Finite Group DatabasebetaFinite Group DatabasebetaOrder 5184.
648 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 648 conjugacy classes of subgroups — too many to draw. Only its 67 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 | #72220561 | C2 | 1 | 4 | yes | maximal |
| 2 | #72220550 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#1572 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#13049818 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C2×#10620211 | — | 2 | 2 | no | — |
| 6 | C2×#10620198 | C22 | 1 | 2 | yes | — |
| 7 | C2×#10620198 | C22 | 1 | 2 | yes | — |
| 8 | C2×#10620198 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×C3⋊D3×#157 | C22 | 1 | 2 | yes | — |
| 10 | #13049818 | — | 2 | 2 | no | — |
| 11 | #13049818 | C4 | 1 | 4 | yes | complemented |
| 12 | #1572 | — | 2 | 2 | no | — |
| 13 | #13049467 | — | 2 | 2 | no | — |
| 14 | C2×C3⋊D32 | C23 | 1 | 1 | yes | characteristic |
| 15 | C2×#66472 | — | 2 | 2 | no | — |
| 16 | C2×#66472 | — | 2 | 2 | no | — |
| 17 | C2×#66472 | D4 | 1 | 2 | yes | — |
| 18 | C2×C3⋊(C3⋊#157) | Q8 | 1 | 2 | yes | complemented |
| 19 | C2×C32×#157 | Q8 | 1 | 2 | yes | complemented |
| 20 | #10620211 | — | 2 | 8 | no | — |
| 21 | #10620198 | — | 2 | 2 | no | — |
| 22 | #10620198 | — | 2 | 4 | no | — |
| 23 | #10620198 | C2×C4 | 1 | 4 | yes | — |
| 24 | #10620198 | C2×C4 | 1 | 2 | yes | — |
| 25 | #10620198 | D4 | 1 | 2 | yes | — |
| 26 | C3⋊D3×#157 | — | 2 | 2 | no | — |
| 27 | C3⋊D3×#157 | — | 2 | 2 | no | — |
| 28 | #10593555 | — | 9 | 2 | no | maximal |
| 29 | C2×D3×#157 | — | 12 | 2 | no | — |
| 30 | C2×C3⋊(C3⋊(C3⋊D3)) | C2×D4 | 1 | 1 | yes | characteristic |
| 31 | C2×C32×C3⋊D3 | C2×Q8 | 1 | 2 | yes | — |
| 32 | C3⋊D32 | C22×C4 | 1 | 1 | yes | characteristic |
| 33 | C3⋊D32 | C2×D4 | 1 | 1 | yes | characteristic |
| 34 | C3⋊D32 | #41 | 1 | 2 | yes | — |
| 35 | #66472 | — | 2 | 4 | no | — |
| 36 | #66472 | — | 4 | 2 | no | — |
| 37 | #66472 | C22⋊C4 | 1 | 4 | yes | — |
| 38 | C3⋊(C3⋊#157) | — | 2 | 2 | no | — |
| 39 | C32×#157 | — | 2 | 2 | no | — |
| 40 | C22×PSU(3,𝔽2) | — | 9 | 2 | no | — |
| 41 | C2×C4×#157 | — | 9 | 2 | no | — |
| 42 | C2×#3777 | — | 9 | 2 | no | — |
| 43 | #63781 | — | 9 | 4 | no | — |
| 44 | #63747 | — | 9 | 4 | no | — |
| 45 | C2×D3×C3⋊D3 | — | 12 | 2 | no | — |
| 46 | C2×C3⋊#157 | — | 12 | 2 | no | — |
| 47 | C2×C3×#157 | — | 4 | 2 | no | — |
| 48 | D3×#157 | — | 12 | 4 | no | — |
| 49 | D3×#157 | — | 12 | 4 | no | — |
| 50 | C2×C34 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 51 | C3⋊(C3⋊(C3⋊D3)) | C2×C22⋊C4 | 1 | 1 | yes | characteristic |
| 52 | C3⋊(C3⋊(C3⋊D3)) | #124 | 1 | 1 | yes | characteristic |
| 53 | C32×C3⋊D3 | C4×Q8 | 1 | 2 | yes | — |
| 54 | C32×C3⋊D3 | C22⋊Q8 | 1 | 2 | yes | — |
| 55 | C22×#157 | — | 9 | 2 | no | — |
| 56 | C22×#157 | — | 9 | 4 | no | — |
| 57 | C22×#157 | — | 9 | 2 | no | — |
| 58 | C22×#157 | — | 9 | 4 | no | — |
| 59 | C2×PSU(3,𝔽2) | — | 9 | 8 | no | — |
| 60 | C2×PSU(3,𝔽2) | — | 18 | 4 | no | — |
| 61 | C2×PSU(3,𝔽2) | — | 18 | 4 | no | — |
| 62 | C2×PSU(3,𝔽2) | — | 36 | 2 | no | — |
| 63 | C2×C4×C3⋊D3 | — | 9 | 2 | no | — |
| 64 | C4×#157 | — | 18 | 4 | no | — |
| 65 | #3777 | — | 18 | 4 | no | — |
| 66 | #3777 | — | 36 | 2 | no | — |
| 67 | C2×C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 68 | C2×C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 69 | C2×C3×C3⋊D3 | — | 4 | 2 | no | — |
| 70 | C2×C32×D3 | — | 12 | 2 | no | — |
| 71 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 72 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 73 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 74 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 75 | C3⋊#157 | — | 12 | 4 | no | — |
| 76 | C3×#157 | — | 4 | 4 | no | — |
| 77 | C34 | #391 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 78 | C22×C3⋊D3 | — | 9 | 2 | no | — |
| 79 | C2×D32 | — | 72 | 2 | no | — |
| 80 | C2×#157 | — | 9 | 4 | no | — |
| 81 | C2×#157 | — | 9 | 4 | no | — |
| 82 | C2×#157 | — | 9 | 4 | no | — |
| 83 | C2×#157 | — | 9 | 8 | no | — |
| 84 | C2×#157 | — | 9 | 2 | no | — |
| 85 | C2×#157 | — | 18 | 2 | no | — |
| 86 | C2×#157 | — | 18 | 2 | no | — |
| 87 | C2×#157 | — | 18 | 4 | no | — |
| 88 | C2×#157 | — | 18 | 4 | no | — |
| 89 | C2×#157 | — | 18 | 4 | no | — |
| 90 | C2×#157 | — | 18 | 4 | no | — |
| 91 | C2×#157 | — | 18 | 4 | no | — |
| 92 | C2×#157 | — | 36 | 2 | no | — |
| 93 | C2×#157 | — | 36 | 2 | no | — |
| 94 | C2×#157 | — | 36 | 4 | no | — |
| 95 | C2×#157 | — | 36 | 4 | no | — |
| 96 | C2×#157 | — | 72 | 2 | no | — |
| 97 | C2×#157 | — | 72 | 2 | no | — |
| 98 | C2×#157 | PSU(3,𝔽2) | 1 | 2 | yes | complemented |
| 99 | PSU(3,𝔽2) | — | 18 | 16 | no | — |
| 100 | PSU(3,𝔽2) | — | 36 | 8 | no | — |
| 101 | C2×C32×C4 | — | 9 | 2 | no | — |
| 102 | C2×C3⋊Dic3 | — | 9 | 2 | no | — |
| 103 | C4×C3⋊D3 | — | 18 | 2 | no | — |
| 104 | C4×C3⋊D3 | — | 18 | 2 | no | — |
| 105 | #391 | — | 81 | 1 | no | Sylow |
| 106 | C2×C33 | — | 4 | 2 | no | — |
| 107 | C2×C33 | — | 16 | 2 | no | — |
| 108 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 109 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 110 | C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 111 | C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 112 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 113 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 114 | C32×D3 | — | 12 | 2 | no | — |
| 115 | C32×D3 | — | 12 | 2 | no | — |
| 116 | C2×C4×D3 | — | 108 | 2 | no | — |
| 117 | C22×C32 | — | 9 | 2 | no | — |
| 118 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 119 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 120 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 121 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 122 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 123 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 124 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 125 | C2×C3⋊D3 | — | 72 | 2 | no | — |
| 126 | C2×C3⋊D3 | — | 72 | 2 | no | — |
| 127 | C2×C3⋊D3 | — | 72 | 2 | no | — |
| 128 | C2×C3⋊D3 | — | 144 | 4 | no | — |
| 129 | C2×C3⋊D3 | C2×PSU(3,𝔽2) | 1 | 2 | yes | — |
| 130 | C2×C3×D3 | — | 24 | 4 | no | — |
| 131 | D32 | — | 72 | 2 | no | — |
| 132 | D32 | — | 72 | 4 | no | — |
| 133 | D32 | — | 72 | 2 | no | — |
| 134 | #157 | — | 2 | 2 | no | — |
| 135 | #157 | — | 9 | 8 | no | — |
| 136 | #157 | — | 18 | 8 | no | — |
| 137 | #157 | — | 18 | 2 | no | — |
| 138 | #157 | — | 36 | 4 | no | — |
| 139 | #157 | — | 36 | 4 | no | — |
| 140 | #157 | — | 36 | 4 | no | — |
| 141 | #157 | — | 36 | 8 | no | — |
| 142 | #157 | — | 72 | 4 | no | — |
| 143 | #157 | — | 72 | 4 | no | — |
| 144 | #157 | — | 72 | 4 | no | — |
| 145 | C32×C4 | — | 18 | 2 | no | — |
| 146 | C3⋊Dic3 | — | 18 | 2 | no | — |
| 147 | C22×Q8 | — | 81 | 1 | no | — |
| 148 | C2×C4⋊C4 | — | 81 | 1 | no | — |
| 149 | C2×C42 | — | 81 | 1 | no | — |
| 150 | #95 | — | 81 | 4 | no | — |
| 151 | C33 | — | 4 | 2 | no | — |
| 152 | C33 | — | 16 | 2 | no | — |
| 153 | C22×D3 | — | 108 | 2 | no | — |
| 154 | C2×C3×C4 | — | 36 | 2 | no | — |
| 155 | C2×Dic3 | — | 108 | 2 | no | — |
| 156 | C4×D3 | — | 108 | 4 | no | — |
| 157 | C4×D3 | — | 108 | 4 | no | — |
| 158 | C2×C32 | — | 4 | 2 | no | — |
| 159 | C2×C32 | — | 4 | 2 | no | — |
| 160 | C2×C32 | — | 8 | 2 | no | — |
| 161 | C2×C32 | — | 8 | 2 | no | — |
| 162 | C2×C32 | — | 8 | 2 | no | — |
| 163 | C2×C32 | — | 9 | 2 | no | — |
| 164 | C2×C32 | — | 9 | 2 | no | — |
| 165 | C2×C32 | — | 16 | 4 | no | — |
| 166 | C2×C32 | C4⋊PSU(3,𝔽2) | 1 | 2 | yes | — |
| 167 | C3⋊D3 | — | 9 | 2 | no | — |
| 168 | C3⋊D3 | — | 9 | 2 | no | — |
| 169 | C3⋊D3 | — | 36 | 2 | no | — |
| 170 | C3⋊D3 | — | 36 | 2 | no | — |
| 171 | C3⋊D3 | — | 36 | 2 | no | — |
| 172 | C3⋊D3 | — | 36 | 2 | no | — |
| 173 | C3⋊D3 | — | 72 | 2 | no | — |
| 174 | C3⋊D3 | — | 72 | 2 | no | — |
| 175 | C3⋊D3 | — | 72 | 2 | no | — |
| 176 | C3⋊D3 | — | 72 | 2 | no | — |
| 177 | C3⋊D3 | — | 72 | 2 | no | — |
| 178 | C3⋊D3 | — | 72 | 2 | no | — |
| 179 | C3⋊D3 | — | 144 | 4 | no | — |
| 180 | C3⋊D3 | — | 144 | 4 | no | — |
| 181 | C3⋊D3 | C4×PSU(3,𝔽2) | 1 | 2 | yes | — |
| 182 | C3⋊D3 | C22⋊PSU(3,𝔽2) | 1 | 2 | yes | — |
| 183 | C3×D3 | — | 24 | 4 | no | — |
| 184 | C3×D3 | — | 24 | 4 | no | — |
| 185 | C2×Q8 | — | 81 | 4 | no | — |
| 186 | C2×Q8 | — | 162 | 2 | no | — |
| 187 | C2×Q8 | — | 162 | 2 | no | — |
| 188 | C22×C4 | — | 81 | 2 | no | — |
| 189 | C22×C4 | — | 81 | 1 | no | — |
| 190 | C22×C4 | — | 81 | 2 | no | — |
| 191 | C22×C4 | — | 81 | 2 | no | — |
| 192 | C4⋊C4 | — | 162 | 2 | no | — |
| 193 | C42 | — | 162 | 2 | no | — |
| 194 | C22×C3 | — | 36 | 2 | no | — |
| 195 | C2×D3 | — | 12 | 2 | no | — |
| 196 | C2×D3 | — | 108 | 2 | no | — |
| 197 | C2×D3 | — | 108 | 2 | no | — |
| 198 | C2×D3 | — | 108 | 2 | no | — |
| 199 | C2×D3 | — | 108 | 2 | no | — |
| 200 | C2×D3 | — | 108 | 2 | no | — |
| 201 | C2×D3 | — | 432 | 2 | no | — |
| 202 | C3×C4 | — | 36 | 4 | no | — |
| 203 | Dic3 | — | 108 | 4 | no | — |
| 204 | C32 | — | 4 | 2 | no | — |
| 205 | C32 | — | 4 | 2 | no | — |
| 206 | C32 | — | 8 | 2 | no | — |
| 207 | C32 | — | 8 | 2 | no | — |
| 208 | C32 | — | 8 | 2 | no | — |
| 209 | C32 | — | 16 | 4 | no | — |
| 210 | C32 | #10593555 | 1 | 2 | yes | complemented |
| 211 | C23 | — | 81 | 1 | no | — |
| 212 | Q8 | — | 162 | 8 | no | — |
| 213 | C2×C4 | — | 9 | 2 | no | — |
| 214 | C2×C4 | — | 81 | 2 | no | — |
| 215 | C2×C4 | — | 81 | 2 | no | — |
| 216 | C2×C4 | — | 81 | 2 | no | — |
| 217 | C2×C4 | — | 81 | 2 | no | — |
| 218 | C2×C4 | — | 81 | 4 | no | — |
| 219 | C2×C4 | — | 162 | 2 | no | — |
| 220 | C2×C4 | — | 162 | 2 | no | — |
| 221 | C2×C4 | — | 162 | 2 | no | — |
| 222 | C2×C4 | — | 162 | 4 | no | — |
| 223 | C2×C4 | — | 162 | 2 | no | — |
| 224 | C2×C4 | — | 162 | 2 | no | — |
| 225 | C2×C3 | — | 4 | 2 | no | — |
| 226 | C2×C3 | — | 16 | 2 | no | — |
| 227 | C2×C3 | — | 36 | 2 | no | — |
| 228 | C2×C3 | — | 36 | 2 | no | — |
| 229 | D3 | — | 12 | 2 | no | — |
| 230 | D3 | — | 12 | 2 | no | — |
| 231 | D3 | — | 108 | 2 | no | — |
| 232 | D3 | — | 108 | 2 | no | — |
| 233 | D3 | — | 432 | 2 | no | — |
| 234 | D3 | — | 432 | 2 | no | — |
| 235 | C22 | — | 9 | 2 | no | — |
| 236 | C22 | — | 81 | 1 | no | — |
| 237 | C22 | — | 81 | 1 | no | — |
| 238 | C22 | — | 81 | 2 | no | — |
| 239 | C22 | — | 81 | 1 | no | — |
| 240 | C4 | — | 18 | 2 | no | — |
| 241 | C4 | — | 324 | 2 | no | — |
| 242 | C4 | — | 81 | 4 | no | — |
| 243 | C4 | — | 162 | 2 | no | — |
| 244 | C4 | — | 162 | 4 | no | — |
| 245 | C3 | — | 4 | 2 | no | — |
| 246 | C3 | — | 16 | 2 | no | — |
| 247 | C2 | — | 9 | 2 | no | — |
| 248 | C2 | — | 9 | 2 | no | — |
| 249 | C2 | — | 81 | 1 | no | — |
| 250 | C2 | — | 81 | 1 | no | — |
| 251 | C2 | #72213277 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #72213277 |
| commutator | C3⋊D32 | C22×C4 |
| Fitting, socle | C2×C34 | C4⋊Q8 |
| prime radical, cosocle | C2×C3⋊D32 | C23 |
| hypocenter | C34 | #391 |
| LCS | G ▷ C3⋊D32 ▷ C34 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 11. #13049818 | C4 | semidirect |
| 18. C2×C3⋊(C3⋊#157) | Q8 | semidirect |
| 19. C2×C32×#157 | Q8 | semidirect |
| 77. C34 | #391 | semidirect |
| 98. C2×#157 | PSU(3,𝔽2) | semidirect |
| 210. C32 | #10593555 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72220561 | 2 |
| #72220550 | 2 |
| C2×#1572 | 2 |
| C2×#13049818 | 2 |
| #10593555 | 9 |