |
| 171 | C2×C3×C4 | — | 36 | 1 | no | — |
| 172 | C2×C3×C4 | — | 72 | 1 | no | — |
| 173 | #67 | — | 72 | 1 | no | — |
| 174 | D12 | — | 12 | 1 | no | — |
| 175 | D12 | — | 36 | 2 | no | — |
| 176 | D12 | — | 72 | 1 | no | — |
| 177 | C4×D3 | — | 36 | 1 | no | — |
| 178 | Dic6 | — | 36 | 1 | no | — |
| 179 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 180 | SL(2,𝔽3) | AΓL(1,𝔽9) | 1 | 1 | yes | characteristic, complemented |
| 181 | C3×C8 | — | 36 | 1 | no | — |
| 182 | C3⋊C8 | — | 12 | 1 | no | — |
| 183 | C3⋊C8 | — | 36 | 2 | no | — |
| 184 | C2×C32 | — | 16 | 1 | no | — |
| 185 | C2×C32 | — | 32 | 1 | no | — |
| 186 | C2×C32 | #5417 | 1 | 1 | yes | characteristic |
| 187 | C3⋊D3 | — | 16 | 2 | no | — |
| 188 | C3⋊D3 | #5421 | 1 | 1 | yes | characteristic |
| 189 | C3⋊D3 | #5429 | 1 | 1 | yes | characteristic |
| 190 | C3×D3 | — | 48 | 1 | no | — |
| 191 | C3×D3 | — | 48 | 1 | no | — |
| 192 | C3×D3 | — | 48 | 2 | no | — |
| 193 | C3×D3 | — | 48 | 1 | no | — |
| 194 | C3×D3 | — | 96 | 2 | no | — |
| 195 | C2×Q8 | — | 9 | 1 | no | — |
| 196 | C2×Q8 | — | 9 | 1 | no | — |
| 197 | C2×Q8 | — | 27 | 2 | no | — |
| 198 | C2×D4 | — | 54 | 1 | no | — |
| 199 | C2×D4 | — | 54 | 1 | no | — |
| 200 | C2×D4 | — | 54 | 1 | no | — |
| 201 | SD16 | — | 36 | 1 | no | — |
| 202 | SD16 | — | 108 | 1 | no | — |
| 203 | SD16 | — | 108 | 4 | no | — |
| 204 | SD16 | — | 108 | 2 | no | — |
| 205 | C2×C8 | — | 54 | 1 | no | — |
| 206 | C2×C8 | — | 54 | 1 | no | — |
| 207 | C2×C8 | — | 54 | 1 | no | — |
| 208 | C4⋊C4 | — | 27 | 1 | no | — |
| 209 | C4⋊C4 | — | 27 | 1 | no | — |
| 210 | C4⋊C4 | — | 54 | 1 | no | — |
| 211 | C4⋊C4 | — | 54 | 1 | no | — |
| 212 | C42 | — | 27 | 1 | no | — |
| 213 | C42 | — | 54 | 1 | no | — |
| 214 | C22×C3 | — | 36 | 1 | no | — |
| 215 | C22×C3 | — | 72 | 1 | no | — |
| 216 | C2×D3 | — | 12 | 1 | no | — |
| 217 | C2×D3 | — | 24 | 1 | no | — |
| 218 | C2×D3 | — | 48 | 1 | no | — |
| 219 | C2×D3 | — | 72 | 2 | no | — |
| 220 | C2×D3 | — | 96 | 1 | no | — |
| 221 | C2×D3 | — | 144 | 1 | no | — |
| 222 | C2×D3 | — | 144 | 1 | no | — |
| 223 | C2×D3 | — | 144 | 1 | no | — |
| 224 | C3×C4 | — | 12 | 1 | no | — |
| 225 | C3×C4 | — | 36 | 1 | no | — |
| 226 | C3×C4 | — | 36 | 1 | no | — |
| 227 | C3×C4 | — | 72 | 2 | no | — |
| 228 | Dic3 | — | 36 | 1 | no | — |
| 229 | C32 | — | 16 | 1 | no | — |
| 230 | C32 | — | 32 | 1 | no | — |
| 231 | C32 | #73100 | 1 | 1 | yes | characteristic, complemented |
| 232 | C23 | — | 108 | 1 | no | — |
| 233 | Q8 | — | 9 | 2 | no | — |
| 234 | Q8 | — | 18 | 1 | no | — |
| 235 | Q8 | — | 27 | 1 | no | — |
| 236 | Q8 | — | 108 | 2 | no | — |
| 237 | Q8 | #89092 | 1 | 1 | yes | characteristic, complemented |
| 238 | D4 | — | 36 | 1 | no | — |
| 239 | D4 | — | 108 | 1 | no | — |
| 240 | D4 | — | 108 | 1 | no | — |
| 241 | D4 | — | 108 | 1 | no | — |
| 242 | D4 | — | 108 | 2 | no | — |
| 243 | C2×C4 | — | 9 | 1 | no | — |
| 244 | C2×C4 | — | 18 | 1 | no | — |
| 245 | C2×C4 | — | 27 | 1 | no | — |
| 246 | C2×C4 | — | 27 | 1 | no | — |
| 247 | C2×C4 | — | 54 | 1 | no | — |
| 248 | C8 | — | 36 | 1 | no | — |
| 249 | C8 | — | 54 | 2 | no | — |
| 250 | C8 | — | 108 | 1 | no | — |
| 251 | C2×C3 | — | 4 | 1 | no | — |
| 252 | C2×C3 | — | 4 | 1 | no | — |
| 253 | C2×C3 | — | 32 | 1 | no | — |
| 254 | C2×C3 | — | 36 | 1 | no | — |
| 255 | C2×C3 | — | 36 | 1 | no | — |
| 256 | C2×C3 | — | 144 | 1 | no | — |
| 257 | D3 | — | 12 | 1 | no | — |
| 258 | D3 | — | 12 | 1 | no | — |
| 259 | D3 | — | 48 | 1 | no | — |
| 260 | D3 | — | 48 | 2 | no | — |
| 261 | D3 | — | 96 | 2 | no | — |
| 262 | C22 | — | 9 | 1 | no | — |
| 263 | C22 | — | 72 | 1 | no | — |
| 264 | C22 | — | 216 | 1 | no | — |
| 265 | C22 | — | 216 | 1 | no | — |
| 266 | C4 | — | 3 | 1 | no | — |
| 267 | C4 | — | 9 | 1 | no | — |
| 268 | C4 | — | 9 | 1 | no | — |
| 269 | C4 | — | 18 | 2 | no | — |
| 270 | C4 | — | 27 | 1 | no | — |
| 271 | C4 | — | 54 | 1 | no | — |
| 272 | C4 | — | 108 | 1 | no | — |
| 273 | C3 | — | 4 | 1 | no | — |
| 274 | C3 | — | 4 | 1 | no | — |
| 275 | C3 | — | 32 | 1 | no | — |
| 276 | C2 | — | 9 | 1 | no | — |
| 277 | C2 | — | 9 | 1 | no | — |
| 278 | C2 | — | 144 | 1 | no | — |
| 279 | C2 | #13181780 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. SL(2,𝔽3)×PSU(3,𝔽2) | C2 | semidirect |
| 3. SL(2,𝔽3)⋊AGL(1,𝔽9) | C2 | semidirect |
| 9. Q8×PSU(3,𝔽2) | D3 | semidirect |
| 46. C32×SL(2,𝔽3) | SD16 | semidirect |
| 94. PSU(3,𝔽2) | GL(2,𝔽3) | semidirect |
| 100. C32×Q8 | #215 | semidirect |
| 180. SL(2,𝔽3) | AΓL(1,𝔽9) | semidirect |
| 231. C32 | #73100 | semidirect |
| 237. Q8 | #89092 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽3)×PSU(3,𝔽2) | 2 |
| #157⋊GL(2,𝔽3) | 2 |
| SL(2,𝔽3)⋊AGL(1,𝔽9) | 2 |
| #11779068 | 3 |
| C2×#89092 | 4 |
| #73100 | 9 |