Finite Group DatabasebetaFinite Group DatabasebetaOrder 1152.
2371 conjugacy classes of subgroups; only the 97 normal subgroups are shown.
This group has 2371 conjugacy classes of subgroups — too many to draw. Only its 97 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 | C2×#63703 | C2 | 1 | 3 | yes | maximal, direct product factor |
| 2 | C22×C3×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic |
| 3 | C23×C22⋊C9 | — | 4 | 3 | no | maximal |
| 4 | #63703 | C22 | 1 | 1 | yes | characteristic, direct product factor |
| 5 | C22×Q8⋊C9 | — | 4 | 2 | no | maximal |
| 6 | C2×C3×H(5,𝔽2) | — | 3 | 6 | no | — |
| 7 | C2×C3×H(5,𝔽2) | — | 3 | 9 | no | — |
| 8 | C2×C3×H(5,𝔽2) | C2×C3 | 1 | 3 | yes | — |
| 9 | C22×C3×#41 | — | 3 | 2 | no | — |
| 10 | C23×C3×D4 | — | 3 | 3 | no | — |
| 11 | C22×C22⋊C9 | — | 4 | 12 | no | — |
| 12 | C22×C22⋊C9 | — | 4 | 9 | no | — |
| 13 | C2×Q8⋊C9 | — | 4 | 6 | no | — |
| 14 | C22×H(5,𝔽2) | C9 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | C25×C3 | — | 3 | 1 | no | — |
| 16 | C25×C3 | A4 | 1 | 3 | yes | — |
| 17 | C3×H(5,𝔽2) | — | 3 | 6 | no | — |
| 18 | C3×H(5,𝔽2) | — | 3 | 18 | no | — |
| 19 | C3×H(5,𝔽2) | — | 3 | 36 | no | — |
| 20 | C3×H(5,𝔽2) | — | 3 | 6 | no | — |
| 21 | C3×H(5,𝔽2) | — | 3 | 4 | no | — |
| 22 | C3×H(5,𝔽2) | — | 3 | 6 | no | — |
| 23 | C3×H(5,𝔽2) | — | 3 | 9 | no | — |
| 24 | C3×H(5,𝔽2) | C22×C3 | 1 | 1 | yes | characteristic |
| 25 | C2×C3×#41 | — | 3 | 18 | no | — |
| 26 | C2×C3×#41 | — | 3 | 18 | no | — |
| 27 | C2×C3×#41 | — | 3 | 6 | no | — |
| 28 | C2×C3×#41 | — | 3 | 6 | no | — |
| 29 | C22×C3×Q8 | A4 | 1 | 2 | yes | — |
| 30 | C22×C3×D4 | — | 3 | 6 | no | — |
| 31 | C22×C3×D4 | — | 3 | 9 | no | — |
| 32 | C22×C3×D4 | — | 3 | 36 | no | — |
| 33 | C22×C3×D4 | — | 3 | 9 | no | — |
| 34 | C22×C3×D4 | — | 3 | 9 | no | — |
| 35 | C22×C3×D4 | — | 3 | 9 | no | — |
| 36 | C23×C3×C4 | — | 3 | 3 | no | — |
| 37 | C23×C9 | — | 16 | 1 | no | — |
| 38 | C2×C22⋊C9 | — | 4 | 18 | no | — |
| 39 | C2×C22⋊C9 | — | 4 | 3 | no | — |
| 40 | Q8⋊C9 | — | 4 | 2 | no | — |
| 41 | C2×H(5,𝔽2) | — | 3 | 6 | no | — |
| 42 | C2×H(5,𝔽2) | — | 3 | 9 | no | — |
| 43 | C2×H(5,𝔽2) | C2×C9 | 1 | 3 | yes | complemented |
| 44 | C22×#41 | — | 3 | 2 | no | — |
| 45 | C23×D4 | — | 3 | 3 | no | — |
| 46 | C24×C3 | — | 3 | 9 | no | — |
| 47 | C24×C3 | — | 3 | 3 | no | — |
| 48 | C24×C3 | — | 3 | 3 | no | — |
| 49 | C24×C3 | — | 3 | 9 | no | — |
| 50 | C24×C3 | — | 4 | 12 | no | — |
| 51 | C24×C3 | — | 12 | 4 | no | — |
| 52 | C24×C3 | C2×A4 | 1 | 9 | yes | — |
| 53 | C3×#41 | — | 3 | 36 | no | — |
| 54 | C3×#41 | — | 3 | 18 | no | — |
| 55 | C3×#41 | — | 3 | 18 | no | — |
| 56 | C3×#41 | — | 3 | 36 | no | — |
| 57 | C3×#41 | — | 3 | 12 | no | — |
| 58 | C3×#41 | — | 3 | 6 | no | — |
| 59 | C3×#41 | — | 3 | 2 | no | — |
| 60 | C2×C3×Q8 | — | 3 | 6 | no | — |
| 61 | C2×C3×Q8 | C2×A4 | 1 | 6 | yes | — |
| 62 | C2×C3×D4 | — | 3 | 36 | no | — |
| 63 | C2×C3×D4 | — | 3 | 18 | no | — |
| 64 | C2×C3×D4 | — | 3 | 18 | no | — |
| 65 | C2×C3×D4 | — | 3 | 36 | no | — |
| 66 | C2×C3×D4 | — | 3 | 36 | no | — |
| 67 | C2×C3×D4 | — | 3 | 36 | no | — |
| 68 | C2×C3×D4 | — | 3 | 18 | no | — |
| 69 | C2×C3×D4 | — | 3 | 9 | no | — |
| 70 | C2×C3×D4 | — | 3 | 36 | no | — |
| 71 | C2×C3×D4 | — | 3 | 9 | no | — |
| 72 | C2×C3×D4 | — | 3 | 9 | no | — |
| 73 | C2×C3×D4 | — | 3 | 3 | no | — |
| 74 | C22×C3×C4 | — | 3 | 9 | no | — |
| 75 | C22×C3×C4 | — | 3 | 18 | no | — |
| 76 | C22×C3×C4 | — | 3 | 9 | no | — |
| 77 | C22×C3×C4 | — | 3 | 2 | no | — |
| 78 | C22×C9 | — | 16 | 4 | no | — |
| 79 | C22×C9 | — | 16 | 3 | no | — |
| 80 | C22⋊C9 | — | 4 | 3 | no | — |
| 81 | C25 | — | 3 | 1 | no | — |
| 82 | C25 | C22⋊C9 | 1 | 3 | yes | complemented |
| 83 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 84 | H(5,𝔽2) | — | 3 | 18 | no | — |
| 85 | H(5,𝔽2) | — | 3 | 36 | no | — |
| 86 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 87 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 88 | H(5,𝔽2) | — | 3 | 6 | no | — |
| 89 | H(5,𝔽2) | — | 3 | 9 | no | — |
| 90 | H(5,𝔽2) | C22×C9 | 1 | 1 | yes | characteristic, complemented |
| 91 | C2×#41 | — | 3 | 18 | no | — |
| 92 | C2×#41 | — | 3 | 18 | no | — |
| 93 | C2×#41 | — | 3 | 6 | no | — |
| 94 | C2×#41 | — | 3 | 6 | no | — |
| 95 | C22×Q8 | C22⋊C9 | 1 | 2 | yes | complemented |
| 96 | C22×D4 | — | 3 | 6 | no | — |
| 97 | C22×D4 | — | 3 | 9 | no | — |
| 98 | C22×D4 | — | 3 | 36 | no | — |
| 99 | C22×D4 | — | 3 | 9 | no | — |
| 100 | C22×D4 | — | 3 | 9 | no | — |
| 101 | C22×D4 | — | 3 | 9 | no | — |
| 102 | C23×C4 | — | 3 | 3 | no | — |
| 103 | C23×C3 | — | 3 | 6 | no | — |
| 104 | C23×C3 | — | 3 | 6 | no | — |
| 105 | C23×C3 | — | 3 | 9 | no | — |
| 106 | C23×C3 | — | 3 | 1 | no | — |
| 107 | C23×C3 | — | 3 | 9 | no | — |
| 108 | C23×C3 | — | 3 | 9 | no | — |
| 109 | C23×C3 | — | 3 | 9 | no | — |
| 110 | C23×C3 | — | 4 | 18 | no | — |
| 111 | C23×C3 | — | 6 | 12 | no | — |
| 112 | C23×C3 | — | 12 | 6 | no | — |
| 113 | C23×C3 | — | 12 | 18 | no | — |
| 114 | C23×C3 | — | 12 | 18 | no | — |
| 115 | C23×C3 | C22×A4 | 1 | 3 | yes | — |
| 116 | C23×C3 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 117 | C3×Q8 | — | 3 | 6 | no | — |
| 118 | C3×Q8 | — | 3 | 4 | no | — |
| 119 | C3×Q8 | C22×A4 | 1 | 2 | yes | — |
| 120 | C3×D4 | — | 3 | 36 | no | — |
| 121 | C3×D4 | — | 3 | 36 | no | — |
| 122 | C3×D4 | — | 3 | 18 | no | — |
| 123 | C3×D4 | — | 3 | 6 | no | — |
| 124 | C2×C3×C4 | — | 3 | 18 | no | — |
| 125 | C2×C3×C4 | — | 3 | 9 | no | — |
| 126 | C2×C3×C4 | — | 3 | 18 | no | — |
| 127 | C2×C3×C4 | — | 3 | 3 | no | — |
| 128 | C2×C3×C4 | — | 3 | 6 | no | — |
| 129 | C2×C3×C4 | — | 3 | 6 | no | — |
| 130 | C2×C9 | — | 16 | 6 | no | — |
| 131 | C2×C9 | — | 16 | 1 | no | — |
| 132 | C24 | — | 3 | 3 | no | — |
| 133 | C24 | — | 3 | 9 | no | — |
| 134 | C24 | — | 3 | 3 | no | — |
| 135 | C24 | — | 3 | 9 | no | — |
| 136 | C24 | — | 4 | 12 | no | — |
| 137 | C24 | — | 12 | 4 | no | — |
| 138 | C24 | C2×C22⋊C9 | 1 | 9 | yes | complemented |
| 139 | #41 | — | 3 | 18 | no | — |
| 140 | #41 | — | 3 | 18 | no | — |
| 141 | #41 | — | 3 | 36 | no | — |
| 142 | #41 | — | 3 | 12 | no | — |
| 143 | #41 | — | 3 | 36 | no | — |
| 144 | #41 | — | 3 | 2 | no | — |
| 145 | #41 | — | 3 | 6 | no | — |
| 146 | C2×Q8 | — | 3 | 6 | no | — |
| 147 | C2×Q8 | C2×C22⋊C9 | 1 | 6 | yes | complemented |
| 148 | C2×D4 | — | 3 | 18 | no | — |
| 149 | C2×D4 | — | 3 | 18 | no | — |
| 150 | C2×D4 | — | 3 | 36 | no | — |
| 151 | C2×D4 | — | 3 | 36 | no | — |
| 152 | C2×D4 | — | 3 | 36 | no | — |
| 153 | C2×D4 | — | 3 | 9 | no | — |
| 154 | C2×D4 | — | 3 | 36 | no | — |
| 155 | C2×D4 | — | 3 | 9 | no | — |
| 156 | C2×D4 | — | 3 | 18 | no | — |
| 157 | C2×D4 | — | 3 | 36 | no | — |
| 158 | C2×D4 | — | 3 | 3 | no | — |
| 159 | C2×D4 | — | 3 | 9 | no | — |
| 160 | C22×C4 | — | 3 | 9 | no | — |
| 161 | C22×C4 | — | 3 | 18 | no | — |
| 162 | C22×C4 | — | 3 | 9 | no | — |
| 163 | C22×C4 | — | 3 | 2 | no | — |
| 164 | C22×C3 | — | 3 | 9 | no | — |
| 165 | C22×C3 | — | 3 | 3 | no | — |
| 166 | C22×C3 | — | 4 | 3 | no | — |
| 167 | C22×C3 | — | 6 | 18 | no | — |
| 168 | C22×C3 | — | 6 | 18 | no | — |
| 169 | C22×C3 | — | 12 | 9 | no | — |
| 170 | C22×C3 | — | 12 | 1 | no | — |
| 171 | C22×C3 | — | 12 | 9 | no | — |
| 172 | C22×C3 | — | 12 | 6 | no | — |
| 173 | C22×C3 | — | 12 | 6 | no | — |
| 174 | C22×C3 | #997 | 1 | 4 | yes | — |
| 175 | C22×C3 | C2×C22⋊A4 | 1 | 3 | yes | — |
| 176 | C3×C4 | — | 3 | 6 | no | — |
| 177 | C3×C4 | — | 3 | 2 | no | — |
| 178 | C9 | — | 16 | 1 | no | Sylow |
| 179 | C23 | — | 3 | 1 | no | — |
| 180 | C23 | — | 3 | 9 | no | — |
| 181 | C23 | — | 3 | 6 | no | — |
| 182 | C23 | — | 3 | 6 | no | — |
| 183 | C23 | — | 3 | 9 | no | — |
| 184 | C23 | — | 3 | 9 | no | — |
| 185 | C23 | — | 3 | 9 | no | — |
| 186 | C23 | — | 4 | 18 | no | — |
| 187 | C23 | — | 6 | 12 | no | — |
| 188 | C23 | — | 12 | 6 | no | — |
| 189 | C23 | — | 12 | 18 | no | — |
| 190 | C23 | — | 12 | 18 | no | — |
| 191 | C23 | C22×C22⋊C9 | 1 | 3 | yes | complemented |
| 192 | C23 | C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| 193 | Q8 | — | 3 | 6 | no | — |
| 194 | Q8 | — | 3 | 4 | no | — |
| 195 | Q8 | C22×C22⋊C9 | 1 | 2 | yes | complemented |
| 196 | D4 | — | 3 | 6 | no | — |
| 197 | D4 | — | 3 | 18 | no | — |
| 198 | D4 | — | 3 | 36 | no | — |
| 199 | D4 | — | 3 | 36 | no | — |
| 200 | C2×C4 | — | 3 | 3 | no | — |
| 201 | C2×C4 | — | 3 | 18 | no | — |
| 202 | C2×C4 | — | 3 | 9 | no | — |
| 203 | C2×C4 | — | 3 | 18 | no | — |
| 204 | C2×C4 | — | 3 | 6 | no | — |
| 205 | C2×C4 | — | 3 | 6 | no | — |
| 206 | C2×C3 | — | 6 | 3 | no | — |
| 207 | C2×C3 | — | 6 | 9 | no | — |
| 208 | C2×C3 | C2×#997 | 1 | 6 | yes | — |
| 209 | C2×C3 | C22×C22⋊A4 | 1 | 1 | yes | characteristic |
| 210 | C22 | — | 3 | 3 | no | — |
| 211 | C22 | — | 3 | 9 | no | — |
| 212 | C22 | — | 4 | 3 | no | — |
| 213 | C22 | — | 6 | 18 | no | — |
| 214 | C22 | — | 6 | 18 | no | — |
| 215 | C22 | — | 12 | 9 | no | — |
| 216 | C22 | — | 12 | 1 | no | — |
| 217 | C22 | — | 12 | 9 | no | — |
| 218 | C22 | — | 12 | 6 | no | — |
| 219 | C22 | — | 12 | 6 | no | — |
| 220 | C22 | #63703 | 1 | 4 | yes | direct product factor |
| 221 | C22 | C2×C22⋊(C22⋊C9) | 1 | 3 | yes | — |
| 222 | C4 | — | 3 | 2 | no | — |
| 223 | C4 | — | 3 | 6 | no | — |
| 224 | C3 | C22×#997 | 1 | 1 | yes | characteristic |
| 225 | C2 | — | 6 | 3 | no | — |
| 226 | C2 | — | 6 | 9 | no | — |
| 227 | C2 | C2×#63703 | 1 | 6 | yes | direct product factor |
| 228 | C2 | C22×C22⋊(C22⋊C9) | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C23×C3 | C22⋊A4 |
| commutator, hypocenter | H(5,𝔽2) | C22×C9 |
| Frattini | C2×C3 | C22×C22⋊A4 |
| Fitting, prime radical | C22×C3×H(5,𝔽2) | C3 |
| cosocle | C3×H(5,𝔽2) | C22×C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#63703 | C2 | direct (N × H) |
| 4. #63703 | C22 | direct (N × H) |
| 14. C22×H(5,𝔽2) | C9 | semidirect |
| 43. C2×H(5,𝔽2) | C2×C9 | semidirect |
| 82. C25 | C22⋊C9 | semidirect |
| 90. H(5,𝔽2) | C22×C9 | semidirect |
| 95. C22×Q8 | C22⋊C9 | semidirect |
| 138. C24 | C2×C22⋊C9 | semidirect |
| 147. C2×Q8 | C2×C22⋊C9 | semidirect |
| 191. C23 | C22×C22⋊C9 | semidirect |
| 195. Q8 | C22×C22⋊C9 | semidirect |
| 220. C22 | #63703 | direct (N × H) |
| 227. C2 | C2×#63703 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#63703 | 2 |
| C22×C3×H(5,𝔽2) | 3 |
| C23×C22⋊C9 | 4 |
| C22×Q8⋊C9 | 4 |