Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
1158 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 216 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 | C9×#585 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10589932 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #10588924 | C2 | 1 | 8 | yes | maximal, complemented |
| 4 | #10588920 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #84126 | — | 3 | 1 | no | maximal |
| 6 | C9×2−1+4 | — | 2 | 3 | no | — |
| 7 | C9×H(5,𝔽2) | — | 2 | 3 | no | — |
| 8 | C9×H(5,𝔽2) | C22 | 1 | 4 | yes | — |
| 9 | C2×C9×#41 | — | 2 | 6 | no | — |
| 10 | C2×C9×#41 | C22 | 1 | 3 | yes | — |
| 11 | C22×D36 | C22 | 1 | 1 | yes | characteristic, complemented |
| 12 | #63509 | — | 2 | 3 | no | — |
| 13 | #63502 | — | 2 | 3 | no | — |
| 14 | #63499 | C22 | 1 | 12 | yes | complemented |
| 15 | #63492 | — | 2 | 6 | no | — |
| 16 | #63488 | C22 | 1 | 3 | yes | complemented |
| 17 | #63393 | C22 | 1 | 12 | yes | complemented |
| 18 | C3×#585 | D3 | 1 | 1 | yes | characteristic |
| 19 | #5830 | — | 3 | 3 | no | — |
| 20 | #5244 | — | 3 | 8 | no | — |
| 21 | #5240 | — | 3 | 3 | no | — |
| 22 | C23×D9 | C23 | 1 | 2 | yes | — |
| 23 | C9×#41 | — | 2 | 4 | no | — |
| 24 | C9×#41 | — | 2 | 12 | no | — |
| 25 | C9×#41 | — | 2 | 6 | no | — |
| 26 | C9×#41 | — | 2 | 12 | no | — |
| 27 | C9×#41 | — | 2 | 6 | no | — |
| 28 | C2×Q8×C9 | — | 2 | 6 | no | — |
| 29 | C2×Q8×C9 | D4 | 1 | 3 | yes | complemented |
| 30 | C2×D4×C9 | — | 2 | 6 | no | — |
| 31 | C2×D4×C9 | — | 2 | 12 | no | — |
| 32 | C2×D4×C9 | D4 | 1 | 3 | yes | complemented |
| 33 | C2×D4×C9 | C23 | 1 | 6 | yes | — |
| 34 | C22×C4×C9 | — | 2 | 4 | no | — |
| 35 | C22×C4×C9 | D4 | 1 | 6 | yes | complemented |
| 36 | C22×C4×C9 | C23 | 1 | 1 | yes | characteristic |
| 37 | C2×#611 | — | 2 | 12 | no | — |
| 38 | C2×D36 | — | 2 | 3 | no | — |
| 39 | C2×D36 | — | 2 | 3 | no | — |
| 40 | #3676 | C23 | 1 | 6 | yes | — |
| 41 | (C2×D9)⋊C4 | — | 2 | 12 | no | — |
| 42 | C4⋊Dic9 | — | 2 | 3 | no | — |
| 43 | C4×Dic9 | — | 2 | 3 | no | — |
| 44 | #3027 | — | 9 | 1 | no | Sylow |
| 45 | C3×2−1+4 | — | 2 | 3 | no | — |
| 46 | C3×H(5,𝔽2) | — | 2 | 3 | no | — |
| 47 | C3×H(5,𝔽2) | C2×D3 | 1 | 4 | yes | — |
| 48 | C2×C3×#41 | — | 2 | 6 | no | — |
| 49 | C2×C3×#41 | C2×D3 | 1 | 3 | yes | — |
| 50 | C22×D12 | — | 3 | 1 | no | — |
| 51 | #947 | — | 6 | 3 | no | — |
| 52 | #940 | — | 6 | 3 | no | — |
| 53 | #937 | — | 3 | 12 | no | — |
| 54 | #930 | — | 6 | 6 | no | — |
| 55 | #926 | — | 3 | 3 | no | — |
| 56 | #834 | — | 3 | 12 | no | — |
| 57 | C23×C9 | — | 2 | 4 | no | — |
| 58 | C23×C9 | C2×D4 | 1 | 6 | yes | — |
| 59 | C23×C9 | C24 | 1 | 1 | yes | characteristic |
| 60 | C22×D9 | — | 2 | 6 | no | — |
| 61 | C22×D9 | — | 8 | 2 | no | — |
| 62 | Q8×C9 | — | 2 | 4 | no | — |
| 63 | Q8×C9 | — | 2 | 6 | no | — |
| 64 | D4×C9 | — | 2 | 12 | no | — |
| 65 | D4×C9 | — | 2 | 12 | no | — |
| 66 | D4×C9 | — | 2 | 6 | no | — |
| 67 | C2×C4×C9 | — | 2 | 12 | no | — |
| 68 | C2×C4×C9 | — | 2 | 6 | no | — |
| 69 | C2×C4×C9 | — | 2 | 6 | no | — |
| 70 | C2×C4×C9 | C2×D4 | 1 | 6 | yes | — |
| 71 | C2×C4×C9 | C2×D4 | 1 | 3 | yes | — |
| 72 | C2×C4×C9 | C2×D4 | 1 | 3 | yes | — |
| 73 | #611 | — | 8 | 12 | no | — |
| 74 | C2×Dic9 | — | 2 | 6 | no | — |
| 75 | D36 | — | 4 | 1 | no | — |
| 76 | D36 | — | 4 | 3 | no | — |
| 77 | #585 | D9 | 1 | 1 | yes | characteristic, complemented |
| 78 | #535 | — | 9 | 3 | no | — |
| 79 | B(4,𝔽2) | — | 9 | 8 | no | — |
| 80 | #410 | — | 9 | 3 | no | — |
| 81 | C23×D3 | — | 3 | 2 | no | — |
| 82 | C3×#41 | — | 2 | 4 | no | — |
| 83 | C3×#41 | — | 2 | 12 | no | — |
| 84 | C3×#41 | — | 2 | 6 | no | — |
| 85 | C3×#41 | — | 2 | 12 | no | — |
| 86 | C3×#41 | — | 2 | 6 | no | — |
| 87 | C2×C3×Q8 | — | 2 | 6 | no | — |
| 88 | C2×C3×Q8 | #67 | 1 | 3 | yes | — |
| 89 | C2×C3×D4 | — | 2 | 6 | no | — |
| 90 | C2×C3×D4 | — | 2 | 12 | no | — |
| 91 | C2×C3×D4 | #67 | 1 | 3 | yes | — |
| 92 | C2×C3×D4 | C22×D3 | 1 | 6 | yes | — |
| 93 | C22×C3×C4 | — | 2 | 4 | no | — |
| 94 | C22×C3×C4 | #67 | 1 | 6 | yes | — |
| 95 | C22×C3×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 96 | C2×#67 | — | 6 | 12 | no | — |
| 97 | C2×D12 | — | 6 | 3 | no | — |
| 98 | C2×D12 | — | 6 | 3 | no | — |
| 99 | #217 | — | 3 | 6 | no | — |
| 100 | (C2×D3)⋊C4 | — | 6 | 12 | no | — |
| 101 | C4⋊Dic3 | — | 6 | 3 | no | — |
| 102 | C4×Dic3 | — | 6 | 3 | no | — |
| 103 | C22×C9 | — | 2 | 6 | no | — |
| 104 | C22×C9 | — | 4 | 1 | no | — |
| 105 | C22×C9 | — | 4 | 6 | no | — |
| 106 | C22×C9 | — | 8 | 4 | no | — |
| 107 | C22×C9 | C22×D4 | 1 | 3 | yes | — |
| 108 | C2×D9 | — | 4 | 12 | no | — |
| 109 | C2×D9 | — | 4 | 2 | no | — |
| 110 | C4×C9 | — | 2 | 6 | no | — |
| 111 | C4×C9 | #120 | 1 | 1 | yes | characteristic |
| 112 | C4×C9 | #120 | 1 | 3 | yes | — |
| 113 | Dic9 | — | 4 | 6 | no | — |
| 114 | 2−1+4 | — | 2 | 3 | no | — |
| 115 | H(5,𝔽2) | — | 2 | 3 | no | — |
| 116 | H(5,𝔽2) | C2×D9 | 1 | 4 | yes | — |
| 117 | C2×#41 | — | 2 | 6 | no | — |
| 118 | C2×#41 | C2×D9 | 1 | 3 | yes | — |
| 119 | C22×D4 | — | 9 | 1 | no | — |
| 120 | #127 | — | 18 | 3 | no | — |
| 121 | #124 | — | 18 | 3 | no | — |
| 122 | #121 | — | 18 | 6 | no | — |
| 123 | #120 | — | 9 | 12 | no | — |
| 124 | #117 | — | 9 | 3 | no | — |
| 125 | #99 | — | 9 | 12 | no | — |
| 126 | C23×C3 | — | 2 | 4 | no | — |
| 127 | C23×C3 | C2×#67 | 1 | 6 | yes | — |
| 128 | C23×C3 | C23×D3 | 1 | 1 | yes | characteristic |
| 129 | C22×D3 | — | 6 | 6 | no | — |
| 130 | C22×D3 | — | 24 | 2 | no | — |
| 131 | C3×Q8 | — | 2 | 4 | no | — |
| 132 | C3×Q8 | — | 2 | 6 | no | — |
| 133 | C3×D4 | — | 2 | 12 | no | — |
| 134 | C3×D4 | — | 2 | 12 | no | — |
| 135 | C3×D4 | — | 2 | 6 | no | — |
| 136 | C2×C3×C4 | — | 2 | 12 | no | — |
| 137 | C2×C3×C4 | — | 2 | 6 | no | — |
| 138 | C2×C3×C4 | — | 2 | 6 | no | — |
| 139 | C2×C3×C4 | C2×#67 | 1 | 6 | yes | — |
| 140 | C2×C3×C4 | C2×#67 | 1 | 3 | yes | — |
| 141 | C2×C3×C4 | C2×#67 | 1 | 3 | yes | — |
| 142 | #67 | — | 24 | 12 | no | — |
| 143 | C2×Dic3 | — | 6 | 6 | no | — |
| 144 | D12 | — | 12 | 1 | no | — |
| 145 | D12 | — | 12 | 3 | no | — |
| 146 | C2×C9 | — | 2 | 3 | no | — |
| 147 | C2×C9 | — | 4 | 6 | no | — |
| 148 | C2×C9 | C2×#120 | 1 | 1 | yes | characteristic |
| 149 | D9 | — | 8 | 2 | no | — |
| 150 | C24 | — | 9 | 2 | no | — |
| 151 | #41 | — | 2 | 4 | no | — |
| 152 | #41 | — | 2 | 12 | no | — |
| 153 | #41 | — | 2 | 6 | no | — |
| 154 | #41 | — | 2 | 12 | no | — |
| 155 | #41 | — | 2 | 6 | no | — |
| 156 | C2×Q8 | — | 2 | 6 | no | — |
| 157 | C2×Q8 | #611 | 1 | 3 | yes | complemented |
| 158 | C2×D4 | — | 2 | 6 | no | — |
| 159 | C2×D4 | — | 2 | 12 | no | — |
| 160 | C2×D4 | — | 18 | 12 | no | — |
| 161 | C2×D4 | — | 18 | 3 | no | — |
| 162 | C2×D4 | — | 18 | 3 | no | — |
| 163 | C2×D4 | #611 | 1 | 3 | yes | complemented |
| 164 | C2×D4 | C22×D9 | 1 | 6 | yes | — |
| 165 | C22×C4 | — | 2 | 4 | no | — |
| 166 | C22×C4 | #611 | 1 | 6 | yes | complemented |
| 167 | C22×C4 | C22×D9 | 1 | 1 | yes | characteristic |
| 168 | C4⋊C4 | — | 18 | 3 | no | — |
| 169 | C22⋊C4 | — | 9 | 6 | no | — |
| 170 | C22⋊C4 | — | 18 | 12 | no | — |
| 171 | C42 | — | 18 | 3 | no | — |
| 172 | C22×C3 | — | 2 | 6 | no | — |
| 173 | C22×C3 | — | 4 | 1 | no | — |
| 174 | C22×C3 | — | 4 | 6 | no | — |
| 175 | C22×C3 | — | 8 | 4 | no | — |
| 176 | C22×C3 | C22×#67 | 1 | 3 | yes | — |
| 177 | C2×D3 | — | 12 | 12 | no | — |
| 178 | C2×D3 | — | 12 | 2 | no | — |
| 179 | C3×C4 | — | 2 | 6 | no | — |
| 180 | C3×C4 | #953 | 1 | 1 | yes | characteristic |
| 181 | C3×C4 | #953 | 1 | 3 | yes | — |
| 182 | Dic3 | — | 12 | 6 | no | — |
| 183 | C9 | #3027 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 184 | C23 | — | 2 | 4 | no | — |
| 185 | C23 | — | 18 | 6 | no | — |
| 186 | C23 | — | 72 | 2 | no | — |
| 187 | C23 | C2×#611 | 1 | 6 | yes | — |
| 188 | C23 | C23×D9 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 2 | 4 | no | — |
| 190 | Q8 | — | 2 | 6 | no | — |
| 191 | D4 | — | 2 | 12 | no | — |
| 192 | D4 | — | 2 | 12 | no | — |
| 193 | D4 | — | 2 | 6 | no | — |
| 194 | D4 | — | 36 | 1 | no | — |
| 195 | D4 | — | 36 | 3 | no | — |
| 196 | D4 | — | 72 | 12 | no | — |
| 197 | C2×C4 | — | 2 | 12 | no | — |
| 198 | C2×C4 | — | 2 | 6 | no | — |
| 199 | C2×C4 | — | 2 | 6 | no | — |
| 200 | C2×C4 | — | 18 | 6 | no | — |
| 201 | C2×C4 | C2×#611 | 1 | 6 | yes | — |
| 202 | C2×C4 | C2×#611 | 1 | 3 | yes | — |
| 203 | C2×C4 | C2×#611 | 1 | 3 | yes | — |
| 204 | C2×C3 | — | 2 | 3 | no | — |
| 205 | C2×C3 | — | 4 | 6 | no | — |
| 206 | C2×C3 | C2×#953 | 1 | 1 | yes | characteristic |
| 207 | D3 | — | 24 | 2 | no | — |
| 208 | C22 | — | 2 | 6 | no | — |
| 209 | C22 | — | 4 | 1 | no | — |
| 210 | C22 | — | 4 | 6 | no | — |
| 211 | C22 | — | 8 | 4 | no | — |
| 212 | C22 | — | 36 | 12 | no | — |
| 213 | C22 | — | 36 | 2 | no | — |
| 214 | C22 | C22×#611 | 1 | 3 | yes | — |
| 215 | C4 | — | 2 | 6 | no | — |
| 216 | C4 | — | 36 | 6 | no | — |
| 217 | C4 | #63515 | 1 | 1 | yes | characteristic |
| 218 | C4 | #63515 | 1 | 3 | yes | — |
| 219 | C3 | #84126 | 1 | 1 | yes | characteristic |
| 220 | C2 | — | 2 | 3 | no | — |
| 221 | C2 | — | 4 | 6 | no | — |
| 222 | C2 | — | 72 | 2 | no | — |
| 223 | C2 | C2×#63515 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#63515 |
| commutator, cosocle | C23×C9 | C24 |
| Frattini | C23×C3 | C23×D3 |
| Fitting | C9×#585 | C2 |
| socle | C2×C3 | C2×#953 |
| hypercenter | #585 | D9 |
| hypocenter | C9 | #3027 |
| LCS | G ▷ C23×C9 ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #585 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C9×#585 | C2 | semidirect |
| 2. #10589932 | C2 | semidirect |
| 3. #10588924 | C2 | semidirect |
| 4. #10588920 | C2 | semidirect |
| 11. C22×D36 | C22 | semidirect |
| 14. #63499 | C22 | semidirect |
| 16. #63488 | C22 | semidirect |
| 17. #63393 | C22 | semidirect |
| 29. C2×Q8×C9 | D4 | semidirect |
| 32. C2×D4×C9 | D4 | semidirect |
| 35. C22×C4×C9 | D4 | semidirect |
| 77. #585 | D9 | semidirect |
| 157. C2×Q8 | #611 | semidirect |
| 163. C2×D4 | #611 | semidirect |
| 166. C22×C4 | #611 | semidirect |
| 183. C9 | #3027 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C9×#585 | 2 |
| #10589932 | 2 |
| #10588924 | 2 |
| #10588920 | 2 |
| #84126 | 3 |