Finite Group DatabasebetaFinite Group DatabasebetaOrder 7776.
567 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 567 conjugacy classes of subgroups — too many to draw. Only its 17 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 | #72210346 | — | 3 | 1 | no | maximal |
| 2 | #72210299 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #72210295 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #72210292 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #14519481 | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | ((((C3⋊D3)⋊C3)⋊D3)⋊C3)⋊C2 | — | 4 | 1 | no | maximal |
| 7 | #13049818 | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | #13049813 | — | 3 | 1 | no | — |
| 9 | #13049804 | — | 3 | 1 | no | — |
| 10 | #13049804 | — | 3 | 1 | no | — |
| 11 | #13049803 | — | 3 | 1 | no | — |
| 12 | #13049803 | — | 3 | 1 | no | — |
| 13 | #13049788 | — | 3 | 1 | no | — |
| 14 | #11757749 | — | 4 | 1 | no | — |
| 15 | #11757731 | — | 4 | 1 | no | — |
| 16 | #11757730 | — | 4 | 2 | no | — |
| 17 | (((C3⋊D3)⋊C3)⋊D3)⋊C3 | — | 4 | 2 | no | — |
| 18 | #11757702 | — | 4 | 1 | no | — |
| 19 | C2×AGL(2,𝔽3) | — | 9 | 2 | no | maximal |
| 20 | #10620215 | — | 6 | 1 | no | — |
| 21 | #10620211 | — | 3 | 1 | no | — |
| 22 | #10620211 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | #10620206 | — | 3 | 1 | no | — |
| 24 | #10620206 | — | 3 | 1 | no | — |
| 25 | #10620206 | — | 6 | 1 | no | — |
| 26 | #10620198 | — | 3 | 1 | no | — |
| 27 | #10620193 | — | 3 | 1 | no | — |
| 28 | #10620192 | — | 3 | 1 | no | — |
| 29 | D3×((C3⋊D3)⋊C3)⋊C2 | — | 12 | 2 | no | — |
| 30 | #92338 | — | 4 | 1 | no | — |
| 31 | #92301 | — | 4 | 1 | no | — |
| 32 | #92300 | — | 4 | 2 | no | — |
| 33 | #92258 | — | 4 | 1 | no | — |
| 34 | (C3⋊D3)⋊H(3,𝔽3) | — | 4 | 2 | no | — |
| 35 | C2×PSU(3,𝔽2)⋊C3 | — | 9 | 2 | no | — |
| 36 | AGL(2,𝔽3) | — | 9 | 2 | no | — |
| 37 | AGL(2,𝔽3) | — | 9 | 2 | no | — |
| 38 | AGL(2,𝔽3) | — | 18 | 1 | no | — |
| 39 | C3⋊D32 | — | 6 | 1 | no | — |
| 40 | C3⋊D32 | — | 6 | 1 | no | — |
| 41 | C3⋊D32 | S4 | 1 | 1 | yes | characteristic |
| 42 | C3×#1113 | — | 12 | 2 | no | — |
| 43 | #66472 | — | 3 | 1 | no | — |
| 44 | #66472 | — | 3 | 1 | no | — |
| 45 | #66432 | — | 12 | 2 | no | — |
| 46 | #66432 | — | 24 | 1 | no | — |
| 47 | D3×#272 | — | 12 | 2 | no | — |
| 48 | ((C3⋊D3)⋊C3)⋊D3 | — | 4 | 2 | no | — |
| 49 | ((C3⋊D3)⋊C3)⋊D3 | — | 12 | 2 | no | — |
| 50 | C3×((C3⋊D3)⋊C3)⋊C2 | — | 12 | 2 | no | — |
| 51 | C3×((C3⋊D3)⋊C3)⋊C2 | — | 24 | 2 | no | — |
| 52 | D3×(C3⋊D3)⋊C3 | — | 12 | 2 | no | — |
| 53 | D3×(C3⋊D3)⋊C3 | — | 12 | 2 | no | — |
| 54 | C2×AΓL(1,𝔽9) | — | 27 | 2 | no | — |
| 55 | C32⋊H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 56 | D33 | — | 36 | 1 | no | — |
| 57 | D33 | — | 36 | 2 | no | — |
| 58 | PSU(3,𝔽2)⋊C3 | — | 9 | 2 | no | — |
| 59 | PSU(3,𝔽2)⋊C3 | — | 18 | 1 | no | — |
| 60 | C2×((C3⋊D3)⋊C3)⋊C2 | — | 36 | 2 | no | — |
| 61 | C3⋊(C3⋊(C3⋊D3)) | C2×S4 | 1 | 1 | yes | characteristic |
| 62 | C32×C3⋊D3 | — | 12 | 1 | no | — |
| 63 | C32×C3⋊D3 | — | 12 | 1 | no | — |
| 64 | C32×C3⋊D3 | GL(2,𝔽3) | 1 | 2 | yes | complemented |
| 65 | C3⋊#272 | — | 4 | 2 | no | — |
| 66 | C3×#272 | — | 12 | 2 | no | — |
| 67 | C3×#272 | — | 24 | 2 | no | — |
| 68 | C3×#272 | — | 24 | 1 | no | — |
| 69 | C3×#272 | — | 48 | 1 | no | — |
| 70 | C3⋊((C3⋊D3)⋊C3) | — | 4 | 2 | no | — |
| 71 | C3⋊((C3⋊D3)⋊C3) | — | 8 | 1 | no | — |
| 72 | C3⋊((C3⋊D3)⋊C3) | — | 12 | 2 | no | — |
| 73 | C3⋊((C3⋊D3)⋊C3) | — | 24 | 1 | no | — |
| 74 | D3×H(3,𝔽3) | — | 12 | 2 | no | — |
| 75 | C3×(C3⋊D3)⋊C3 | — | 4 | 2 | no | — |
| 76 | C3×(C3⋊D3)⋊C3 | — | 8 | 2 | no | — |
| 77 | C3×(C3⋊D3)⋊C3 | — | 12 | 2 | no | — |
| 78 | C3×(C3⋊D3)⋊C3 | — | 24 | 2 | no | — |
| 79 | C2×PSU(3,𝔽2) | — | 9 | 2 | no | — |
| 80 | C2×GO+(4,𝔽2) | — | 27 | 2 | no | — |
| 81 | C2×AGL(1,𝔽9) | — | 27 | 2 | no | — |
| 82 | AΓL(1,𝔽9) | — | 27 | 2 | no | — |
| 83 | AΓL(1,𝔽9) | — | 27 | 2 | no | — |
| 84 | AΓL(1,𝔽9) | — | 27 | 2 | no | — |
| 85 | AΓL(1,𝔽9) | — | 27 | 2 | no | — |
| 86 | AΓL(1,𝔽9) | — | 54 | 1 | no | — |
| 87 | AΓL(1,𝔽9) | — | 54 | 1 | no | — |
| 88 | #1113 | — | 12 | 2 | no | — |
| 89 | #1113 | — | 36 | 1 | no | — |
| 90 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 91 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 92 | D3×C3⋊D3 | — | 36 | 2 | no | — |
| 93 | D3×C3⋊D3 | — | 36 | 2 | no | — |
| 94 | D3×C3⋊D3 | — | 36 | 1 | no | — |
| 95 | D3×C3⋊D3 | — | 72 | 1 | no | — |
| 96 | D3×C3⋊D3 | — | 72 | 1 | no | — |
| 97 | C3×D32 | — | 36 | 1 | no | — |
| 98 | C3×D32 | — | 36 | 2 | no | — |
| 99 | C3×D32 | — | 36 | 2 | no | — |
| 100 | C3×D32 | — | 36 | 2 | no | — |
| 101 | C3×D32 | — | 36 | 2 | no | — |
| 102 | C3×D32 | — | 72 | 2 | no | — |
| 103 | C2×#272 | — | 36 | 2 | no | — |
| 104 | C2×(C3⋊D3)⋊C3 | — | 36 | 2 | no | — |
| 105 | C2×(C3⋊D3)⋊C3 | — | 36 | 2 | no | — |
| 106 | ((C3⋊D3)⋊C3)⋊C2 | — | 12 | 2 | no | — |
| 107 | ((C3⋊D3)⋊C3)⋊C2 | — | 24 | 2 | no | — |
| 108 | ((C3⋊D3)⋊C3)⋊C2 | — | 36 | 2 | no | — |
| 109 | ((C3⋊D3)⋊C3)⋊C2 | — | 36 | 2 | no | — |
| 110 | ((C3⋊D3)⋊C3)⋊C2 | — | 36 | 2 | no | — |
| 111 | ((C3⋊D3)⋊C3)⋊C2 | — | 72 | 2 | no | — |
| 112 | ((C3⋊D3)⋊C3)⋊C2 | — | 72 | 1 | no | — |
| 113 | C2×GL(2,𝔽3) | — | 81 | 1 | no | — |
| 114 | C34 | C2×GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 115 | C3×H(3,𝔽3) | — | 4 | 2 | no | — |
| 116 | C3×H(3,𝔽3) | — | 8 | 2 | no | — |
| 117 | C3×H(3,𝔽3) | — | 8 | 1 | no | — |
| 118 | C3×H(3,𝔽3) | — | 16 | 1 | no | — |
| 119 | C2×D32 | — | 54 | 2 | no | — |
| 120 | C2×D32 | — | 108 | 1 | no | — |
| 121 | C2×D32 | — | 108 | 1 | no | — |
| 122 | C2×D32 | — | 108 | 2 | no | — |
| 123 | C2×#157 | — | 27 | 2 | no | — |
| 124 | PSU(3,𝔽2) | — | 9 | 2 | no | — |
| 125 | PSU(3,𝔽2) | — | 18 | 1 | no | — |
| 126 | PSU(3,𝔽2) | — | 27 | 2 | no | — |
| 127 | PSU(3,𝔽2) | — | 54 | 1 | no | — |
| 128 | GO+(4,𝔽2) | — | 27 | 2 | no | — |
| 129 | GO+(4,𝔽2) | — | 27 | 2 | no | — |
| 130 | GO+(4,𝔽2) | — | 54 | 2 | no | — |
| 131 | GO+(4,𝔽2) | — | 54 | 1 | no | — |
| 132 | GO+(4,𝔽2) | — | 54 | 1 | no | — |
| 133 | GO+(4,𝔽2) | — | 108 | 1 | no | — |
| 134 | AGL(1,𝔽9) | — | 27 | 2 | no | — |
| 135 | AGL(1,𝔽9) | — | 27 | 2 | no | — |
| 136 | AGL(1,𝔽9) | — | 54 | 1 | no | — |
| 137 | AGL(1,𝔽9) | — | 54 | 1 | no | — |
| 138 | AGL(1,𝔽9) | — | 108 | 1 | no | — |
| 139 | C3⋊(C3⋊D3) | — | 4 | 1 | no | — |
| 140 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 141 | C3⋊(C3⋊D3) | — | 24 | 1 | no | — |
| 142 | C3⋊(C3⋊D3) | — | 72 | 1 | no | — |
| 143 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 144 | C3×C3⋊D3 | — | 12 | 2 | no | — |
| 145 | C3×C3⋊D3 | — | 12 | 2 | no | — |
| 146 | C3×C3⋊D3 | — | 12 | 2 | no | — |
| 147 | C3×C3⋊D3 | — | 24 | 2 | no | — |
| 148 | C3×C3⋊D3 | — | 24 | 1 | no | — |
| 149 | C3×C3⋊D3 | — | 24 | 1 | no | — |
| 150 | C3×C3⋊D3 | — | 36 | 1 | no | — |
| 151 | C32×D3 | — | 12 | 2 | no | — |
| 152 | C32×D3 | — | 12 | 2 | no | — |
| 153 | C32×D3 | — | 24 | 2 | no | — |
| 154 | C32×D3 | — | 36 | 2 | no | — |
| 155 | C32×D3 | — | 36 | 1 | no | — |
| 156 | C32×D3 | — | 36 | 2 | no | — |
| 157 | C32×D3 | — | 72 | 1 | no | — |
| 158 | C32×D3 | — | 72 | 1 | no | — |
| 159 | C32×D3 | — | 72 | 2 | no | — |
| 160 | C32×D3 | — | 144 | 1 | no | — |
| 161 | C2×H(3,𝔽3) | — | 36 | 2 | no | — |
| 162 | #272 | — | 12 | 2 | no | — |
| 163 | #272 | — | 24 | 2 | no | — |
| 164 | #272 | — | 24 | 1 | no | — |
| 165 | #272 | — | 36 | 2 | no | — |
| 166 | #272 | — | 48 | 1 | no | — |
| 167 | #272 | — | 72 | 2 | no | — |
| 168 | (C3⋊D3)⋊C3 | — | 12 | 2 | no | — |
| 169 | (C3⋊D3)⋊C3 | — | 12 | 2 | no | — |
| 170 | (C3⋊D3)⋊C3 | — | 24 | 2 | no | — |
| 171 | (C3⋊D3)⋊C3 | — | 24 | 2 | no | — |
| 172 | (C3⋊D3)⋊C3 | — | 24 | 2 | no | — |
| 173 | (C3⋊D3)⋊C3 | — | 36 | 2 | no | — |
| 174 | (C3⋊D3)⋊C3 | — | 36 | 2 | no | — |
| 175 | (C3⋊D3)⋊C3 | — | 48 | 2 | no | — |
| 176 | (C3⋊D3)⋊C3 | — | 72 | 1 | no | — |
| 177 | (C3⋊D3)⋊C3 | — | 72 | 2 | no | — |
| 178 | (C3⋊D3)⋊C3 | — | 72 | 1 | no | — |
| 179 | (C3⋊D3)⋊C3 | — | 144 | 1 | no | — |
| 180 | C2×SL(2,𝔽3) | — | 81 | 1 | no | — |
| 181 | GL(2,𝔽3) | — | 81 | 1 | no | — |
| 182 | GL(2,𝔽3) | — | 81 | 1 | no | — |
| 183 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 184 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 185 | C2×C3⋊D3 | — | 108 | 1 | no | — |
| 186 | C2×C3⋊D3 | — | 108 | 1 | no | — |
| 187 | C2×C3×D3 | — | 108 | 2 | no | — |
| 188 | C2×C3×D3 | — | 108 | 2 | no | — |
| 189 | C2×C3×D3 | — | 108 | 2 | no | — |
| 190 | C2×C3×D3 | — | 108 | 2 | no | — |
| 191 | C2×C3×D3 | — | 108 | 2 | no | — |
| 192 | C2×C3×D3 | — | 216 | 2 | no | — |
| 193 | D32 | — | 36 | 1 | no | — |
| 194 | D32 | — | 36 | 2 | no | — |
| 195 | D32 | — | 36 | 2 | no | — |
| 196 | D32 | — | 36 | 2 | no | — |
| 197 | D32 | — | 36 | 2 | no | — |
| 198 | D32 | — | 54 | 2 | no | — |
| 199 | D32 | — | 54 | 2 | no | — |
| 200 | D32 | — | 72 | 2 | no | — |
| 201 | D32 | — | 108 | 1 | no | — |
| 202 | D32 | — | 108 | 1 | no | — |
| 203 | D32 | — | 108 | 2 | no | — |
| 204 | D32 | — | 108 | 2 | no | — |
| 205 | D32 | — | 108 | 2 | no | — |
| 206 | D32 | — | 108 | 1 | no | — |
| 207 | D32 | — | 108 | 1 | no | — |
| 208 | D32 | — | 108 | 1 | no | — |
| 209 | D32 | — | 108 | 1 | no | — |
| 210 | D32 | — | 108 | 1 | no | — |
| 211 | D32 | — | 216 | 2 | no | — |
| 212 | D32 | — | 216 | 2 | no | — |
| 213 | D32 | — | 216 | 1 | no | — |
| 214 | #157 | — | 27 | 2 | no | — |
| 215 | #157 | — | 27 | 2 | no | — |
| 216 | #157 | — | 54 | 1 | no | — |
| 217 | #157 | — | 54 | 1 | no | — |
| 218 | #157 | — | 108 | 1 | no | — |
| 219 | #157 | — | 108 | 1 | no | — |
| 220 | #157 | — | 108 | 1 | no | — |
| 221 | C2×SD16 | — | 243 | 1 | no | Sylow |
| 222 | C33 | — | 4 | 2 | no | — |
| 223 | C33 | — | 4 | 1 | no | — |
| 224 | C33 | — | 8 | 1 | no | — |
| 225 | C33 | — | 8 | 2 | no | — |
| 226 | C33 | — | 16 | 1 | no | — |
| 227 | C33 | — | 24 | 1 | no | — |
| 228 | H(3,𝔽3) | — | 12 | 2 | no | — |
| 229 | H(3,𝔽3) | — | 24 | 1 | no | — |
| 230 | H(3,𝔽3) | — | 24 | 2 | no | — |
| 231 | H(3,𝔽3) | — | 24 | 2 | no | — |
| 232 | H(3,𝔽3) | — | 48 | 1 | no | — |
| 233 | H(3,𝔽3) | — | 48 | 2 | no | — |
| 234 | H(3,𝔽3) | — | 48 | 1 | no | — |
| 235 | H(3,𝔽3) | — | 96 | 1 | no | — |
| 236 | C22×D3 | — | 324 | 1 | no | — |
| 237 | C22×D3 | — | 324 | 2 | no | — |
| 238 | SL(2,𝔽3) | — | 81 | 1 | no | — |
| 239 | C2×C32 | — | 9 | 2 | no | — |
| 240 | C2×C32 | — | 36 | 2 | no | — |
| 241 | C2×C32 | — | 72 | 2 | no | — |
| 242 | C2×C32 | — | 108 | 1 | no | — |
| 243 | C2×C32 | — | 108 | 1 | no | — |
| 244 | C3⋊D3 | — | 9 | 2 | no | — |
| 245 | C3⋊D3 | — | 12 | 1 | no | — |
| 246 | C3⋊D3 | — | 12 | 1 | no | — |
| 247 | C3⋊D3 | — | 12 | 2 | no | — |
| 248 | C3⋊D3 | — | 18 | 1 | no | — |
| 249 | C3⋊D3 | — | 24 | 2 | no | — |
| 250 | C3⋊D3 | — | 36 | 2 | no | — |
| 251 | C3⋊D3 | — | 36 | 1 | no | — |
| 252 | C3⋊D3 | — | 54 | 1 | no | — |
| 253 | C3⋊D3 | — | 72 | 1 | no | — |
| 254 | C3⋊D3 | — | 72 | 2 | no | — |
| 255 | C3⋊D3 | — | 108 | 1 | no | — |
| 256 | C3⋊D3 | — | 108 | 1 | no | — |
| 257 | C3⋊D3 | — | 108 | 1 | no | — |
| 258 | C3⋊D3 | — | 144 | 1 | no | — |
| 259 | C3⋊D3 | — | 216 | 2 | no | — |
| 260 | C3⋊D3 | AGL(2,𝔽3) | 1 | 2 | yes | complemented |
| 261 | C3×D3 | — | 12 | 2 | no | — |
| 262 | C3×D3 | — | 36 | 2 | no | — |
| 263 | C3×D3 | — | 36 | 2 | no | — |
| 264 | C3×D3 | — | 36 | 2 | no | — |
| 265 | C3×D3 | — | 36 | 2 | no | — |
| 266 | C3×D3 | — | 36 | 2 | no | — |
| 267 | C3×D3 | — | 36 | 2 | no | — |
| 268 | C3×D3 | — | 36 | 2 | no | — |
| 269 | C3×D3 | — | 72 | 2 | no | — |
| 270 | C3×D3 | — | 72 | 2 | no | — |
| 271 | C3×D3 | — | 72 | 1 | no | — |
| 272 | C3×D3 | — | 72 | 2 | no | — |
| 273 | C3×D3 | — | 72 | 2 | no | — |
| 274 | C3×D3 | — | 72 | 2 | no | — |
| 275 | C3×D3 | — | 72 | 2 | no | — |
| 276 | C3×D3 | — | 72 | 2 | no | — |
| 277 | C3×D3 | — | 72 | 1 | no | — |
| 278 | C3×D3 | — | 72 | 1 | no | — |
| 279 | C3×D3 | — | 72 | 1 | no | — |
| 280 | C3×D3 | — | 72 | 1 | no | — |
| 281 | C3×D3 | — | 108 | 2 | no | — |
| 282 | C3×D3 | — | 108 | 2 | no | — |
| 283 | C3×D3 | — | 144 | 2 | no | — |
| 284 | C3×D3 | — | 144 | 2 | no | — |
| 285 | C3×D3 | — | 144 | 1 | no | — |
| 286 | C3×D3 | — | 144 | 1 | no | — |
| 287 | C3×D3 | — | 216 | 1 | no | — |
| 288 | C3×D3 | — | 216 | 2 | no | — |
| 289 | C2×Q8 | — | 81 | 1 | no | — |
| 290 | C2×D4 | — | 243 | 1 | no | — |
| 291 | SD16 | — | 243 | 1 | no | — |
| 292 | SD16 | — | 243 | 1 | no | — |
| 293 | SD16 | — | 243 | 1 | no | — |
| 294 | SD16 | — | 243 | 1 | no | — |
| 295 | C2×C8 | — | 243 | 1 | no | — |
| 296 | C22×C3 | — | 324 | 2 | no | — |
| 297 | C22×C3 | — | 324 | 1 | no | — |
| 298 | C2×D3 | — | 324 | 2 | no | — |
| 299 | C2×D3 | — | 324 | 1 | no | — |
| 300 | C2×D3 | — | 324 | 2 | no | — |
| 301 | C2×D3 | — | 324 | 2 | no | — |
| 302 | C2×D3 | — | 324 | 1 | no | — |
| 303 | C2×D3 | — | 108 | 2 | no | — |
| 304 | C2×D3 | — | 108 | 2 | no | — |
| 305 | C2×D3 | — | 108 | 2 | no | — |
| 306 | C2×D3 | — | 108 | 2 | no | — |
| 307 | C2×D3 | — | 108 | 2 | no | — |
| 308 | C2×D3 | — | 648 | 1 | no | — |
| 309 | C2×D3 | — | 648 | 1 | no | — |
| 310 | C2×D3 | — | 216 | 2 | no | — |
| 311 | C32 | — | 2 | 1 | no | — |
| 312 | C32 | — | 4 | 1 | no | — |
| 313 | C32 | — | 6 | 1 | no | — |
| 314 | C32 | — | 8 | 2 | no | — |
| 315 | C32 | — | 12 | 1 | no | — |
| 316 | C32 | — | 12 | 2 | no | — |
| 317 | C32 | — | 12 | 1 | no | — |
| 318 | C32 | — | 12 | 1 | no | — |
| 319 | C32 | — | 16 | 1 | no | — |
| 320 | C32 | — | 24 | 2 | no | — |
| 321 | C32 | — | 24 | 2 | no | — |
| 322 | C32 | — | 24 | 2 | no | — |
| 323 | C32 | — | 24 | 1 | no | — |
| 324 | C32 | — | 48 | 2 | no | — |
| 325 | C32 | — | 48 | 2 | no | — |
| 326 | C32 | — | 48 | 1 | no | — |
| 327 | C32 | — | 48 | 1 | no | — |
| 328 | C32 | C2×AGL(2,𝔽3) | 1 | 2 | yes | complemented |
| 329 | C23 | — | 486 | 1 | no | — |
| 330 | Q8 | — | 81 | 1 | no | — |
| 331 | Q8 | — | 243 | 1 | no | — |
| 332 | D4 | — | 486 | 1 | no | — |
| 333 | D4 | — | 243 | 1 | no | — |
| 334 | D4 | — | 243 | 1 | no | — |
| 335 | C2×C4 | — | 243 | 1 | no | — |
| 336 | C8 | — | 243 | 1 | no | — |
| 337 | C8 | — | 243 | 1 | no | — |
| 338 | C2×C3 | — | 36 | 2 | no | — |
| 339 | C2×C3 | — | 324 | 1 | no | — |
| 340 | C2×C3 | — | 108 | 2 | no | — |
| 341 | C2×C3 | — | 108 | 2 | no | — |
| 342 | C2×C3 | — | 108 | 2 | no | — |
| 343 | C2×C3 | — | 216 | 2 | no | — |
| 344 | C2×C3 | — | 216 | 1 | no | — |
| 345 | C2×C3 | — | 216 | 1 | no | — |
| 346 | D3 | — | 12 | 2 | no | — |
| 347 | D3 | — | 36 | 2 | no | — |
| 348 | D3 | — | 36 | 2 | no | — |
| 349 | D3 | — | 36 | 1 | no | — |
| 350 | D3 | — | 324 | 1 | no | — |
| 351 | D3 | — | 72 | 1 | no | — |
| 352 | D3 | — | 72 | 1 | no | — |
| 353 | D3 | — | 72 | 2 | no | — |
| 354 | D3 | — | 108 | 2 | no | — |
| 355 | D3 | — | 108 | 2 | no | — |
| 356 | D3 | — | 648 | 1 | no | — |
| 357 | D3 | — | 144 | 1 | no | — |
| 358 | D3 | — | 216 | 1 | no | — |
| 359 | D3 | — | 216 | 2 | no | — |
| 360 | C22 | — | 324 | 2 | no | — |
| 361 | C22 | — | 81 | 1 | no | — |
| 362 | C22 | — | 486 | 1 | no | — |
| 363 | C22 | — | 486 | 1 | no | — |
| 364 | C4 | — | 243 | 1 | no | — |
| 365 | C4 | — | 243 | 1 | no | — |
| 366 | C3 | — | 4 | 2 | no | — |
| 367 | C3 | — | 8 | 1 | no | — |
| 368 | C3 | — | 24 | 1 | no | — |
| 369 | C3 | — | 36 | 1 | no | — |
| 370 | C3 | — | 72 | 2 | no | — |
| 371 | C3 | — | 144 | 1 | no | — |
| 372 | C2 | — | 9 | 2 | no | — |
| 373 | C2 | — | 81 | 1 | no | — |
| 374 | C2 | — | 108 | 1 | no | — |
| 375 | C2 | — | 108 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | #14519481 | C22 |
| Fitting, socle | C34 | C2×GL(2,𝔽3) |
| LCS | G ▷ #14519481 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #72210299 | C2 | semidirect |
| 3. #72210295 | C2 | semidirect |
| 4. #72210292 | C2 | semidirect |
| 5. #14519481 | C22 | semidirect |
| 7. #13049818 | D3 | semidirect |
| 22. #10620211 | C2×D3 | semidirect |
| 64. C32×C3⋊D3 | GL(2,𝔽3) | semidirect |
| 114. C34 | C2×GL(2,𝔽3) | semidirect |
| 260. C3⋊D3 | AGL(2,𝔽3) | semidirect |
| 328. C32 | C2×AGL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72210346 | 3 |
| #72210299 | 2 |
| #72210295 | 2 |
| #72210292 | 2 |
| ((((C3⋊D3)⋊C3)⋊D3)⋊C3)⋊C2 | 4 |
| C2×AGL(2,𝔽3) | 9 |