Finite Group DatabasebetaFinite Group DatabasebetaOrder 2916.
332 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 28 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 | #13091149 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #13091149 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #13091148 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11758087 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | D3×#4176 | — | 3 | 1 | no | maximal |
| 6 | D3×#4176 | — | 3 | 1 | no | maximal |
| 7 | #10624760 | C22 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 8 | #92404 | C2×C3 | 1 | 1 | yes | characteristic |
| 9 | #92392 | C2×C3 | 1 | 1 | yes | characteristic |
| 10 | #92392 | C2×C3 | 1 | 1 | yes | characteristic |
| 11 | C3⋊#4178 | — | 6 | 1 | no | — |
| 12 | C3⋊#4176 | — | 3 | 1 | no | — |
| 13 | C3⋊#4176 | — | 3 | 1 | no | — |
| 14 | #92285 | — | 2 | 1 | no | — |
| 15 | #92285 | — | 2 | 1 | no | — |
| 16 | D3×#731 | — | 3 | 1 | no | — |
| 17 | D3×#731 | — | 3 | 1 | no | — |
| 18 | C3×#4176 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | C3×#4176 | D3 | 1 | 1 | yes | characteristic, complemented |
| 20 | D3×#272 | — | 3 | 1 | no | — |
| 21 | D3×#272 | — | 3 | 1 | no | — |
| 22 | D3×#272 | — | 9 | 1 | no | — |
| 23 | D3×#272 | — | 9 | 1 | no | — |
| 24 | D32×C9 | — | 9 | 1 | no | — |
| 25 | C2×#4176 | — | 9 | 1 | no | — |
| 26 | C2×#4176 | — | 9 | 1 | no | — |
| 27 | H(5,𝔽3) | C22×C3 | 1 | 1 | yes | characteristic |
| 28 | #6911 | — | 2 | 1 | no | — |
| 29 | #6911 | — | 2 | 1 | no | — |
| 30 | #6909 | — | 4 | 1 | no | — |
| 31 | C3×#732 | — | 2 | 1 | no | — |
| 32 | C3×#731 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 33 | C3×#731 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 34 | C3⋊#272 | — | 3 | 1 | no | — |
| 35 | C3⋊#272 | — | 3 | 1 | no | — |
| 36 | C3⋊#272 | — | 6 | 1 | no | — |
| 37 | C3⋊#272 | — | 9 | 1 | no | — |
| 38 | C3⋊#272 | — | 9 | 1 | no | — |
| 39 | C3⋊#272 | — | 18 | 1 | no | — |
| 40 | C3⋊#272 | — | 18 | 1 | no | — |
| 41 | C3⋊#272 | — | 18 | 1 | no | — |
| 42 | C3⋊#272 | — | 18 | 1 | no | — |
| 43 | C3⋊#272 | — | 18 | 1 | no | — |
| 44 | C3×#272 | — | 3 | 1 | no | — |
| 45 | C3×#272 | — | 3 | 1 | no | — |
| 46 | C3×#272 | C3×D3 | 1 | 1 | yes | characteristic |
| 47 | C3×#272 | C3×D3 | 1 | 1 | yes | characteristic |
| 48 | C9×C3⋊D3 | — | 9 | 1 | no | — |
| 49 | D3×3−1+2 | — | 6 | 1 | no | — |
| 50 | D3×3−1+2 | — | 6 | 1 | no | — |
| 51 | D3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 52 | D3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 53 | D3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 54 | D3×H(3,𝔽3) | — | 9 | 1 | no | — |
| 55 | C3×D3×C9 | — | 3 | 1 | no | — |
| 56 | C3×D3×C9 | — | 3 | 1 | no | — |
| 57 | C2×#731 | — | 9 | 1 | no | — |
| 58 | C2×#731 | — | 9 | 1 | no | — |
| 59 | #4178 | — | 6 | 1 | no | — |
| 60 | #4178 | — | 6 | 1 | no | — |
| 61 | #4178 | — | 18 | 1 | no | — |
| 62 | #4178 | — | 18 | 1 | no | — |
| 63 | #4178 | — | 18 | 1 | no | — |
| 64 | #4176 | — | 3 | 1 | no | — |
| 65 | #4176 | — | 6 | 1 | no | — |
| 66 | #4176 | — | 9 | 1 | no | — |
| 67 | #4176 | — | 9 | 1 | no | — |
| 68 | #4176 | — | 18 | 1 | no | — |
| 69 | #4176 | — | 18 | 1 | no | — |
| 70 | C3×D32 | — | 9 | 1 | no | — |
| 71 | C3×D32 | — | 27 | 1 | no | — |
| 72 | C3×D32 | — | 27 | 1 | no | — |
| 73 | C3×D32 | — | 27 | 1 | no | — |
| 74 | C3×D32 | — | 27 | 1 | no | — |
| 75 | C3×D32 | — | 27 | 1 | no | — |
| 76 | C2×#272 | — | 9 | 1 | no | — |
| 77 | C2×#272 | — | 9 | 1 | no | — |
| 78 | C2×D3×C9 | — | 27 | 1 | no | — |
| 79 | C2×D3×C9 | — | 27 | 1 | no | — |
| 80 | C3×3−1+2 | — | 2 | 1 | no | — |
| 81 | C3×3−1+2 | — | 2 | 1 | no | — |
| 82 | C3×H(3,𝔽3) | — | 2 | 1 | no | — |
| 83 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 84 | C3×H(3,𝔽3) | — | 3 | 1 | no | — |
| 85 | C3×H(3,𝔽3) | — | 4 | 1 | no | — |
| 86 | C3×H(3,𝔽3) | — | 6 | 1 | no | — |
| 87 | C3×H(3,𝔽3) | — | 6 | 1 | no | — |
| 88 | C3×H(3,𝔽3) | — | 6 | 1 | no | — |
| 89 | C3×H(3,𝔽3) | — | 6 | 1 | no | — |
| 90 | C3×H(3,𝔽3) | — | 6 | 1 | no | — |
| 91 | C3×H(3,𝔽3) | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 92 | C3×H(3,𝔽3) | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 93 | C32×C9 | D32 | 1 | 1 | yes | characteristic, complemented |
| 94 | #732 | — | 6 | 1 | no | — |
| 95 | #732 | — | 6 | 1 | no | — |
| 96 | #732 | — | 6 | 1 | no | — |
| 97 | #732 | — | 12 | 1 | no | — |
| 98 | #732 | — | 12 | 1 | no | — |
| 99 | #732 | — | 12 | 1 | no | — |
| 100 | #732 | — | 12 | 1 | no | — |
| 101 | #731 | — | 3 | 1 | no | — |
| 102 | #731 | — | 6 | 1 | no | — |
| 103 | #731 | — | 6 | 1 | no | — |
| 104 | #731 | — | 6 | 1 | no | — |
| 105 | AΣL(1,𝔽27) | — | 12 | 1 | no | — |
| 106 | AΣL(1,𝔽27) | — | 12 | 1 | no | — |
| 107 | AΣL(1,𝔽27) | — | 12 | 1 | no | — |
| 108 | C3×C3⋊D3 | — | 9 | 1 | no | — |
| 109 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 110 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 111 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 112 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 113 | C3×C3⋊D3 | — | 27 | 1 | no | — |
| 114 | C3×C3⋊D3 | — | 54 | 1 | no | — |
| 115 | C3×C3⋊D3 | — | 54 | 1 | no | — |
| 116 | C3×C3⋊D3 | — | 54 | 1 | no | — |
| 117 | C3×C3⋊D3 | — | 54 | 1 | no | — |
| 118 | C32×D3 | — | 3 | 1 | no | — |
| 119 | C32×D3 | — | 3 | 1 | no | — |
| 120 | C32×D3 | — | 9 | 1 | no | — |
| 121 | C32×D3 | — | 9 | 1 | no | — |
| 122 | C32×D3 | — | 9 | 1 | no | — |
| 123 | C32×D3 | — | 9 | 1 | no | — |
| 124 | C32×D3 | — | 9 | 1 | no | — |
| 125 | C32×D3 | — | 9 | 1 | no | — |
| 126 | C32×D3 | — | 9 | 1 | no | — |
| 127 | C32×D3 | — | 9 | 1 | no | — |
| 128 | C32×D3 | — | 9 | 1 | no | — |
| 129 | C32×D3 | — | 9 | 1 | no | — |
| 130 | C2×3−1+2 | — | 18 | 1 | no | — |
| 131 | C2×3−1+2 | — | 18 | 1 | no | — |
| 132 | C2×H(3,𝔽3) | — | 9 | 1 | no | — |
| 133 | C2×H(3,𝔽3) | — | 9 | 1 | no | — |
| 134 | C2×C3×C9 | — | 9 | 1 | no | — |
| 135 | C2×C3×C9 | — | 9 | 1 | no | — |
| 136 | #272 | — | 9 | 1 | no | — |
| 137 | #272 | — | 9 | 1 | no | — |
| 138 | #272 | — | 18 | 1 | no | — |
| 139 | #272 | — | 18 | 1 | no | — |
| 140 | #272 | — | 18 | 1 | no | — |
| 141 | #272 | — | 18 | 1 | no | — |
| 142 | #272 | — | 18 | 1 | no | — |
| 143 | #272 | — | 54 | 1 | no | — |
| 144 | #272 | — | 54 | 1 | no | — |
| 145 | #272 | — | 54 | 1 | no | — |
| 146 | #272 | — | 54 | 1 | no | — |
| 147 | #272 | — | 54 | 1 | no | — |
| 148 | #272 | — | 54 | 1 | no | — |
| 149 | #272 | — | 54 | 1 | no | — |
| 150 | #272 | — | 54 | 1 | no | — |
| 151 | #272 | — | 54 | 1 | no | — |
| 152 | #272 | — | 54 | 1 | no | — |
| 153 | #272 | — | 54 | 1 | no | — |
| 154 | #272 | — | 54 | 1 | no | — |
| 155 | #272 | — | 54 | 1 | no | — |
| 156 | #272 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 157 | #272 | (C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 158 | D3×C9 | — | 18 | 1 | no | — |
| 159 | D3×C9 | — | 18 | 1 | no | — |
| 160 | D3×C9 | — | 18 | 1 | no | — |
| 161 | D3×C9 | — | 18 | 1 | no | — |
| 162 | D3×C9 | — | 27 | 1 | no | — |
| 163 | D3×C9 | — | 27 | 1 | no | — |
| 164 | D3×C9 | — | 54 | 1 | no | — |
| 165 | C2×C3×D3 | — | 27 | 1 | no | — |
| 166 | C2×C3×D3 | — | 27 | 1 | no | — |
| 167 | C2×C3×D3 | — | 81 | 1 | no | — |
| 168 | C2×C3×D3 | — | 81 | 1 | no | — |
| 169 | D32 | — | 9 | 1 | no | — |
| 170 | D32 | — | 27 | 1 | no | — |
| 171 | D32 | — | 27 | 1 | no | — |
| 172 | D32 | — | 27 | 1 | no | — |
| 173 | D32 | — | 27 | 1 | no | — |
| 174 | D32 | — | 27 | 1 | no | — |
| 175 | C22×C9 | — | 81 | 1 | no | — |
| 176 | C33 | — | 3 | 1 | no | — |
| 177 | C33 | — | 3 | 1 | no | — |
| 178 | C33 | — | 3 | 1 | no | — |
| 179 | C33 | — | 3 | 1 | no | — |
| 180 | C33 | — | 3 | 1 | no | — |
| 181 | C33 | — | 6 | 1 | no | — |
| 182 | C33 | — | 6 | 1 | no | — |
| 183 | C33 | — | 6 | 1 | no | — |
| 184 | C33 | — | 6 | 1 | no | — |
| 185 | C33 | — | 12 | 1 | no | — |
| 186 | C33 | C3×D32 | 1 | 1 | yes | characteristic |
| 187 | 3−1+2 | — | 6 | 1 | no | — |
| 188 | 3−1+2 | — | 12 | 1 | no | — |
| 189 | 3−1+2 | — | 12 | 1 | no | — |
| 190 | 3−1+2 | — | 12 | 1 | no | — |
| 191 | H(3,𝔽3) | — | 2 | 1 | no | — |
| 192 | H(3,𝔽3) | — | 2 | 1 | no | — |
| 193 | H(3,𝔽3) | — | 2 | 1 | no | — |
| 194 | H(3,𝔽3) | — | 2 | 1 | no | — |
| 195 | H(3,𝔽3) | — | 2 | 1 | no | — |
| 196 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 197 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 198 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 199 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 200 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 201 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 202 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 203 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 204 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 205 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 206 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 207 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 208 | H(3,𝔽3) | — | 6 | 1 | no | — |
| 209 | H(3,𝔽3) | — | 12 | 1 | no | — |
| 210 | H(3,𝔽3) | — | 12 | 1 | no | — |
| 211 | H(3,𝔽3) | — | 12 | 1 | no | — |
| 212 | H(3,𝔽3) | C2×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 213 | H(3,𝔽3) | C2×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 214 | C3×C9 | — | 6 | 1 | no | — |
| 215 | C3×C9 | — | 6 | 1 | no | — |
| 216 | C3×C9 | — | 6 | 1 | no | — |
| 217 | C3×C9 | — | 6 | 1 | no | — |
| 218 | C3×C9 | — | 6 | 1 | no | — |
| 219 | C2×C32 | — | 9 | 1 | no | — |
| 220 | C2×C32 | — | 9 | 1 | no | — |
| 221 | C2×C32 | — | 27 | 1 | no | — |
| 222 | C2×C32 | — | 27 | 1 | no | — |
| 223 | C3⋊D3 | — | 9 | 1 | no | — |
| 224 | C3⋊D3 | — | 27 | 1 | no | — |
| 225 | C3⋊D3 | — | 27 | 1 | no | — |
| 226 | C3⋊D3 | — | 27 | 1 | no | — |
| 227 | C3⋊D3 | — | 27 | 1 | no | — |
| 228 | C3⋊D3 | — | 27 | 1 | no | — |
| 229 | C3⋊D3 | — | 54 | 1 | no | — |
| 230 | C3⋊D3 | — | 54 | 1 | no | — |
| 231 | C3⋊D3 | — | 54 | 1 | no | — |
| 232 | C3⋊D3 | — | 54 | 1 | no | — |
| 233 | C3×D3 | — | 3 | 1 | no | — |
| 234 | C3×D3 | — | 3 | 1 | no | — |
| 235 | C3×D3 | — | 9 | 1 | no | — |
| 236 | C3×D3 | — | 9 | 1 | no | — |
| 237 | C3×D3 | — | 9 | 1 | no | — |
| 238 | C3×D3 | — | 18 | 1 | no | — |
| 239 | C3×D3 | — | 27 | 1 | no | — |
| 240 | C3×D3 | — | 27 | 1 | no | — |
| 241 | C3×D3 | — | 54 | 1 | no | — |
| 242 | C3×D3 | — | 54 | 1 | no | — |
| 243 | C3×D3 | — | 54 | 1 | no | — |
| 244 | C3×D3 | — | 54 | 1 | no | — |
| 245 | C3×D3 | — | 54 | 1 | no | — |
| 246 | C3×D3 | — | 54 | 1 | no | — |
| 247 | C3×D3 | — | 54 | 1 | no | — |
| 248 | C3×D3 | — | 54 | 1 | no | — |
| 249 | C3×D3 | — | 54 | 1 | no | — |
| 250 | C3×D3 | — | 54 | 1 | no | — |
| 251 | C3×D3 | — | 54 | 1 | no | — |
| 252 | C3×D3 | — | 81 | 1 | no | — |
| 253 | C3×D3 | — | 81 | 1 | no | — |
| 254 | C3×D3 | — | 162 | 1 | no | — |
| 255 | C3×D3 | — | 162 | 1 | no | — |
| 256 | C3×D3 | — | 162 | 1 | no | — |
| 257 | C3×D3 | — | 162 | 1 | no | — |
| 258 | C3×D3 | — | 162 | 1 | no | — |
| 259 | C2×C9 | — | 54 | 1 | no | — |
| 260 | C2×C9 | — | 54 | 1 | no | — |
| 261 | C2×C9 | — | 54 | 1 | no | — |
| 262 | C2×C9 | — | 54 | 1 | no | — |
| 263 | C2×C9 | — | 81 | 1 | no | — |
| 264 | C22×C3 | — | 81 | 1 | no | — |
| 265 | C2×D3 | — | 27 | 1 | no | — |
| 266 | C2×D3 | — | 27 | 1 | no | — |
| 267 | C2×D3 | — | 81 | 1 | no | — |
| 268 | C2×D3 | — | 81 | 1 | no | — |
| 269 | C32 | — | 2 | 1 | no | — |
| 270 | C32 | — | 3 | 1 | no | — |
| 271 | C32 | — | 3 | 1 | no | — |
| 272 | C32 | — | 6 | 1 | no | — |
| 273 | C32 | — | 6 | 1 | no | — |
| 274 | C32 | — | 6 | 1 | no | — |
| 275 | C32 | — | 6 | 1 | no | — |
| 276 | C32 | — | 6 | 1 | no | — |
| 277 | C32 | — | 36 | 1 | no | — |
| 278 | C32 | — | 36 | 1 | no | — |
| 279 | C32 | — | 54 | 1 | no | — |
| 280 | C32 | — | 54 | 1 | no | — |
| 281 | C32 | — | 54 | 1 | no | — |
| 282 | C32 | — | 54 | 1 | no | — |
| 283 | C32 | — | 108 | 1 | no | — |
| 284 | C32 | — | 108 | 1 | no | — |
| 285 | C32 | — | 108 | 1 | no | — |
| 286 | C32 | — | 108 | 1 | no | — |
| 287 | C32 | — | 108 | 1 | no | — |
| 288 | C32 | — | 108 | 1 | no | — |
| 289 | C32 | D3×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 290 | C32 | D3×(C3⋊D3)⋊C3 | 1 | 1 | yes | characteristic |
| 291 | C9 | — | 18 | 1 | no | — |
| 292 | C9 | — | 18 | 1 | no | — |
| 293 | C9 | — | 36 | 1 | no | — |
| 294 | C2×C3 | — | 9 | 1 | no | — |
| 295 | C2×C3 | — | 9 | 1 | no | — |
| 296 | C2×C3 | — | 27 | 1 | no | — |
| 297 | C2×C3 | — | 54 | 1 | no | — |
| 298 | C2×C3 | — | 81 | 1 | no | — |
| 299 | C2×C3 | — | 81 | 1 | no | — |
| 300 | C2×C3 | — | 162 | 1 | no | — |
| 301 | D3 | — | 3 | 1 | no | — |
| 302 | D3 | — | 3 | 1 | no | — |
| 303 | D3 | — | 9 | 1 | no | — |
| 304 | D3 | — | 9 | 1 | no | — |
| 305 | D3 | — | 27 | 1 | no | — |
| 306 | D3 | — | 27 | 1 | no | — |
| 307 | D3 | — | 54 | 1 | no | — |
| 308 | D3 | — | 81 | 1 | no | — |
| 309 | D3 | — | 81 | 1 | no | — |
| 310 | D3 | — | 162 | 1 | no | — |
| 311 | D3 | — | 162 | 1 | no | — |
| 312 | D3 | — | 162 | 1 | no | — |
| 313 | D3 | — | 162 | 1 | no | — |
| 314 | D3 | — | 162 | 1 | no | — |
| 315 | C22 | — | 81 | 1 | no | Sylow |
| 316 | C3 | — | 3 | 1 | no | — |
| 317 | C3 | — | 6 | 1 | no | — |
| 318 | C3 | — | 9 | 1 | no | — |
| 319 | C3 | — | 12 | 1 | no | — |
| 320 | C3 | — | 18 | 1 | no | — |
| 321 | C3 | — | 18 | 1 | no | — |
| 322 | C3 | — | 36 | 1 | no | — |
| 323 | C3 | — | 36 | 1 | no | — |
| 324 | C3 | — | 36 | 1 | no | — |
| 325 | C3 | — | 36 | 1 | no | — |
| 326 | C3 | — | 108 | 1 | no | — |
| 327 | C3 | #11757702 | 1 | 1 | yes | characteristic |
| 328 | C2 | — | 9 | 1 | no | — |
| 329 | C2 | — | 9 | 1 | no | — |
| 330 | C2 | — | 81 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C3 | #11757702 |
| commutator, cosocle, hypocenter | H(5,𝔽3) | C22×C3 |
| Frattini | C33 | C3×D32 |
| Fitting | #10624760 | C22 |
| LCS | G ▷ H(5,𝔽3) | |
| UCS | C1 ◁ C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #13091149 | C2 | semidirect |
| 2. #13091149 | C2 | semidirect |
| 3. #13091148 | C2 | semidirect |
| 4. #11758087 | C3 | semidirect |
| 7. #10624760 | C22 | semidirect |
| 18. C3×#4176 | D3 | semidirect |
| 19. C3×#4176 | D3 | semidirect |
| 32. C3×#731 | C2×D3 | semidirect |
| 33. C3×#731 | C2×D3 | semidirect |
| 93. C32×C9 | D32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #13091149 | 2 |
| #13091148 | 2 |
| #11758087 | 3 |
| D3×#4176 | 3 |