Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
5133 conjugacy classes of subgroups; only the 310 normal subgroups are shown.
This group has 5133 conjugacy classes of subgroups — too many to draw. Only its 310 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 | C25×D9 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#63515 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#63444 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | C9×#379 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #10588635 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #73861 | — | 3 | 1 | no | maximal |
| 7 | C25×C9 | C22 | 1 | 1 | yes | characteristic |
| 8 | C24×D9 | — | 2 | 4 | no | — |
| 9 | C24×D9 | — | 2 | 24 | no | — |
| 10 | C24×D9 | C22 | 1 | 6 | yes | — |
| 11 | C22×#611 | C22 | 1 | 3 | yes | complemented |
| 12 | C22×D36 | C22 | 1 | 4 | yes | complemented |
| 13 | C2×C9×C22⋊C4 | C22 | 1 | 6 | yes | complemented |
| 14 | #63515 | — | 2 | 4 | no | — |
| 15 | C2×#3676 | C22 | 1 | 3 | yes | complemented |
| 16 | C2×(C2×D9)⋊C4 | C22 | 1 | 12 | yes | complemented |
| 17 | #63444 | — | 2 | 24 | no | — |
| 18 | C25×D3 | — | 3 | 1 | no | — |
| 19 | C2×#953 | — | 3 | 1 | no | — |
| 20 | C2×#882 | — | 3 | 6 | no | — |
| 21 | C3×#379 | D3 | 1 | 1 | yes | characteristic |
| 22 | #4955 | — | 3 | 6 | no | — |
| 23 | C24×C9 | — | 2 | 12 | no | — |
| 24 | C24×C9 | D4 | 1 | 4 | yes | complemented |
| 25 | C24×C9 | C23 | 1 | 3 | yes | — |
| 26 | C23×D9 | — | 2 | 96 | no | — |
| 27 | C23×D9 | — | 2 | 128 | no | — |
| 28 | C23×D9 | — | 2 | 12 | no | — |
| 29 | C23×D9 | — | 2 | 12 | no | — |
| 30 | C23×D9 | — | 2 | 48 | no | — |
| 31 | C23×D9 | D4 | 1 | 24 | yes | complemented |
| 32 | C23×D9 | C23 | 1 | 4 | yes | — |
| 33 | C23×D9 | C23 | 1 | 1 | yes | characteristic, complemented |
| 34 | C22×C4×C9 | C23 | 1 | 4 | yes | complemented |
| 35 | C2×#611 | — | 2 | 6 | no | — |
| 36 | C2×#611 | — | 2 | 12 | no | — |
| 37 | C22×Dic9 | C23 | 1 | 3 | yes | complemented |
| 38 | C2×D36 | — | 2 | 24 | no | — |
| 39 | C9×C22⋊C4 | — | 2 | 12 | no | — |
| 40 | #3676 | — | 2 | 6 | no | — |
| 41 | (C2×D9)⋊C4 | — | 2 | 24 | no | — |
| 42 | #2846 | — | 9 | 1 | no | Sylow |
| 43 | C25×C3 | C2×D3 | 1 | 1 | yes | characteristic |
| 44 | C24×D3 | — | 3 | 6 | no | — |
| 45 | C24×D3 | — | 6 | 24 | no | — |
| 46 | C24×D3 | — | 6 | 4 | no | — |
| 47 | C22×#67 | — | 3 | 3 | no | — |
| 48 | C22×D12 | — | 3 | 4 | no | — |
| 49 | C2×C3×C22⋊C4 | C2×D3 | 1 | 6 | yes | — |
| 50 | #953 | — | 6 | 4 | no | — |
| 51 | C2×#217 | — | 3 | 3 | no | — |
| 52 | C2×(C2×D3)⋊C4 | — | 3 | 12 | no | — |
| 53 | #882 | — | 6 | 24 | no | — |
| 54 | C23×C9 | — | 2 | 32 | no | — |
| 55 | C23×C9 | — | 2 | 24 | no | — |
| 56 | C23×C9 | — | 2 | 12 | no | — |
| 57 | C23×C9 | C2×D4 | 1 | 6 | yes | — |
| 58 | C23×C9 | C2×D4 | 1 | 12 | yes | — |
| 59 | C23×C9 | C24 | 1 | 1 | yes | characteristic |
| 60 | C22×D9 | — | 2 | 256 | no | — |
| 61 | C22×D9 | — | 2 | 96 | no | — |
| 62 | C22×D9 | — | 2 | 96 | no | — |
| 63 | C22×D9 | — | 2 | 48 | no | — |
| 64 | C22×D9 | — | 2 | 4 | no | — |
| 65 | C22×D9 | — | 2 | 12 | no | — |
| 66 | C22×D9 | — | 2 | 96 | no | — |
| 67 | C22×D9 | — | 2 | 6 | no | — |
| 68 | C22×D9 | — | 2 | 1 | no | — |
| 69 | C22×D9 | C2×D4 | 1 | 24 | yes | — |
| 70 | C2×C4×C9 | — | 2 | 12 | no | — |
| 71 | #611 | — | 4 | 12 | no | — |
| 72 | C2×Dic9 | — | 2 | 6 | no | — |
| 73 | C2×Dic9 | — | 2 | 3 | no | — |
| 74 | D36 | — | 4 | 16 | no | — |
| 75 | C26 | — | 9 | 1 | no | — |
| 76 | C2×#120 | — | 9 | 1 | no | — |
| 77 | C2×#120 | — | 9 | 6 | no | — |
| 78 | #379 | — | 9 | 6 | no | — |
| 79 | #379 | D9 | 1 | 1 | yes | characteristic, complemented |
| 80 | C24×C3 | — | 2 | 12 | no | — |
| 81 | C24×C3 | D12 | 1 | 4 | yes | — |
| 82 | C24×C3 | C22×D3 | 1 | 3 | yes | — |
| 83 | C23×D3 | — | 3 | 4 | no | — |
| 84 | C23×D3 | — | 3 | 24 | no | — |
| 85 | C23×D3 | — | 3 | 1 | no | — |
| 86 | C23×D3 | — | 6 | 48 | no | — |
| 87 | C23×D3 | — | 6 | 128 | no | — |
| 88 | C23×D3 | — | 6 | 96 | no | — |
| 89 | C23×D3 | — | 6 | 12 | no | — |
| 90 | C23×D3 | — | 6 | 12 | no | — |
| 91 | C22×C3×C4 | C22×D3 | 1 | 4 | yes | — |
| 92 | C2×#67 | — | 6 | 6 | no | — |
| 93 | C2×#67 | — | 6 | 12 | no | — |
| 94 | C22×Dic3 | — | 3 | 3 | no | — |
| 95 | C2×D12 | — | 6 | 24 | no | — |
| 96 | C3×C22⋊C4 | — | 2 | 12 | no | — |
| 97 | #217 | — | 6 | 6 | no | — |
| 98 | (C2×D3)⋊C4 | — | 6 | 24 | no | — |
| 99 | C22×C9 | — | 2 | 32 | no | — |
| 100 | C22×C9 | — | 2 | 12 | no | — |
| 101 | C22×C9 | — | 2 | 24 | no | — |
| 102 | C22×C9 | #120 | 1 | 12 | yes | complemented |
| 103 | C22×C9 | C22×D4 | 1 | 1 | yes | characteristic |
| 104 | C22×C9 | C22×D4 | 1 | 6 | yes | — |
| 105 | C2×D9 | — | 2 | 192 | no | — |
| 106 | C2×D9 | — | 2 | 24 | no | — |
| 107 | C2×D9 | — | 2 | 24 | no | — |
| 108 | C2×D9 | — | 4 | 4 | no | — |
| 109 | C2×D9 | — | 4 | 3 | no | — |
| 110 | C2×D9 | #120 | 1 | 16 | yes | complemented |
| 111 | C4×C9 | — | 4 | 4 | no | — |
| 112 | Dic9 | — | 4 | 3 | no | — |
| 113 | C25 | — | 9 | 6 | no | — |
| 114 | C25 | — | 18 | 24 | no | — |
| 115 | C25 | — | 18 | 4 | no | — |
| 116 | C25 | C2×D9 | 1 | 1 | yes | characteristic |
| 117 | C22×D4 | — | 9 | 3 | no | — |
| 118 | C22×D4 | — | 9 | 4 | no | — |
| 119 | #120 | — | 18 | 24 | no | — |
| 120 | #120 | — | 18 | 4 | no | — |
| 121 | C2×C22⋊C4 | — | 9 | 3 | no | — |
| 122 | C2×C22⋊C4 | — | 9 | 12 | no | — |
| 123 | C2×C22⋊C4 | C2×D9 | 1 | 6 | yes | complemented |
| 124 | C23×C3 | — | 2 | 32 | no | — |
| 125 | C23×C3 | — | 2 | 24 | no | — |
| 126 | C23×C3 | — | 2 | 12 | no | — |
| 127 | C23×C3 | C2×D12 | 1 | 6 | yes | — |
| 128 | C23×C3 | D3×D4 | 1 | 12 | yes | — |
| 129 | C23×C3 | C23×D3 | 1 | 1 | yes | characteristic |
| 130 | C22×D3 | — | 3 | 24 | no | — |
| 131 | C22×D3 | — | 6 | 96 | no | — |
| 132 | C22×D3 | — | 6 | 256 | no | — |
| 133 | C22×D3 | — | 6 | 96 | no | — |
| 134 | C22×D3 | — | 6 | 96 | no | — |
| 135 | C22×D3 | — | 6 | 48 | no | — |
| 136 | C22×D3 | — | 6 | 4 | no | — |
| 137 | C22×D3 | — | 6 | 12 | no | — |
| 138 | C22×D3 | — | 6 | 1 | no | — |
| 139 | C22×D3 | — | 6 | 6 | no | — |
| 140 | C2×C3×C4 | — | 2 | 12 | no | — |
| 141 | #67 | — | 12 | 12 | no | — |
| 142 | C2×Dic3 | — | 6 | 3 | no | — |
| 143 | C2×Dic3 | — | 6 | 6 | no | — |
| 144 | D12 | — | 12 | 16 | no | — |
| 145 | C2×C9 | — | 2 | 12 | no | — |
| 146 | C2×C9 | C2×#120 | 1 | 3 | yes | — |
| 147 | C2×C9 | C2×#120 | 1 | 4 | yes | — |
| 148 | D9 | — | 2 | 16 | no | — |
| 149 | D9 | — | 8 | 1 | no | — |
| 150 | C24 | — | 2 | 12 | no | — |
| 151 | C24 | — | 9 | 4 | no | — |
| 152 | C24 | — | 9 | 24 | no | — |
| 153 | C24 | — | 9 | 1 | no | — |
| 154 | C24 | — | 18 | 48 | no | — |
| 155 | C24 | — | 18 | 128 | no | — |
| 156 | C24 | — | 18 | 96 | no | — |
| 157 | C24 | — | 18 | 12 | no | — |
| 158 | C24 | — | 18 | 12 | no | — |
| 159 | C24 | D36 | 1 | 4 | yes | complemented |
| 160 | C24 | C22×D9 | 1 | 3 | yes | — |
| 161 | C2×D4 | — | 18 | 24 | no | — |
| 162 | C2×D4 | — | 18 | 6 | no | — |
| 163 | C2×D4 | — | 18 | 12 | no | — |
| 164 | C22×C4 | — | 9 | 3 | no | — |
| 165 | C22×C4 | C22×D9 | 1 | 4 | yes | complemented |
| 166 | C22⋊C4 | — | 2 | 12 | no | — |
| 167 | C22⋊C4 | — | 18 | 24 | no | — |
| 168 | C22⋊C4 | — | 18 | 6 | no | — |
| 169 | C22×C3 | — | 2 | 32 | no | — |
| 170 | C22×C3 | — | 2 | 12 | no | — |
| 171 | C22×C3 | — | 2 | 24 | no | — |
| 172 | C22×C3 | #882 | 1 | 12 | yes | — |
| 173 | C22×C3 | C22×D12 | 1 | 1 | yes | characteristic |
| 174 | C22×C3 | C2×D3×D4 | 1 | 6 | yes | — |
| 175 | C2×D3 | — | 3 | 16 | no | — |
| 176 | C2×D3 | — | 6 | 192 | no | — |
| 177 | C2×D3 | — | 6 | 24 | no | — |
| 178 | C2×D3 | — | 6 | 24 | no | — |
| 179 | C2×D3 | — | 12 | 3 | no | — |
| 180 | C2×D3 | — | 12 | 4 | no | — |
| 181 | C3×C4 | — | 4 | 4 | no | — |
| 182 | Dic3 | — | 12 | 3 | no | — |
| 183 | C9 | #2846 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 184 | C23 | — | 2 | 32 | no | — |
| 185 | C23 | — | 2 | 24 | no | — |
| 186 | C23 | — | 2 | 12 | no | — |
| 187 | C23 | — | 9 | 24 | no | — |
| 188 | C23 | — | 18 | 96 | no | — |
| 189 | C23 | — | 18 | 256 | no | — |
| 190 | C23 | — | 18 | 96 | no | — |
| 191 | C23 | — | 18 | 96 | no | — |
| 192 | C23 | — | 18 | 48 | no | — |
| 193 | C23 | — | 18 | 4 | no | — |
| 194 | C23 | — | 18 | 12 | no | — |
| 195 | C23 | — | 18 | 1 | no | — |
| 196 | C23 | — | 18 | 6 | no | — |
| 197 | C23 | C2×D36 | 1 | 6 | yes | — |
| 198 | C23 | D4×D9 | 1 | 12 | yes | — |
| 199 | C23 | C23×D9 | 1 | 1 | yes | characteristic |
| 200 | D4 | — | 36 | 16 | no | — |
| 201 | D4 | — | 36 | 12 | no | — |
| 202 | C2×C4 | — | 2 | 12 | no | — |
| 203 | C2×C4 | — | 18 | 3 | no | — |
| 204 | C2×C4 | — | 18 | 6 | no | — |
| 205 | C2×C3 | — | 2 | 12 | no | — |
| 206 | C2×C3 | C2×#882 | 1 | 3 | yes | — |
| 207 | C2×C3 | D3×#120 | 1 | 4 | yes | — |
| 208 | D3 | — | 6 | 16 | no | — |
| 209 | D3 | — | 24 | 1 | no | — |
| 210 | C22 | — | 2 | 32 | no | — |
| 211 | C22 | — | 2 | 12 | no | — |
| 212 | C22 | — | 2 | 24 | no | — |
| 213 | C22 | — | 9 | 16 | no | — |
| 214 | C22 | — | 18 | 192 | no | — |
| 215 | C22 | — | 18 | 24 | no | — |
| 216 | C22 | — | 18 | 24 | no | — |
| 217 | C22 | — | 36 | 3 | no | — |
| 218 | C22 | — | 36 | 4 | no | — |
| 219 | C22 | #63444 | 1 | 12 | yes | complemented |
| 220 | C22 | C22×D36 | 1 | 1 | yes | characteristic |
| 221 | C22 | C2×D4×D9 | 1 | 6 | yes | — |
| 222 | C4 | — | 4 | 4 | no | — |
| 223 | C4 | — | 36 | 3 | no | — |
| 224 | C3 | #73861 | 1 | 1 | yes | characteristic |
| 225 | C2 | — | 2 | 12 | no | — |
| 226 | C2 | — | 18 | 16 | no | — |
| 227 | C2 | — | 72 | 1 | no | — |
| 228 | C2 | C2×#63444 | 1 | 3 | yes | — |
| 229 | C2 | D9×#120 | 1 | 4 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D9 |
| commutator, cosocle | C23×C9 | C24 |
| Frattini, socle | C23×C3 | C23×D3 |
| Fitting | C9×#379 | C2 |
| hypercenter | #379 | D9 |
| hypocenter | C9 | #2846 |
| LCS | G ▷ C23×C9 ▷ C9 | |
| UCS | C1 ◁ C23 ◁ #379 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×D9 | C2 | semidirect |
| 2. C2×#63515 | C2 | semidirect |
| 3. C2×#63444 | C2 | semidirect |
| 4. C9×#379 | C2 | semidirect |
| 5. #10588635 | C2 | semidirect |
| 11. C22×#611 | C22 | semidirect |
| 12. C22×D36 | C22 | semidirect |
| 13. C2×C9×C22⋊C4 | C22 | semidirect |
| 15. C2×#3676 | C22 | semidirect |
| 16. C2×(C2×D9)⋊C4 | C22 | semidirect |
| 24. C24×C9 | D4 | semidirect |
| 31. C23×D9 | D4 | semidirect |
| 33. C23×D9 | C23 | semidirect |
| 34. C22×C4×C9 | C23 | semidirect |
| 37. C22×Dic9 | C23 | semidirect |
| 79. #379 | D9 | semidirect |
| 102. C22×C9 | #120 | semidirect |
| 110. C2×D9 | #120 | semidirect |
| 123. C2×C22⋊C4 | C2×D9 | semidirect |
| 159. C24 | D36 | semidirect |
| 165. C22×C4 | C22×D9 | semidirect |
| 183. C9 | #2846 | semidirect |
| 219. C22 | #63444 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×D9 | 2 |
| C2×#63515 | 2 |
| C2×#63444 | 2 |
| C9×#379 | 2 |
| #10588635 | 2 |
| #73861 | 3 |