Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
217 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 35 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 | C3×#576⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5752⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C3⋊(#504⋊C3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #84007 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #3415⋊C3 | — | 3 | 1 | no | maximal |
| 6 | C3×#994 | — | 2 | 1 | no | — |
| 7 | C3⋊#994 | — | 2 | 1 | no | — |
| 8 | C3×#131⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | C3×#829 | — | 4 | 1 | no | maximal |
| 10 | C3×#576 | C2×C3 | 1 | 1 | yes | characteristic |
| 11 | #5752 | C2×C3 | 1 | 1 | yes | characteristic |
| 12 | #576⋊C3 | — | 2 | 1 | no | — |
| 13 | #576⋊C3 | — | 3 | 1 | no | — |
| 14 | #576⋊C3 | D3 | 1 | 1 | yes | characteristic, complemented |
| 15 | #5194 | — | 3 | 1 | no | — |
| 16 | #5193 | — | 3 | 1 | no | — |
| 17 | #5147 | — | 3 | 1 | no | — |
| 18 | C2×#826 | — | 3 | 1 | no | — |
| 19 | #5140 | C2×C3 | 1 | 1 | yes | characteristic |
| 20 | #504⋊C3 | — | 3 | 1 | no | — |
| 21 | C3×#41⋊C3 | — | 4 | 1 | no | — |
| 22 | C3×#41⋊C3 | D4 | 1 | 1 | yes | characteristic |
| 23 | C3⋊(#41⋊C3) | — | 4 | 1 | no | — |
| 24 | C32×Q16 | — | 4 | 1 | no | — |
| 25 | C3×Dic12 | — | 4 | 1 | no | — |
| 26 | C3×C3⋊C16 | — | 4 | 1 | no | — |
| 27 | #3415 | — | 3 | 1 | no | Sylow |
| 28 | C3×H(5,𝔽2) | — | 2 | 1 | no | — |
| 29 | #1009 | — | 2 | 1 | no | — |
| 30 | #994 | — | 2 | 2 | no | — |
| 31 | #994 | — | 2 | 1 | no | — |
| 32 | #994 | — | 6 | 1 | no | — |
| 33 | C3×AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 34 | C3×#135 | — | 3 | 1 | no | — |
| 35 | C2×C3×D8 | — | 3 | 1 | no | — |
| 36 | C3×#131 | C22×C3 | 1 | 1 | yes | characteristic |
| 37 | #908 | — | 6 | 1 | no | — |
| 38 | #904 | — | 3 | 1 | no | — |
| 39 | C2×D24 | — | 3 | 1 | no | — |
| 40 | #131⋊C3 | — | 2 | 1 | no | — |
| 41 | #131⋊C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 42 | C3×Q32 | — | 12 | 1 | no | — |
| 43 | #829 | A4 | 1 | 1 | yes | characteristic |
| 44 | #828 | — | 3 | 1 | no | — |
| 45 | #827 | — | 3 | 1 | no | — |
| 46 | #826 | — | 3 | 2 | no | — |
| 47 | #826 | — | 3 | 1 | no | — |
| 48 | #812 | — | 3 | 1 | no | — |
| 49 | C2×C3⋊C16 | — | 3 | 1 | no | — |
| 50 | C32×Q8 | — | 8 | 1 | no | — |
| 51 | C3×Dic6 | — | 8 | 1 | no | — |
| 52 | C3×SL(2,𝔽3) | D8 | 1 | 1 | yes | characteristic |
| 53 | C32×C8 | — | 4 | 1 | no | — |
| 54 | #576 | — | 3 | 1 | no | — |
| 55 | #576 | C3×D3 | 1 | 1 | yes | characteristic |
| 56 | #509 | — | 9 | 1 | no | — |
| 57 | #509 | — | 9 | 1 | no | — |
| 58 | #508 | — | 9 | 1 | no | — |
| 59 | C2×D16 | — | 9 | 1 | no | — |
| 60 | #504 | — | 3 | 1 | no | — |
| 61 | C3×#41 | — | 4 | 1 | no | — |
| 62 | C3×#41 | — | 6 | 1 | no | — |
| 63 | C3×#41 | C3×D4 | 1 | 1 | yes | characteristic |
| 64 | C2×C3×D4 | — | 6 | 1 | no | — |
| 65 | C2×C3×D4 | — | 12 | 1 | no | — |
| 66 | #239 | — | 4 | 1 | no | — |
| 67 | D3×D4 | — | 12 | 1 | no | — |
| 68 | #235 | — | 6 | 1 | no | — |
| 69 | C2×D12 | — | 6 | 1 | no | — |
| 70 | #41⋊C3 | — | 2 | 1 | no | — |
| 71 | #41⋊C3 | — | 4 | 2 | no | — |
| 72 | #41⋊C3 | — | 4 | 1 | no | — |
| 73 | #41⋊C3 | — | 12 | 1 | no | — |
| 74 | #41⋊C3 | #67 | 1 | 1 | yes | characteristic |
| 75 | C3×Q16 | — | 4 | 1 | no | — |
| 76 | C3×Q16 | — | 8 | 1 | no | — |
| 77 | C3×Q16 | — | 12 | 1 | no | — |
| 78 | C3×Q16 | C2×A4 | 1 | 1 | yes | characteristic |
| 79 | C3×SD16 | — | 6 | 1 | no | — |
| 80 | C3×D8 | — | 3 | 1 | no | — |
| 81 | C3×D8 | — | 6 | 1 | no | — |
| 82 | C3×M16 | — | 3 | 1 | no | — |
| 83 | C2×C3×C8 | — | 3 | 1 | no | — |
| 84 | Dic12 | C2×A4 | 1 | 1 | yes | characteristic |
| 85 | D24 | — | 3 | 1 | no | — |
| 86 | D24 | — | 6 | 1 | no | — |
| 87 | #204 | — | 6 | 1 | no | — |
| 88 | C3×C16 | — | 12 | 1 | no | — |
| 89 | C3⋊C16 | — | 3 | 1 | no | — |
| 90 | C3⋊C16 | C2×A4 | 1 | 1 | yes | characteristic |
| 91 | C32×C4 | — | 4 | 1 | no | — |
| 92 | C32×C4 | — | 16 | 1 | no | — |
| 93 | C3×Dic3 | — | 16 | 1 | no | — |
| 94 | H(5,𝔽2) | — | 2 | 1 | no | — |
| 95 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 96 | AGL(1,ℤ/8) | — | 6 | 1 | no | — |
| 97 | AGL(1,ℤ/8) | — | 18 | 1 | no | — |
| 98 | #135 | — | 3 | 1 | no | — |
| 99 | #135 | — | 9 | 1 | no | — |
| 100 | C2×D8 | — | 3 | 1 | no | — |
| 101 | C2×D8 | — | 9 | 1 | no | — |
| 102 | #131 | C2×C3×D3 | 1 | 1 | yes | characteristic |
| 103 | Q32 | — | 3 | 1 | no | — |
| 104 | SD32 | — | 9 | 1 | no | — |
| 105 | SD32 | — | 9 | 1 | no | — |
| 106 | D16 | — | 9 | 1 | no | — |
| 107 | D16 | — | 9 | 2 | no | — |
| 108 | M32 | — | 9 | 1 | no | — |
| 109 | C2×C16 | — | 9 | 1 | no | — |
| 110 | C23×C3 | — | 12 | 1 | no | — |
| 111 | C22×D3 | — | 12 | 1 | no | — |
| 112 | C3×Q8 | — | 2 | 1 | no | — |
| 113 | C3×Q8 | — | 8 | 1 | no | — |
| 114 | C3×Q8 | — | 8 | 2 | no | — |
| 115 | C3×Q8 | — | 24 | 1 | no | — |
| 116 | C3×Q8 | C3×D8 | 1 | 1 | yes | characteristic |
| 117 | C3×D4 | — | 3 | 1 | no | — |
| 118 | C3×D4 | — | 6 | 1 | no | — |
| 119 | C3×D4 | — | 12 | 1 | no | — |
| 120 | C3×D4 | — | 12 | 1 | no | — |
| 121 | C2×C3×C4 | — | 3 | 1 | no | — |
| 122 | C2×C3×C4 | — | 12 | 1 | no | — |
| 123 | #67 | — | 12 | 1 | no | — |
| 124 | D12 | — | 6 | 1 | no | — |
| 125 | D12 | — | 12 | 1 | no | — |
| 126 | C4×D3 | — | 12 | 1 | no | — |
| 127 | Dic6 | — | 2 | 1 | no | — |
| 128 | SL(2,𝔽3) | — | 2 | 1 | no | — |
| 129 | SL(2,𝔽3) | #213 | 1 | 1 | yes | characteristic |
| 130 | C3×C8 | — | 3 | 1 | no | — |
| 131 | C3×C8 | — | 4 | 1 | no | — |
| 132 | C3×C8 | — | 8 | 1 | no | — |
| 133 | C3×C8 | C22×A4 | 1 | 1 | yes | characteristic |
| 134 | C2×C32 | — | 4 | 1 | no | — |
| 135 | #41 | — | 4 | 1 | no | — |
| 136 | #41 | — | 6 | 1 | no | — |
| 137 | #41 | — | 12 | 1 | no | — |
| 138 | #41 | — | 18 | 1 | no | — |
| 139 | #41 | C3×#67 | 1 | 1 | yes | characteristic |
| 140 | C2×D4 | — | 6 | 1 | no | — |
| 141 | C2×D4 | — | 12 | 1 | no | — |
| 142 | C2×D4 | — | 18 | 1 | no | — |
| 143 | C2×D4 | — | 36 | 1 | no | — |
| 144 | Q16 | — | 3 | 1 | no | — |
| 145 | Q16 | D3×A4 | 1 | 1 | yes | characteristic |
| 146 | SD16 | — | 6 | 1 | no | — |
| 147 | SD16 | — | 18 | 1 | no | — |
| 148 | D8 | — | 3 | 1 | no | — |
| 149 | D8 | — | 6 | 1 | no | — |
| 150 | D8 | — | 9 | 1 | no | — |
| 151 | D8 | — | 18 | 1 | no | — |
| 152 | M16 | — | 3 | 1 | no | — |
| 153 | C2×C8 | — | 3 | 1 | no | — |
| 154 | C16 | — | 3 | 1 | no | — |
| 155 | C16 | — | 9 | 1 | no | — |
| 156 | C22×C3 | — | 3 | 1 | no | — |
| 157 | C22×C3 | — | 12 | 1 | no | — |
| 158 | C22×C3 | — | 48 | 1 | no | — |
| 159 | C2×D3 | — | 12 | 1 | no | — |
| 160 | C2×D3 | — | 48 | 1 | no | — |
| 161 | C3×C4 | — | 3 | 1 | no | — |
| 162 | C3×C4 | — | 4 | 1 | no | — |
| 163 | C3×C4 | — | 4 | 1 | no | — |
| 164 | C3×C4 | — | 8 | 1 | no | — |
| 165 | C3×C4 | — | 16 | 1 | no | — |
| 166 | C3×C4 | — | 16 | 2 | no | — |
| 167 | C3×C4 | — | 48 | 1 | no | — |
| 168 | C3×C4 | D4×A4 | 1 | 1 | yes | characteristic |
| 169 | Dic3 | — | 4 | 1 | no | — |
| 170 | C32 | — | 4 | 1 | no | Sylow |
| 171 | C23 | — | 12 | 1 | no | — |
| 172 | C23 | — | 36 | 1 | no | — |
| 173 | Q8 | — | 2 | 1 | no | — |
| 174 | Q8 | — | 6 | 1 | no | — |
| 175 | Q8 | C3×#213 | 1 | 1 | yes | characteristic |
| 176 | D4 | — | 3 | 1 | no | — |
| 177 | D4 | — | 6 | 1 | no | — |
| 178 | D4 | — | 12 | 1 | no | — |
| 179 | D4 | — | 12 | 1 | no | — |
| 180 | D4 | — | 18 | 1 | no | — |
| 181 | D4 | — | 36 | 1 | no | — |
| 182 | D4 | — | 36 | 1 | no | — |
| 183 | C2×C4 | — | 3 | 1 | no | — |
| 184 | C2×C4 | — | 12 | 1 | no | — |
| 185 | C2×C4 | — | 36 | 1 | no | — |
| 186 | C8 | — | 3 | 1 | no | — |
| 187 | C8 | C2×D3×A4 | 1 | 1 | yes | characteristic |
| 188 | C2×C3 | — | 4 | 1 | no | — |
| 189 | C2×C3 | — | 6 | 1 | no | — |
| 190 | C2×C3 | — | 8 | 1 | no | — |
| 191 | C2×C3 | — | 24 | 1 | no | — |
| 192 | C2×C3 | A4×D8 | 1 | 1 | yes | characteristic |
| 193 | D3 | — | 24 | 1 | no | — |
| 194 | C22 | — | 3 | 1 | no | — |
| 195 | C22 | — | 12 | 1 | no | — |
| 196 | C22 | — | 36 | 1 | no | — |
| 197 | C22 | — | 48 | 1 | no | — |
| 198 | C22 | — | 144 | 1 | no | — |
| 199 | C4 | — | 3 | 1 | no | — |
| 200 | C4 | — | 4 | 1 | no | — |
| 201 | C4 | — | 12 | 1 | no | — |
| 202 | C4 | A4×#67 | 1 | 1 | yes | characteristic |
| 203 | C3 | — | 4 | 1 | no | — |
| 204 | C3 | — | 8 | 1 | no | — |
| 205 | C3 | #3415⋊C3 | 1 | 1 | yes | characteristic, complemented |
| 206 | C2 | — | 6 | 1 | no | — |
| 207 | C2 | — | 24 | 1 | no | — |
| 208 | C2 | — | 72 | 1 | no | — |
| 209 | C2 | A4×#213 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#576⋊C3 | C2 | semidirect |
| 2. #5752⋊C3 | C2 | semidirect |
| 3. C3⋊(#504⋊C3) | C2 | semidirect |
| 4. #84007 | C3 | semidirect |
| 14. #576⋊C3 | D3 | semidirect |
| 205. C3 | #3415⋊C3 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#576⋊C3 | 2 |
| #5752⋊C3 | 2 |
| C3⋊(#504⋊C3) | 2 |
| #84007 | 3 |
| #3415⋊C3 | 3 |
| C3×#829 | 4 |