Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
555 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 555 conjugacy classes of subgroups — too many to draw. Only its 98 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 | #10592784 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#5088 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | #10590352 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10590350 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #10589079 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #71194 | — | 3 | 2 | no | maximal |
| 7 | C32×C4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#3755 | — | 2 | 2 | no | — |
| 9 | #64083 | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#874 | C22 | 1 | 2 | yes | — |
| 11 | C2×#3714 | — | 2 | 4 | no | — |
| 12 | C2×#3713 | — | 2 | 2 | no | — |
| 13 | C4×C3⋊(C3⋊C8) | C22 | 1 | 1 | yes | characteristic |
| 14 | C3×#809 | — | 2 | 4 | no | — |
| 15 | C3×C4×C3⋊C8 | C22 | 1 | 2 | yes | — |
| 16 | C3×#486 | — | 3 | 2 | no | — |
| 17 | #5088 | D3 | 1 | 2 | yes | complemented |
| 18 | #5085 | — | 3 | 2 | no | — |
| 19 | #5085 | — | 6 | 1 | no | — |
| 20 | #5073 | — | 3 | 2 | no | — |
| 21 | #5073 | — | 3 | 2 | no | — |
| 22 | #4695 | — | 3 | 2 | no | — |
| 23 | #4454 | — | 3 | 2 | no | — |
| 24 | C2×C32×Q8 | — | 2 | 1 | no | — |
| 25 | C2×C4⋊(C3⋊D3) | — | 2 | 1 | no | — |
| 26 | C2×C4⋊(C3⋊D3) | — | 2 | 2 | no | — |
| 27 | C2×C3×D12 | — | 2 | 2 | no | — |
| 28 | C2×C3×D12 | — | 2 | 2 | no | — |
| 29 | C2×C3×D12 | — | 2 | 2 | no | — |
| 30 | C32×C4⋊C4 | — | 2 | 2 | no | — |
| 31 | C32×C42 | C23 | 1 | 1 | yes | characteristic |
| 32 | #3755 | — | 4 | 4 | no | — |
| 33 | C2×C3⋊(C3⋊C8) | D4 | 1 | 2 | yes | — |
| 34 | C2×C3×C3⋊C8 | D4 | 1 | 4 | yes | — |
| 35 | #3714 | — | 4 | 8 | no | — |
| 36 | #3713 | — | 4 | 4 | no | — |
| 37 | #1721 | — | 9 | 1 | no | Sylow |
| 38 | C3×C4⋊Q8 | — | 2 | 1 | no | — |
| 39 | C3×C4⋊Q8 | C2×D3 | 1 | 2 | yes | — |
| 40 | C3×#127 | — | 3 | 2 | no | — |
| 41 | C2×#215 | — | 6 | 4 | no | — |
| 42 | C2×#215 | — | 6 | 4 | no | — |
| 43 | C2×#213 | — | 6 | 4 | no | — |
| 44 | C2×#213 | — | 6 | 4 | no | — |
| 45 | C2×D24 | — | 6 | 4 | no | — |
| 46 | #874 | — | 3 | 2 | no | — |
| 47 | #874 | — | 6 | 1 | no | — |
| 48 | #874 | C2×D3 | 1 | 2 | yes | — |
| 49 | C3×#102 | — | 6 | 4 | no | — |
| 50 | C3×C4×C8 | — | 3 | 2 | no | — |
| 51 | #809 | — | 2 | 4 | no | — |
| 52 | C4×C3⋊C8 | — | 3 | 2 | no | — |
| 53 | C4×C3⋊C8 | — | 6 | 1 | no | — |
| 54 | C4×C3⋊C8 | C2×D3 | 1 | 2 | yes | — |
| 55 | C22×C3⋊D3 | — | 4 | 1 | no | — |
| 56 | C22×C3×D3 | — | 4 | 2 | no | — |
| 57 | C32×Q8 | — | 4 | 2 | no | — |
| 58 | C2×C32×C4 | — | 4 | 1 | no | — |
| 59 | C2×C32×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 60 | C2×C32×C4 | C2×D4 | 1 | 2 | yes | — |
| 61 | C4⋊(C3⋊D3) | — | 4 | 2 | no | — |
| 62 | C4⋊(C3⋊D3) | — | 4 | 4 | no | — |
| 63 | C3×D12 | — | 4 | 4 | no | — |
| 64 | C3×D12 | — | 4 | 4 | no | — |
| 65 | C3×D12 | — | 4 | 4 | no | — |
| 66 | C3⋊(C3⋊C8) | D8 | 1 | 4 | yes | complemented |
| 67 | C3×C3⋊C8 | — | 2 | 4 | no | — |
| 68 | #493 | — | 9 | 2 | no | — |
| 69 | #493 | — | 9 | 1 | no | — |
| 70 | #492 | — | 9 | 1 | no | — |
| 71 | #486 | — | 3 | 2 | no | — |
| 72 | #322 | — | 9 | 1 | no | — |
| 73 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 74 | C2×C3×Q8 | — | 2 | 2 | no | — |
| 75 | C2×C3×D4 | — | 6 | 2 | no | — |
| 76 | C2×C3×D4 | — | 6 | 2 | no | — |
| 77 | C2×C3×D4 | — | 6 | 2 | no | — |
| 78 | C2×D12 | — | 2 | 2 | no | — |
| 79 | C2×D12 | — | 2 | 2 | no | — |
| 80 | C2×D12 | — | 2 | 2 | no | — |
| 81 | C2×D12 | — | 6 | 2 | no | — |
| 82 | C2×D12 | — | 6 | 2 | no | — |
| 83 | C2×D12 | — | 6 | 2 | no | — |
| 84 | C2×D12 | — | 6 | 2 | no | — |
| 85 | C2×D12 | — | 12 | 2 | no | — |
| 86 | C2×C3×C8 | — | 3 | 4 | no | — |
| 87 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 88 | C3×C4⋊C4 | — | 2 | 2 | no | — |
| 89 | C3×C4⋊C4 | — | 4 | 2 | no | — |
| 90 | C3×C42 | — | 2 | 1 | no | — |
| 91 | C3×C42 | C22×D3 | 1 | 2 | yes | — |
| 92 | #215 | — | 12 | 8 | no | — |
| 93 | #215 | — | 12 | 8 | no | — |
| 94 | #213 | — | 12 | 8 | no | — |
| 95 | #213 | — | 12 | 8 | no | — |
| 96 | C2×C3⋊C8 | — | 3 | 4 | no | — |
| 97 | C2×C3⋊C8 | — | 6 | 2 | no | — |
| 98 | C2×C3⋊C8 | #67 | 1 | 4 | yes | — |
| 99 | D24 | — | 12 | 8 | no | — |
| 100 | C22×C32 | #127 | 1 | 1 | yes | characteristic |
| 101 | C2×C3⋊D3 | — | 8 | 1 | no | — |
| 102 | C2×C3⋊D3 | — | 8 | 2 | no | — |
| 103 | C2×C3×D3 | — | 8 | 2 | no | — |
| 104 | C2×C3×D3 | — | 8 | 2 | no | — |
| 105 | C2×C3×D3 | — | 8 | 2 | no | — |
| 106 | C32×C4 | — | 8 | 1 | no | — |
| 107 | C32×C4 | C2×D8 | 1 | 2 | yes | — |
| 108 | C32×C4 | AGL(1,ℤ/8) | 1 | 4 | yes | — |
| 109 | C2×SD16 | — | 18 | 2 | no | — |
| 110 | C2×D8 | — | 18 | 4 | no | — |
| 111 | C2×D8 | — | 18 | 2 | no | — |
| 112 | C4⋊Q8 | D32 | 1 | 1 | yes | characteristic |
| 113 | #127 | — | 3 | 2 | no | — |
| 114 | #127 | — | 9 | 1 | no | — |
| 115 | #102 | — | 6 | 4 | no | — |
| 116 | C4×C8 | — | 3 | 2 | no | — |
| 117 | C4×C8 | — | 9 | 1 | no | — |
| 118 | C23×C3 | — | 12 | 2 | no | — |
| 119 | C22×D3 | — | 4 | 2 | no | — |
| 120 | C22×D3 | — | 12 | 2 | no | — |
| 121 | C22×D3 | — | 12 | 2 | no | — |
| 122 | C3×Q8 | — | 4 | 4 | no | — |
| 123 | C3×Q8 | — | 4 | 4 | no | — |
| 124 | C3×D4 | — | 12 | 4 | no | — |
| 125 | C3×D4 | — | 12 | 4 | no | — |
| 126 | C3×D4 | — | 12 | 4 | no | — |
| 127 | C2×C3×C4 | — | 2 | 2 | no | — |
| 128 | C2×C3×C4 | — | 2 | 1 | no | — |
| 129 | C2×C3×C4 | — | 4 | 2 | no | — |
| 130 | C2×C3×C4 | — | 4 | 2 | no | — |
| 131 | C2×C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 132 | C2×C3×C4 | D3×D4 | 1 | 2 | yes | — |
| 133 | C2×C3×C4 | C2×#67 | 1 | 2 | yes | — |
| 134 | D12 | — | 4 | 4 | no | — |
| 135 | D12 | — | 4 | 4 | no | — |
| 136 | D12 | — | 4 | 4 | no | — |
| 137 | D12 | — | 12 | 4 | no | — |
| 138 | D12 | — | 12 | 4 | no | — |
| 139 | D12 | — | 12 | 4 | no | — |
| 140 | D12 | — | 12 | 4 | no | — |
| 141 | D12 | — | 24 | 4 | no | — |
| 142 | C3×C8 | — | 6 | 4 | no | — |
| 143 | C3⋊C8 | — | 2 | 4 | no | — |
| 144 | C3⋊C8 | — | 3 | 8 | no | — |
| 145 | C3⋊C8 | — | 6 | 4 | no | — |
| 146 | C2×C32 | #493 | 1 | 1 | yes | characteristic |
| 147 | C2×C32 | #496 | 1 | 2 | yes | — |
| 148 | C3⋊D3 | — | 16 | 1 | no | — |
| 149 | C3×D3 | — | 16 | 2 | no | — |
| 150 | C2×Q8 | — | 2 | 1 | no | — |
| 151 | C2×D4 | — | 6 | 2 | no | — |
| 152 | C2×D4 | — | 6 | 2 | no | — |
| 153 | C2×D4 | — | 6 | 2 | no | — |
| 154 | C2×D4 | — | 18 | 2 | no | — |
| 155 | C2×D4 | — | 18 | 1 | no | — |
| 156 | SD16 | — | 36 | 4 | no | — |
| 157 | D8 | — | 36 | 8 | no | — |
| 158 | D8 | — | 36 | 4 | no | — |
| 159 | C2×C8 | — | 3 | 4 | no | — |
| 160 | C2×C8 | — | 9 | 2 | no | — |
| 161 | C4⋊C4 | — | 2 | 2 | no | — |
| 162 | C42 | C2×D32 | 1 | 1 | yes | characteristic |
| 163 | C22×C3 | — | 2 | 1 | no | — |
| 164 | C22×C3 | — | 24 | 2 | no | — |
| 165 | C22×C3 | — | 24 | 2 | no | — |
| 166 | C22×C3 | — | 24 | 2 | no | — |
| 167 | C22×C3 | #940 | 1 | 2 | yes | — |
| 168 | C2×D3 | — | 8 | 2 | no | — |
| 169 | C2×D3 | — | 8 | 2 | no | — |
| 170 | C2×D3 | — | 8 | 2 | no | — |
| 171 | C2×D3 | — | 24 | 2 | no | — |
| 172 | C2×D3 | — | 24 | 4 | no | — |
| 173 | C2×D3 | — | 24 | 2 | no | — |
| 174 | C2×D3 | — | 24 | 2 | no | — |
| 175 | C2×D3 | — | 24 | 2 | no | — |
| 176 | C3×C4 | — | 2 | 2 | no | — |
| 177 | C3×C4 | — | 2 | 4 | no | — |
| 178 | C3×C4 | — | 8 | 2 | no | — |
| 179 | C3×C4 | — | 8 | 2 | no | — |
| 180 | C3×C4 | D3×D8 | 1 | 4 | yes | — |
| 181 | C3×C4 | #914 | 1 | 4 | yes | — |
| 182 | C3×C4 | #949 | 1 | 4 | yes | — |
| 183 | C32 | #1721 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 184 | C23 | — | 12 | 2 | no | — |
| 185 | C23 | — | 36 | 1 | no | — |
| 186 | Q8 | — | 4 | 2 | no | — |
| 187 | D4 | — | 12 | 4 | no | — |
| 188 | D4 | — | 12 | 4 | no | — |
| 189 | D4 | — | 12 | 4 | no | — |
| 190 | D4 | — | 36 | 2 | no | — |
| 191 | D4 | — | 36 | 4 | no | — |
| 192 | C2×C4 | — | 4 | 1 | no | — |
| 193 | C2×C4 | D3×#67 | 1 | 2 | yes | — |
| 194 | C2×C4 | #3811 | 1 | 1 | yes | characteristic |
| 195 | C8 | — | 6 | 4 | no | — |
| 196 | C8 | — | 9 | 4 | no | — |
| 197 | C2×C3 | — | 2 | 2 | no | — |
| 198 | C2×C3 | — | 2 | 1 | no | — |
| 199 | C2×C3 | — | 48 | 2 | no | — |
| 200 | C2×C3 | #5058 | 1 | 2 | yes | — |
| 201 | C2×C3 | #5151 | 1 | 2 | yes | — |
| 202 | C2×C3 | #5176 | 1 | 2 | yes | — |
| 203 | D3 | — | 16 | 2 | no | — |
| 204 | D3 | — | 48 | 2 | no | — |
| 205 | D3 | — | 48 | 2 | no | — |
| 206 | C22 | — | 24 | 2 | no | — |
| 207 | C22 | — | 24 | 2 | no | — |
| 208 | C22 | — | 24 | 2 | no | — |
| 209 | C22 | — | 72 | 2 | no | — |
| 210 | C22 | — | 72 | 1 | no | — |
| 211 | C22 | #63979 | 1 | 1 | yes | characteristic |
| 212 | C4 | — | 8 | 1 | no | — |
| 213 | C4 | #63926 | 1 | 2 | yes | — |
| 214 | C4 | #63939 | 1 | 4 | yes | — |
| 215 | C3 | — | 2 | 1 | no | — |
| 216 | C3 | #71194 | 1 | 2 | yes | complemented |
| 217 | C2 | — | 48 | 2 | no | — |
| 218 | C2 | — | 144 | 1 | no | — |
| 219 | C2 | #10591484 | 1 | 1 | yes | characteristic |
| 220 | C2 | #10591619 | 1 | 2 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #63979 |
| commutator, cosocle | C32×C42 | C23 |
| Frattini | C42 | C2×D32 |
| Fitting | C32×C4⋊Q8 | C22 |
| socle | C22×C32 | #127 |
| hypercenter | C4⋊Q8 | D32 |
| hypocenter | C32 | #1721 |
| LCS | G ▷ C32×C42 ▷ C22×C32 ▷ C32 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C4⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10592784 | C2 | semidirect |
| 2. C3×#5088 | C2 | semidirect |
| 3. #10590352 | C2 | semidirect |
| 4. #10590350 | C2 | semidirect |
| 5. #10589079 | C2 | semidirect |
| 17. #5088 | D3 | semidirect |
| 66. C3⋊(C3⋊C8) | D8 | semidirect |
| 183. C32 | #1721 | semidirect |
| 216. C3 | #71194 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10592784 | 2 |
| C3×#5088 | 2 |
| #10590352 | 2 |
| #10590350 | 2 |
| #10589079 | 2 |
| #71194 | 3 |