Finite Group DatabasebetaFinite Group DatabasebetaOrder 5184.
636 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 636 conjugacy classes of subgroups — too many to draw. Only its 83 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 | #72220550 | C2 | 1 | 2 | yes | maximal |
| 2 | C2×#13049783 | C2 | 1 | 4 | yes | maximal |
| 3 | C2×#1572 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | C2×#10620198 | C22 | 1 | 4 | yes | — |
| 5 | C2×#10620198 | C22 | 1 | 1 | yes | characteristic |
| 6 | C2×C3⋊D3×#157 | C22 | 1 | 2 | yes | — |
| 7 | #13049783 | — | 2 | 8 | no | — |
| 8 | #1572 | C4 | 1 | 4 | yes | complemented |
| 9 | #13049467 | — | 2 | 4 | no | — |
| 10 | C2×C3⋊D32 | C23 | 1 | 1 | yes | characteristic |
| 11 | C2×#66472 | — | 2 | 4 | no | — |
| 12 | C2×#66472 | D4 | 1 | 2 | yes | — |
| 13 | C2×C3⋊(C3⋊#157) | Q8 | 1 | 2 | yes | — |
| 14 | C2×C32×#157 | Q8 | 1 | 2 | yes | — |
| 15 | #10620198 | — | 2 | 8 | no | — |
| 16 | #10620198 | C2×C4 | 1 | 2 | yes | — |
| 17 | #10620198 | Q8 | 1 | 2 | yes | — |
| 18 | C3⋊D3×#157 | C2×C4 | 1 | 4 | yes | — |
| 19 | C3⋊D3×#157 | D4 | 1 | 4 | yes | — |
| 20 | #10593553 | — | 9 | 2 | no | maximal |
| 21 | C2×D3×#157 | — | 12 | 2 | no | — |
| 22 | C2×C3⋊(C3⋊(C3⋊D3)) | C2×D4 | 1 | 1 | yes | characteristic |
| 23 | C2×C32×C3⋊D3 | C2×Q8 | 1 | 2 | yes | — |
| 24 | C3⋊D32 | C22×C4 | 1 | 1 | yes | characteristic |
| 25 | C3⋊D32 | C2×D4 | 1 | 2 | yes | — |
| 26 | C3⋊D32 | C2×Q8 | 1 | 1 | yes | characteristic |
| 27 | #66472 | — | 4 | 4 | no | — |
| 28 | #66472 | C4⋊C4 | 1 | 4 | yes | complemented |
| 29 | C3⋊(C3⋊#157) | C4⋊C4 | 1 | 4 | yes | complemented |
| 30 | C32×#157 | C4⋊C4 | 1 | 4 | yes | complemented |
| 31 | C2×C4⋊#157 | — | 9 | 4 | no | — |
| 32 | C2×C4×#157 | — | 9 | 2 | no | — |
| 33 | C2×#3777 | — | 9 | 4 | no | — |
| 34 | C2×#3777 | — | 9 | 4 | no | — |
| 35 | C2×D3×C3⋊D3 | — | 12 | 2 | no | — |
| 36 | C2×C3⋊#157 | — | 12 | 2 | no | — |
| 37 | C2×C3×#157 | — | 4 | 2 | no | — |
| 38 | D3×#157 | — | 12 | 4 | no | — |
| 39 | D3×#157 | — | 12 | 4 | no | — |
| 40 | C2×C34 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 41 | C3⋊(C3⋊(C3⋊D3)) | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 42 | C3⋊(C3⋊(C3⋊D3)) | #127 | 1 | 1 | yes | characteristic |
| 43 | C32×C3⋊D3 | C2×C4⋊C4 | 1 | 2 | yes | — |
| 44 | C32×C3⋊D3 | C4⋊Q8 | 1 | 2 | yes | — |
| 45 | C22×#157 | — | 9 | 2 | no | — |
| 46 | C22×#157 | — | 9 | 2 | no | — |
| 47 | C22×#157 | — | 9 | 8 | no | — |
| 48 | C2×C4×C3⋊D3 | — | 9 | 2 | no | — |
| 49 | C4⋊#157 | — | 18 | 8 | no | — |
| 50 | C4×#157 | — | 9 | 8 | no | — |
| 51 | #3777 | — | 18 | 8 | no | — |
| 52 | #3777 | — | 18 | 8 | no | — |
| 53 | #3777 | — | 36 | 4 | no | — |
| 54 | C2×C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 55 | C2×C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 56 | C2×C3×C3⋊D3 | — | 4 | 2 | no | — |
| 57 | C2×C32×D3 | — | 12 | 2 | no | — |
| 58 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 59 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 60 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 61 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 62 | C3⋊#157 | — | 12 | 4 | no | — |
| 63 | C3×#157 | — | 4 | 4 | no | — |
| 64 | C34 | #384 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 65 | C22×C3⋊D3 | — | 9 | 2 | no | — |
| 66 | C2×D32 | — | 72 | 2 | no | — |
| 67 | C2×#157 | — | 9 | 4 | no | — |
| 68 | C2×#157 | — | 9 | 4 | no | — |
| 69 | C2×#157 | — | 9 | 4 | no | — |
| 70 | C2×#157 | — | 9 | 2 | no | — |
| 71 | C2×#157 | — | 9 | 4 | no | — |
| 72 | C2×#157 | — | 9 | 4 | no | — |
| 73 | C2×#157 | — | 18 | 8 | no | — |
| 74 | C2×#157 | — | 18 | 8 | no | — |
| 75 | C2×#157 | — | 18 | 8 | no | — |
| 76 | C2×#157 | — | 36 | 4 | no | — |
| 77 | C2×#157 | — | 36 | 8 | no | — |
| 78 | C2×#157 | — | 72 | 4 | no | — |
| 79 | C2×#157 | PSU(3,𝔽2) | 1 | 2 | yes | — |
| 80 | C2×C32×C4 | — | 9 | 2 | no | — |
| 81 | C2×C3⋊Dic3 | — | 9 | 2 | no | — |
| 82 | C4×C3⋊D3 | — | 9 | 4 | no | — |
| 83 | C4×C3⋊D3 | — | 9 | 4 | no | — |
| 84 | #384 | — | 81 | 1 | no | Sylow |
| 85 | C2×C33 | — | 4 | 2 | no | — |
| 86 | C2×C33 | — | 16 | 2 | no | — |
| 87 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 88 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 89 | C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 90 | C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 91 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 92 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 93 | C32×D3 | — | 12 | 2 | no | — |
| 94 | C32×D3 | — | 12 | 2 | no | — |
| 95 | C2×C4×D3 | — | 108 | 2 | no | — |
| 96 | C22×C32 | — | 9 | 2 | no | — |
| 97 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 98 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 99 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 100 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 101 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 102 | C2×C3⋊D3 | — | 36 | 4 | no | — |
| 103 | C2×C3⋊D3 | — | 72 | 2 | no | — |
| 104 | C2×C3⋊D3 | — | 72 | 4 | no | — |
| 105 | C2×C3⋊D3 | — | 144 | 4 | no | — |
| 106 | C2×C3⋊D3 | C2×PSU(3,𝔽2) | 1 | 2 | yes | — |
| 107 | C2×C3×D3 | — | 24 | 4 | no | — |
| 108 | D32 | — | 72 | 2 | no | — |
| 109 | D32 | — | 72 | 4 | no | — |
| 110 | D32 | — | 72 | 2 | no | — |
| 111 | #157 | — | 9 | 8 | no | — |
| 112 | #157 | — | 9 | 4 | no | — |
| 113 | #157 | — | 36 | 8 | no | — |
| 114 | #157 | — | 36 | 8 | no | — |
| 115 | #157 | — | 72 | 8 | no | — |
| 116 | #157 | — | 72 | 8 | no | — |
| 117 | #157 | #3777 | 1 | 4 | yes | complemented |
| 118 | C32×C4 | — | 9 | 4 | no | — |
| 119 | C3⋊Dic3 | — | 9 | 4 | no | — |
| 120 | C2×C4⋊C4 | — | 81 | 4 | no | — |
| 121 | C2×C4⋊C4 | — | 81 | 2 | no | — |
| 122 | C2×C42 | — | 81 | 1 | no | — |
| 123 | C33 | — | 4 | 2 | no | — |
| 124 | C33 | — | 16 | 2 | no | — |
| 125 | C22×D3 | — | 108 | 2 | no | — |
| 126 | C2×C3×C4 | — | 36 | 2 | no | — |
| 127 | C2×Dic3 | — | 108 | 2 | no | — |
| 128 | C4×D3 | — | 108 | 4 | no | — |
| 129 | C4×D3 | — | 108 | 4 | no | — |
| 130 | C2×C32 | — | 4 | 4 | no | — |
| 131 | C2×C32 | — | 8 | 2 | no | — |
| 132 | C2×C32 | — | 8 | 4 | no | — |
| 133 | C2×C32 | — | 9 | 2 | no | — |
| 134 | C2×C32 | — | 9 | 2 | no | — |
| 135 | C2×C32 | — | 16 | 4 | no | — |
| 136 | C2×C32 | C4⋊PSU(3,𝔽2) | 1 | 2 | yes | — |
| 137 | C3⋊D3 | — | 9 | 2 | no | — |
| 138 | C3⋊D3 | — | 9 | 2 | no | — |
| 139 | C3⋊D3 | — | 36 | 4 | no | — |
| 140 | C3⋊D3 | — | 36 | 4 | no | — |
| 141 | C3⋊D3 | — | 72 | 2 | no | — |
| 142 | C3⋊D3 | — | 72 | 2 | no | — |
| 143 | C3⋊D3 | — | 72 | 4 | no | — |
| 144 | C3⋊D3 | — | 72 | 4 | no | — |
| 145 | C3⋊D3 | — | 144 | 4 | no | — |
| 146 | C3⋊D3 | — | 144 | 4 | no | — |
| 147 | C3⋊D3 | C4⋊PSU(3,𝔽2) | 1 | 2 | yes | — |
| 148 | C3⋊D3 | C2×#3777 | 1 | 2 | yes | — |
| 149 | C3×D3 | — | 24 | 4 | no | — |
| 150 | C3×D3 | — | 24 | 4 | no | — |
| 151 | C22×C4 | — | 81 | 2 | no | — |
| 152 | C22×C4 | — | 81 | 1 | no | — |
| 153 | C22×C4 | — | 81 | 4 | no | — |
| 154 | C4⋊C4 | — | 162 | 4 | no | — |
| 155 | C4⋊C4 | — | 162 | 8 | no | — |
| 156 | C42 | — | 81 | 4 | no | — |
| 157 | C22×C3 | — | 36 | 2 | no | — |
| 158 | C2×D3 | — | 12 | 2 | no | — |
| 159 | C2×D3 | — | 108 | 2 | no | — |
| 160 | C2×D3 | — | 108 | 2 | no | — |
| 161 | C2×D3 | — | 108 | 2 | no | — |
| 162 | C2×D3 | — | 108 | 2 | no | — |
| 163 | C2×D3 | — | 108 | 2 | no | — |
| 164 | C2×D3 | — | 432 | 2 | no | — |
| 165 | C3×C4 | — | 36 | 4 | no | — |
| 166 | Dic3 | — | 108 | 4 | no | — |
| 167 | C32 | — | 4 | 4 | no | — |
| 168 | C32 | — | 8 | 2 | no | — |
| 169 | C32 | — | 8 | 4 | no | — |
| 170 | C32 | — | 16 | 4 | no | — |
| 171 | C32 | #10593553 | 1 | 2 | yes | complemented |
| 172 | C23 | — | 81 | 1 | no | — |
| 173 | C2×C4 | — | 9 | 2 | no | — |
| 174 | C2×C4 | — | 81 | 4 | no | — |
| 175 | C2×C4 | — | 81 | 4 | no | — |
| 176 | C2×C4 | — | 81 | 2 | no | — |
| 177 | C2×C4 | — | 81 | 2 | no | — |
| 178 | C2×C4 | — | 81 | 2 | no | — |
| 179 | C2×C4 | — | 81 | 2 | no | — |
| 180 | C2×C4 | — | 162 | 8 | no | — |
| 181 | C2×C4 | — | 162 | 4 | no | — |
| 182 | C2×C3 | — | 4 | 2 | no | — |
| 183 | C2×C3 | — | 16 | 2 | no | — |
| 184 | C2×C3 | — | 36 | 2 | no | — |
| 185 | C2×C3 | — | 36 | 2 | no | — |
| 186 | D3 | — | 12 | 2 | no | — |
| 187 | D3 | — | 12 | 2 | no | — |
| 188 | D3 | — | 108 | 2 | no | — |
| 189 | D3 | — | 108 | 2 | no | — |
| 190 | D3 | — | 432 | 2 | no | — |
| 191 | D3 | — | 432 | 2 | no | — |
| 192 | C22 | — | 9 | 2 | no | — |
| 193 | C22 | — | 81 | 1 | no | — |
| 194 | C22 | — | 81 | 2 | no | — |
| 195 | C22 | — | 81 | 1 | no | — |
| 196 | C22 | — | 81 | 1 | no | — |
| 197 | C4 | — | 9 | 4 | no | — |
| 198 | C4 | — | 324 | 4 | no | — |
| 199 | C4 | — | 81 | 4 | no | — |
| 200 | C4 | — | 81 | 4 | no | — |
| 201 | C3 | — | 4 | 2 | no | — |
| 202 | C3 | — | 16 | 2 | no | — |
| 203 | C2 | — | 9 | 2 | no | — |
| 204 | C2 | — | 9 | 2 | no | — |
| 205 | C2 | — | 81 | 1 | no | — |
| 206 | C2 | — | 81 | 1 | no | — |
| 207 | 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 | #384 |
| LCS | G ▷ C3⋊D32 ▷ C34 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. #1572 | C4 | semidirect |
| 28. #66472 | C4⋊C4 | semidirect |
| 29. C3⋊(C3⋊#157) | C4⋊C4 | semidirect |
| 30. C32×#157 | C4⋊C4 | semidirect |
| 64. C34 | #384 | semidirect |
| 117. #157 | #3777 | semidirect |
| 171. C32 | #10593553 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72220550 | 2 |
| C2×#13049783 | 2 |
| C2×#1572 | 2 |
| #10593553 | 9 |