Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
748 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 748 conjugacy classes of subgroups — too many to draw. Only its 46 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 | C22×C42⋊C9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#63419 | C2 | 1 | 6 | yes | maximal, direct product factor |
| 3 | C22×#805 | — | 3 | 1 | no | maximal |
| 4 | C22×(C22⋊C9)⋊C2 | — | 4 | 1 | no | maximal |
| 5 | C2×C42⋊C9 | C22 | 1 | 3 | yes | complemented |
| 6 | #63419 | C22 | 1 | 4 | yes | direct product factor |
| 7 | C22×#235 | — | 3 | 1 | no | — |
| 8 | C22×C3×C42 | D3 | 1 | 1 | yes | characteristic |
| 9 | C22×#208 | — | 3 | 1 | no | — |
| 10 | C2×#805 | — | 3 | 6 | no | — |
| 11 | C2×#805 | — | 3 | 6 | no | — |
| 12 | C22×C22⋊C9 | — | 4 | 1 | no | — |
| 13 | C2×(C22⋊C9)⋊C2 | — | 4 | 6 | no | — |
| 14 | C42⋊C9 | C23 | 1 | 1 | yes | characteristic, complemented |
| 15 | C22×#104 | — | 9 | 1 | no | Sylow |
| 16 | C23×C3×C4 | — | 3 | 1 | no | — |
| 17 | C22×#67 | — | 6 | 1 | no | — |
| 18 | C2×#235 | — | 3 | 6 | no | — |
| 19 | C2×#235 | — | 3 | 6 | no | — |
| 20 | C2×#235 | — | 6 | 3 | no | — |
| 21 | C2×#235 | — | 6 | 3 | no | — |
| 22 | C22×D12 | — | 3 | 1 | no | — |
| 23 | C22×C4×D3 | — | 6 | 1 | no | — |
| 24 | C22×Dic6 | — | 3 | 1 | no | — |
| 25 | C2×C3×C42 | — | 3 | 3 | no | — |
| 26 | C2×C3×C42 | C2×D3 | 1 | 3 | yes | — |
| 27 | C2×#208 | — | 3 | 6 | no | — |
| 28 | C2×#208 | — | 6 | 3 | no | — |
| 29 | C22×C3⋊C8 | — | 6 | 1 | no | — |
| 30 | #805 | — | 3 | 12 | no | — |
| 31 | #805 | — | 3 | 4 | no | — |
| 32 | C22×D9 | — | 16 | 1 | no | — |
| 33 | C2×C22⋊C9 | — | 4 | 3 | no | — |
| 34 | (C22⋊C9)⋊C2 | — | 4 | 4 | no | — |
| 35 | C22×#41 | — | 9 | 1 | no | — |
| 36 | C22×M16 | — | 9 | 1 | no | — |
| 37 | C22×C42 | D9 | 1 | 1 | yes | characteristic, complemented |
| 38 | C2×#104 | — | 9 | 6 | no | — |
| 39 | C2×#104 | — | 9 | 6 | no | — |
| 40 | C24×C3 | S4 | 1 | 1 | yes | characteristic |
| 41 | C23×D3 | — | 6 | 1 | no | — |
| 42 | C22×C3×C4 | — | 3 | 1 | no | — |
| 43 | C22×C3×C4 | — | 3 | 3 | no | — |
| 44 | C22×C3×C4 | — | 3 | 1 | no | — |
| 45 | C22×C3×C4 | — | 3 | 3 | no | — |
| 46 | C22×C3×C4 | — | 3 | 3 | no | — |
| 47 | C22×C3×C4 | — | 3 | 3 | no | — |
| 48 | C2×#67 | — | 6 | 6 | no | — |
| 49 | C2×#67 | — | 6 | 6 | no | — |
| 50 | C22×Dic3 | — | 6 | 1 | no | — |
| 51 | #235 | — | 3 | 4 | no | — |
| 52 | #235 | — | 3 | 12 | no | — |
| 53 | #235 | — | 6 | 6 | no | — |
| 54 | #235 | — | 6 | 12 | no | — |
| 55 | #235 | — | 6 | 6 | no | — |
| 56 | C2×D12 | — | 3 | 6 | no | — |
| 57 | C2×D12 | — | 6 | 3 | no | — |
| 58 | C2×C4×D3 | — | 6 | 6 | no | — |
| 59 | C2×C4×D3 | — | 6 | 6 | no | — |
| 60 | C2×Dic6 | — | 3 | 6 | no | — |
| 61 | C2×Dic6 | — | 6 | 3 | no | — |
| 62 | C3×C42 | — | 3 | 3 | no | — |
| 63 | C3×C42 | — | 6 | 1 | no | — |
| 64 | C3×C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 65 | #208 | — | 3 | 4 | no | — |
| 66 | #208 | — | 6 | 6 | no | — |
| 67 | C2×C3⋊C8 | — | 6 | 6 | no | — |
| 68 | C22×C9 | — | 16 | 1 | no | — |
| 69 | C2×D9 | — | 16 | 6 | no | — |
| 70 | C22⋊C9 | — | 4 | 1 | no | — |
| 71 | C2×#41 | — | 9 | 6 | no | — |
| 72 | C2×#41 | — | 9 | 6 | no | — |
| 73 | C2×#41 | — | 18 | 3 | no | — |
| 74 | C2×#41 | — | 18 | 3 | no | — |
| 75 | C22×Q8 | — | 9 | 1 | no | — |
| 76 | C22×D4 | — | 9 | 1 | no | — |
| 77 | C22×D4 | — | 18 | 1 | no | — |
| 78 | C23×C4 | — | 3 | 1 | no | — |
| 79 | C23×C4 | — | 18 | 1 | no | — |
| 80 | C2×M16 | — | 9 | 6 | no | — |
| 81 | C2×M16 | — | 18 | 3 | no | — |
| 82 | C22×C8 | — | 18 | 1 | no | — |
| 83 | C2×C42 | — | 3 | 3 | no | — |
| 84 | C2×C42 | C2×D9 | 1 | 3 | yes | complemented |
| 85 | #104 | — | 9 | 12 | no | — |
| 86 | #104 | — | 9 | 4 | no | — |
| 87 | C23×C3 | — | 3 | 1 | no | — |
| 88 | C23×C3 | — | 3 | 3 | no | — |
| 89 | C23×C3 | C2×S4 | 1 | 3 | yes | — |
| 90 | C22×D3 | — | 6 | 6 | no | — |
| 91 | C22×D3 | — | 12 | 3 | no | — |
| 92 | C22×D3 | — | 12 | 1 | no | — |
| 93 | C2×C3×C4 | — | 3 | 3 | no | — |
| 94 | C2×C3×C4 | — | 3 | 3 | no | — |
| 95 | C2×C3×C4 | — | 3 | 3 | no | — |
| 96 | C2×C3×C4 | — | 3 | 3 | no | — |
| 97 | C2×C3×C4 | — | 3 | 3 | no | — |
| 98 | C2×C3×C4 | — | 3 | 3 | no | — |
| 99 | C2×C3×C4 | — | 3 | 6 | no | — |
| 100 | C2×C3×C4 | — | 3 | 1 | no | — |
| 101 | C2×C3×C4 | — | 3 | 3 | no | — |
| 102 | #67 | — | 6 | 12 | no | — |
| 103 | #67 | — | 6 | 4 | no | — |
| 104 | C2×Dic3 | — | 6 | 6 | no | — |
| 105 | D12 | — | 3 | 4 | no | — |
| 106 | D12 | — | 6 | 6 | no | — |
| 107 | C4×D3 | — | 6 | 4 | no | — |
| 108 | C4×D3 | — | 6 | 12 | no | — |
| 109 | Dic6 | — | 3 | 4 | no | — |
| 110 | Dic6 | — | 6 | 6 | no | — |
| 111 | C3⋊C8 | — | 6 | 4 | no | — |
| 112 | C2×C9 | — | 16 | 3 | no | — |
| 113 | D9 | — | 16 | 4 | no | — |
| 114 | C24 | — | 18 | 1 | no | — |
| 115 | C24 | (C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 116 | #41 | — | 9 | 4 | no | — |
| 117 | #41 | — | 9 | 12 | no | — |
| 118 | #41 | — | 18 | 6 | no | — |
| 119 | #41 | — | 18 | 12 | no | — |
| 120 | #41 | — | 18 | 6 | no | — |
| 121 | C2×Q8 | — | 9 | 6 | no | — |
| 122 | C2×Q8 | — | 18 | 3 | no | — |
| 123 | C2×D4 | — | 9 | 6 | no | — |
| 124 | C2×D4 | — | 18 | 3 | no | — |
| 125 | C2×D4 | — | 18 | 6 | no | — |
| 126 | C2×D4 | — | 18 | 6 | no | — |
| 127 | C22×C4 | — | 3 | 1 | no | — |
| 128 | C22×C4 | — | 3 | 3 | no | — |
| 129 | C22×C4 | — | 3 | 1 | no | — |
| 130 | C22×C4 | — | 3 | 3 | no | — |
| 131 | C22×C4 | — | 3 | 3 | no | — |
| 132 | C22×C4 | — | 3 | 3 | no | — |
| 133 | C22×C4 | — | 18 | 6 | no | — |
| 134 | C22×C4 | — | 18 | 6 | no | — |
| 135 | C22×C4 | — | 18 | 1 | no | — |
| 136 | M16 | — | 9 | 4 | no | — |
| 137 | M16 | — | 18 | 6 | no | — |
| 138 | C2×C8 | — | 18 | 6 | no | — |
| 139 | C42 | — | 3 | 3 | no | — |
| 140 | C42 | — | 6 | 1 | no | — |
| 141 | C42 | C22×D9 | 1 | 1 | yes | characteristic, complemented |
| 142 | C22×C3 | — | 3 | 3 | no | — |
| 143 | C22×C3 | — | 3 | 3 | no | — |
| 144 | C22×C3 | — | 3 | 3 | no | — |
| 145 | C22×C3 | — | 6 | 1 | no | — |
| 146 | C22×C3 | #857 | 1 | 1 | yes | characteristic |
| 147 | C22×C3 | C22×S4 | 1 | 1 | yes | characteristic |
| 148 | C2×D3 | — | 6 | 4 | no | — |
| 149 | C2×D3 | — | 12 | 6 | no | — |
| 150 | C2×D3 | — | 12 | 6 | no | — |
| 151 | C3×C4 | — | 3 | 1 | no | — |
| 152 | C3×C4 | — | 3 | 1 | no | — |
| 153 | C3×C4 | — | 3 | 3 | no | — |
| 154 | C3×C4 | — | 3 | 3 | no | — |
| 155 | Dic3 | — | 6 | 4 | no | — |
| 156 | C9 | — | 16 | 1 | no | Sylow |
| 157 | C23 | — | 3 | 1 | no | — |
| 158 | C23 | — | 3 | 3 | no | — |
| 159 | C23 | — | 18 | 6 | no | — |
| 160 | C23 | — | 36 | 3 | no | — |
| 161 | C23 | — | 36 | 1 | no | — |
| 162 | C23 | C2×(C22⋊C9)⋊C2 | 1 | 3 | yes | — |
| 163 | Q8 | — | 9 | 4 | no | — |
| 164 | Q8 | — | 18 | 6 | no | — |
| 165 | D4 | — | 9 | 4 | no | — |
| 166 | D4 | — | 18 | 6 | no | — |
| 167 | D4 | — | 18 | 12 | no | — |
| 168 | D4 | — | 18 | 4 | no | — |
| 169 | C2×C4 | — | 3 | 3 | no | — |
| 170 | C2×C4 | — | 3 | 3 | no | — |
| 171 | C2×C4 | — | 3 | 3 | no | — |
| 172 | C2×C4 | — | 3 | 3 | no | — |
| 173 | C2×C4 | — | 3 | 3 | no | — |
| 174 | C2×C4 | — | 3 | 3 | no | — |
| 175 | C2×C4 | — | 3 | 6 | no | — |
| 176 | C2×C4 | — | 3 | 1 | no | — |
| 177 | C2×C4 | — | 3 | 3 | no | — |
| 178 | C2×C4 | — | 18 | 4 | no | — |
| 179 | C2×C4 | — | 18 | 12 | no | — |
| 180 | C2×C4 | — | 18 | 6 | no | — |
| 181 | C8 | — | 18 | 4 | no | — |
| 182 | C2×C3 | — | 3 | 1 | no | — |
| 183 | C2×C3 | — | 3 | 3 | no | — |
| 184 | C2×C3 | C2×#857 | 1 | 3 | yes | — |
| 185 | D3 | — | 12 | 4 | no | — |
| 186 | C22 | — | 3 | 3 | no | — |
| 187 | C22 | — | 3 | 3 | no | — |
| 188 | C22 | — | 3 | 3 | no | — |
| 189 | C22 | — | 6 | 1 | no | — |
| 190 | C22 | — | 18 | 4 | no | — |
| 191 | C22 | — | 36 | 6 | no | — |
| 192 | C22 | — | 36 | 6 | no | — |
| 193 | C22 | #63419 | 1 | 1 | yes | characteristic, direct product factor |
| 194 | C22 | C22×(C22⋊C9)⋊C2 | 1 | 1 | yes | characteristic |
| 195 | C4 | — | 3 | 1 | no | — |
| 196 | C4 | — | 3 | 3 | no | — |
| 197 | C4 | — | 3 | 1 | no | — |
| 198 | C4 | — | 3 | 3 | no | — |
| 199 | C4 | — | 18 | 4 | no | — |
| 200 | C3 | C22×#857 | 1 | 1 | yes | characteristic |
| 201 | C2 | — | 3 | 1 | no | — |
| 202 | C2 | — | 3 | 3 | no | — |
| 203 | C2 | — | 36 | 4 | no | — |
| 204 | C2 | C2×#63419 | 1 | 3 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #63419 |
| commutator, cosocle, hypocenter | C42⋊C9 | C23 |
| Frattini | C22×C3 | C22×S4 |
| Fitting | C22×C3×C42 | D3 |
| socle | C24×C3 | S4 |
| LCS | G ▷ C42⋊C9 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C42⋊C9 | C2 | semidirect |
| 2. C2×#63419 | C2 | direct (N × H) |
| 5. C2×C42⋊C9 | C22 | semidirect |
| 6. #63419 | C22 | direct (N × H) |
| 14. C42⋊C9 | C23 | semidirect |
| 37. C22×C42 | D9 | semidirect |
| 84. C2×C42 | C2×D9 | semidirect |
| 141. C42 | C22×D9 | semidirect |
| 193. C22 | #63419 | direct (N × H) |
| 204. C2 | C2×#63419 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C42⋊C9 | 2 |
| C2×#63419 | 2 |
| C22×#805 | 3 |
| C22×(C22⋊C9)⋊C2 | 4 |