Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
766 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 766 conjugacy classes of subgroups — too many to draw. Only its 133 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 | #5966 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5586 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C3×B(4,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5243 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5032 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #4742 | C2 | 1 | 3 | yes | maximal, complemented |
| 7 | #4740 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #3025 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 10 | C23×Dic3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | #1010 | — | 2 | 3 | no | — |
| 12 | D3×#41 | — | 2 | 3 | no | — |
| 13 | #1004 | — | 2 | 3 | no | — |
| 14 | #1004 | C22 | 1 | 3 | yes | — |
| 15 | C2×#237 | — | 2 | 3 | no | — |
| 16 | C2×#237 | C22 | 1 | 3 | yes | — |
| 17 | C3×#120 | C22 | 1 | 3 | yes | complemented |
| 18 | #953 | C22 | 1 | 3 | yes | complemented |
| 19 | #939 | — | 2 | 3 | no | — |
| 20 | #935 | C22 | 1 | 3 | yes | complemented |
| 21 | D4×Dic3 | — | 2 | 3 | no | — |
| 22 | #886 | C22 | 1 | 3 | yes | complemented |
| 23 | #67⋊C4 | — | 2 | 3 | no | — |
| 24 | #878 | — | 2 | 3 | no | — |
| 25 | #877 | C22 | 1 | 3 | yes | complemented |
| 26 | C3×#99 | C22 | 1 | 3 | yes | complemented |
| 27 | #834 | C22 | 1 | 3 | yes | complemented |
| 28 | #806 | C22 | 1 | 6 | yes | complemented |
| 29 | #585 | — | 3 | 1 | no | — |
| 30 | #525 | — | 3 | 3 | no | — |
| 31 | #458 | — | 3 | 1 | no | — |
| 32 | #458 | — | 3 | 3 | no | — |
| 33 | B(4,𝔽2) | — | 3 | 3 | no | — |
| 34 | B(4,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 35 | #410 | — | 3 | 3 | no | — |
| 36 | C24×C3 | C23 | 1 | 1 | yes | characteristic |
| 37 | C3×#41 | — | 2 | 3 | no | — |
| 38 | C2×C3×D4 | — | 2 | 3 | no | — |
| 39 | C2×C3×D4 | — | 2 | 3 | no | — |
| 40 | C2×C3×D4 | C23 | 1 | 3 | yes | — |
| 41 | C2×#67 | — | 2 | 3 | no | — |
| 42 | C2×#67 | — | 2 | 3 | no | — |
| 43 | C2×#67 | — | 2 | 3 | no | — |
| 44 | C2×#67 | D4 | 1 | 3 | yes | complemented |
| 45 | C2×#67 | C23 | 1 | 3 | yes | — |
| 46 | C22×Dic3 | — | 2 | 1 | no | — |
| 47 | C22×Dic3 | — | 2 | 3 | no | — |
| 48 | C22×Dic3 | — | 2 | 3 | no | — |
| 49 | C22×Dic3 | D4 | 1 | 3 | yes | complemented |
| 50 | C22×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 51 | C22×Dic3 | C23 | 1 | 1 | yes | characteristic |
| 52 | #239 | — | 2 | 1 | no | — |
| 53 | D3×Q8 | — | 2 | 3 | no | — |
| 54 | #237 | — | 2 | 3 | no | — |
| 55 | #237 | — | 2 | 6 | no | — |
| 56 | #237 | — | 2 | 6 | no | — |
| 57 | #237 | — | 2 | 6 | no | — |
| 58 | #237 | — | 2 | 6 | no | — |
| 59 | D3×D4 | — | 2 | 3 | no | — |
| 60 | D3×D4 | — | 2 | 6 | no | — |
| 61 | #235 | — | 2 | 6 | no | — |
| 62 | #235 | — | 2 | 3 | no | — |
| 63 | C2×C4×D3 | — | 2 | 3 | no | — |
| 64 | C2×C4×D3 | D4 | 1 | 3 | yes | complemented |
| 65 | C2×Dic6 | — | 2 | 3 | no | — |
| 66 | C2×Dic6 | D4 | 1 | 3 | yes | complemented |
| 67 | C3×C22⋊C4 | — | 2 | 3 | no | — |
| 68 | C3×C22⋊C4 | C23 | 1 | 3 | yes | — |
| 69 | #217 | — | 2 | 3 | no | — |
| 70 | #217 | — | 2 | 3 | no | — |
| 71 | #217 | C23 | 1 | 3 | yes | — |
| 72 | (C2×D3)⋊C4 | — | 2 | 3 | no | — |
| 73 | C4⋊Dic3 | — | 2 | 3 | no | — |
| 74 | Dic3⋊C4 | — | 2 | 3 | no | — |
| 75 | Dic3⋊C4 | — | 2 | 3 | no | — |
| 76 | C4×Dic3 | — | 2 | 3 | no | — |
| 77 | 2−1+4 | — | 6 | 3 | no | — |
| 78 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 79 | H(5,𝔽2) | — | 6 | 3 | no | — |
| 80 | H(5,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic |
| 81 | C2×#41 | — | 3 | 3 | no | — |
| 82 | C2×#41 | — | 6 | 3 | no | — |
| 83 | C2×#41 | — | 6 | 3 | no | — |
| 84 | C23×C4 | — | 3 | 1 | no | — |
| 85 | #123 | — | 3 | 3 | no | — |
| 86 | #123 | — | 3 | 3 | no | — |
| 87 | C22⋊Q8 | — | 6 | 3 | no | — |
| 88 | #121 | — | 6 | 3 | no | — |
| 89 | #120 | — | 3 | 3 | no | — |
| 90 | #120 | C2×D3 | 1 | 3 | yes | complemented |
| 91 | C4×D4 | — | 6 | 3 | no | — |
| 92 | C4×D4 | — | 6 | 3 | no | — |
| 93 | #117 | — | 3 | 3 | no | — |
| 94 | #99 | — | 3 | 3 | no | — |
| 95 | #99 | — | 3 | 6 | no | — |
| 96 | #99 | C2×D3 | 1 | 3 | yes | complemented |
| 97 | C23×C3 | — | 2 | 1 | no | — |
| 98 | C23×C3 | — | 2 | 3 | no | — |
| 99 | C23×C3 | — | 4 | 2 | no | — |
| 100 | C23×C3 | C2×D4 | 1 | 3 | yes | — |
| 101 | C23×C3 | C24 | 1 | 1 | yes | characteristic |
| 102 | C22×D3 | — | 2 | 3 | no | — |
| 103 | C22×D3 | C2×D4 | 1 | 3 | yes | — |
| 104 | C3×Q8 | — | 2 | 1 | no | — |
| 105 | C3×D4 | — | 2 | 3 | no | — |
| 106 | C3×D4 | — | 2 | 6 | no | — |
| 107 | C3×D4 | — | 4 | 6 | no | — |
| 108 | C2×C3×C4 | — | 2 | 3 | no | — |
| 109 | C2×C3×C4 | — | 2 | 3 | no | — |
| 110 | C2×C3×C4 | C2×D4 | 1 | 3 | yes | — |
| 111 | #67 | — | 2 | 6 | no | — |
| 112 | #67 | — | 2 | 6 | no | — |
| 113 | #67 | — | 2 | 6 | no | — |
| 114 | #67 | — | 4 | 6 | no | — |
| 115 | C2×Dic3 | — | 2 | 3 | no | — |
| 116 | C2×Dic3 | — | 2 | 3 | no | — |
| 117 | C2×Dic3 | — | 2 | 3 | no | — |
| 118 | C2×Dic3 | — | 2 | 3 | no | — |
| 119 | C2×Dic3 | — | 2 | 3 | no | — |
| 120 | C2×Dic3 | — | 4 | 3 | no | — |
| 121 | C2×Dic3 | — | 4 | 1 | no | — |
| 122 | C2×Dic3 | C2×D4 | 1 | 3 | yes | — |
| 123 | C2×Dic3 | C2×D4 | 1 | 3 | yes | — |
| 124 | C2×Dic3 | C2×D4 | 1 | 3 | yes | — |
| 125 | D12 | — | 2 | 3 | no | — |
| 126 | C4×D3 | — | 2 | 6 | no | — |
| 127 | C4×D3 | — | 2 | 3 | no | — |
| 128 | C4×D3 | — | 2 | 3 | no | — |
| 129 | Dic6 | — | 2 | 3 | no | — |
| 130 | Dic6 | — | 2 | 6 | no | — |
| 131 | C24 | C22×D3 | 1 | 1 | yes | characteristic |
| 132 | #41 | — | 2 | 3 | no | — |
| 133 | #41 | — | 6 | 6 | no | — |
| 134 | #41 | — | 6 | 6 | no | — |
| 135 | #41 | — | 6 | 6 | no | — |
| 136 | #41 | — | 6 | 6 | no | — |
| 137 | #41 | — | 6 | 3 | no | — |
| 138 | #41 | — | 6 | 6 | no | — |
| 139 | #41 | — | 6 | 1 | no | — |
| 140 | #41 | — | 6 | 3 | no | — |
| 141 | C2×Q8 | — | 3 | 3 | no | — |
| 142 | C2×Q8 | — | 6 | 3 | no | — |
| 143 | C2×Q8 | — | 6 | 3 | no | — |
| 144 | C2×D4 | — | 2 | 3 | no | — |
| 145 | C2×D4 | — | 2 | 3 | no | — |
| 146 | C2×D4 | — | 3 | 3 | no | — |
| 147 | C2×D4 | — | 3 | 3 | no | — |
| 148 | C2×D4 | — | 6 | 6 | no | — |
| 149 | C2×D4 | — | 6 | 3 | no | — |
| 150 | C2×D4 | — | 6 | 3 | no | — |
| 151 | C2×D4 | — | 6 | 3 | no | — |
| 152 | C2×D4 | — | 6 | 3 | no | — |
| 153 | C2×D4 | C22×D3 | 1 | 3 | yes | — |
| 154 | C22×C4 | — | 3 | 1 | no | — |
| 155 | C22×C4 | — | 3 | 3 | no | — |
| 156 | C22×C4 | — | 3 | 3 | no | — |
| 157 | C22×C4 | — | 3 | 1 | no | — |
| 158 | C22×C4 | — | 6 | 3 | no | — |
| 159 | C22×C4 | — | 6 | 3 | no | — |
| 160 | C22×C4 | — | 6 | 3 | no | — |
| 161 | C22×C4 | — | 6 | 1 | no | — |
| 162 | C4⋊C4 | — | 6 | 3 | no | — |
| 163 | C4⋊C4 | — | 6 | 3 | no | — |
| 164 | C4⋊C4 | — | 6 | 3 | no | — |
| 165 | C22⋊C4 | — | 2 | 3 | no | — |
| 166 | C22⋊C4 | — | 3 | 3 | no | — |
| 167 | C22⋊C4 | — | 6 | 3 | no | — |
| 168 | C22⋊C4 | — | 6 | 3 | no | — |
| 169 | C22⋊C4 | — | 6 | 3 | no | — |
| 170 | C22⋊C4 | C22×D3 | 1 | 3 | yes | — |
| 171 | C42 | — | 6 | 3 | no | — |
| 172 | C22×C3 | — | 2 | 6 | no | — |
| 173 | C22×C3 | — | 2 | 3 | no | — |
| 174 | C22×C3 | — | 4 | 1 | no | — |
| 175 | C22×C3 | — | 4 | 3 | no | — |
| 176 | C22×C3 | — | 4 | 3 | no | — |
| 177 | C22×C3 | — | 4 | 1 | no | — |
| 178 | C22×C3 | — | 8 | 1 | no | — |
| 179 | C22×C3 | C22×D4 | 1 | 3 | yes | — |
| 180 | C2×D3 | — | 2 | 3 | no | — |
| 181 | C2×D3 | — | 4 | 3 | no | — |
| 182 | C2×D3 | — | 8 | 3 | no | — |
| 183 | C3×C4 | — | 2 | 3 | no | — |
| 184 | C3×C4 | — | 4 | 3 | no | — |
| 185 | Dic3 | — | 2 | 3 | no | — |
| 186 | Dic3 | — | 4 | 3 | no | — |
| 187 | Dic3 | — | 4 | 1 | no | — |
| 188 | Dic3 | #120 | 1 | 3 | yes | — |
| 189 | Dic3 | #120 | 1 | 1 | yes | characteristic |
| 190 | C23 | — | 2 | 1 | no | — |
| 191 | C23 | — | 2 | 3 | no | — |
| 192 | C23 | — | 3 | 3 | no | — |
| 193 | C23 | — | 4 | 2 | no | — |
| 194 | C23 | — | 6 | 3 | no | — |
| 195 | C23 | D3×D4 | 1 | 3 | yes | — |
| 196 | C23 | C23×D3 | 1 | 1 | yes | characteristic |
| 197 | Q8 | — | 2 | 1 | no | — |
| 198 | Q8 | — | 6 | 6 | no | — |
| 199 | Q8 | — | 6 | 3 | no | — |
| 200 | D4 | — | 2 | 3 | no | — |
| 201 | D4 | — | 2 | 6 | no | — |
| 202 | D4 | — | 4 | 6 | no | — |
| 203 | D4 | — | 6 | 3 | no | — |
| 204 | D4 | — | 6 | 6 | no | — |
| 205 | D4 | — | 6 | 6 | no | — |
| 206 | D4 | — | 6 | 6 | no | — |
| 207 | D4 | — | 12 | 6 | no | — |
| 208 | C2×C4 | — | 2 | 3 | no | — |
| 209 | C2×C4 | — | 2 | 3 | no | — |
| 210 | C2×C4 | — | 3 | 3 | no | — |
| 211 | C2×C4 | — | 3 | 3 | no | — |
| 212 | C2×C4 | — | 3 | 3 | no | — |
| 213 | C2×C4 | — | 6 | 3 | no | — |
| 214 | C2×C4 | — | 6 | 3 | no | — |
| 215 | C2×C4 | — | 6 | 6 | no | — |
| 216 | C2×C4 | — | 6 | 3 | no | — |
| 217 | C2×C4 | — | 6 | 3 | no | — |
| 218 | C2×C4 | — | 6 | 3 | no | — |
| 219 | C2×C4 | — | 6 | 3 | no | — |
| 220 | C2×C4 | — | 6 | 3 | no | — |
| 221 | C2×C4 | — | 12 | 1 | no | — |
| 222 | C2×C4 | — | 12 | 3 | no | — |
| 223 | C2×C4 | D3×D4 | 1 | 3 | yes | — |
| 224 | C2×C3 | — | 2 | 3 | no | — |
| 225 | C2×C3 | — | 4 | 3 | no | — |
| 226 | C2×C3 | — | 4 | 2 | no | — |
| 227 | C2×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 228 | D3 | — | 4 | 3 | no | — |
| 229 | C22 | — | 2 | 6 | no | — |
| 230 | C22 | — | 2 | 3 | no | — |
| 231 | C22 | — | 4 | 1 | no | — |
| 232 | C22 | — | 4 | 3 | no | — |
| 233 | C22 | — | 4 | 3 | no | — |
| 234 | C22 | — | 4 | 1 | no | — |
| 235 | C22 | — | 6 | 3 | no | — |
| 236 | C22 | — | 8 | 1 | no | — |
| 237 | C22 | — | 12 | 3 | no | — |
| 238 | C22 | — | 24 | 3 | no | — |
| 239 | C22 | C2×D3×D4 | 1 | 3 | yes | — |
| 240 | C4 | — | 2 | 3 | no | — |
| 241 | C4 | — | 3 | 1 | no | — |
| 242 | C4 | — | 3 | 3 | no | — |
| 243 | C4 | — | 4 | 3 | no | — |
| 244 | C4 | — | 6 | 3 | no | — |
| 245 | C4 | — | 12 | 3 | no | — |
| 246 | C4 | — | 12 | 1 | no | — |
| 247 | C3 | #3025 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 248 | C2 | — | 2 | 3 | no | — |
| 249 | C2 | — | 4 | 2 | no | — |
| 250 | C2 | — | 4 | 3 | no | — |
| 251 | C2 | — | 12 | 3 | no | — |
| 252 | C2 | D3×#120 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#120 |
| commutator, cosocle | C23×C3 | C24 |
| Frattini | C23 | C23×D3 |
| Fitting | C3×B(4,𝔽2) | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | B(4,𝔽2) | D3 |
| hypocenter | C3 | #3025 |
| LCS | G ▷ C23×C3 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ B(4,𝔽2) | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5966 | C2 | semidirect |
| 2. #5586 | C2 | semidirect |
| 3. C3×B(4,𝔽2) | C2 | semidirect |
| 4. #5243 | C2 | semidirect |
| 5. #5032 | C2 | semidirect |
| 6. #4742 | C2 | semidirect |
| 7. #4740 | C2 | semidirect |
| 10. C23×Dic3 | C22 | semidirect |
| 17. C3×#120 | C22 | semidirect |
| 18. #953 | C22 | semidirect |
| 20. #935 | C22 | semidirect |
| 22. #886 | C22 | semidirect |
| 25. #877 | C22 | semidirect |
| 26. C3×#99 | C22 | semidirect |
| 27. #834 | C22 | semidirect |
| 28. #806 | C22 | semidirect |
| 34. B(4,𝔽2) | D3 | semidirect |
| 44. C2×#67 | D4 | semidirect |
| 49. C22×Dic3 | D4 | semidirect |
| 64. C2×C4×D3 | D4 | semidirect |
| 66. C2×Dic6 | D4 | semidirect |
| 90. #120 | C2×D3 | semidirect |
| 96. #99 | C2×D3 | semidirect |
| 247. C3 | #3025 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5966 | 2 |
| #5586 | 2 |
| C3×B(4,𝔽2) | 2 |
| #5243 | 2 |
| #5032 | 2 |
| #4742 | 2 |
| #4740 | 2 |
| #3025 | 3 |