Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
860 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 860 conjugacy classes of subgroups — too many to draw. Only its 102 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 | A4×#41 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5919 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | D4×S4 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #5912 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C2×C4⋊S4 | C2 | 1 | 3 | yes | maximal, complemented |
| 6 | #3488 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C22×S4 | C22 | 1 | 6 | yes | complemented |
| 8 | #1009 | — | 4 | 1 | no | maximal |
| 9 | Q8×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | D4×A4 | C22 | 1 | 3 | yes | complemented |
| 11 | C2×C4×A4 | C22 | 1 | 3 | yes | complemented |
| 12 | GL(2,ℤ/4) | C22 | 1 | 6 | yes | complemented |
| 13 | C4⋊S4 | C22 | 1 | 3 | yes | complemented |
| 14 | C4⋊S4 | C22 | 1 | 6 | yes | complemented |
| 15 | C4×S4 | C22 | 1 | 6 | yes | complemented |
| 16 | A4⋊Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C2×H(5,𝔽2) | — | 3 | 1 | no | — |
| 18 | C22×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 19 | #550 | — | 3 | 2 | no | — |
| 20 | #548 | — | 3 | 2 | no | — |
| 21 | #546 | — | 3 | 6 | no | — |
| 22 | D42 | — | 3 | 6 | no | — |
| 23 | #537 | — | 3 | 3 | no | — |
| 24 | #535 | — | 3 | 3 | no | — |
| 25 | #525 | — | 3 | 3 | no | — |
| 26 | C2×#121 | — | 3 | 3 | no | — |
| 27 | #520 | — | 3 | 1 | no | — |
| 28 | C22×A4 | C23 | 1 | 3 | yes | — |
| 29 | C2×S4 | — | 4 | 6 | no | — |
| 30 | C2×S4 | C23 | 1 | 6 | yes | — |
| 31 | C3×#41 | — | 4 | 1 | no | — |
| 32 | #239 | — | 4 | 2 | no | — |
| 33 | D3×D4 | — | 4 | 6 | no | — |
| 34 | #235 | — | 4 | 3 | no | — |
| 35 | C2×D12 | — | 4 | 3 | no | — |
| 36 | C4×A4 | C23 | 1 | 1 | yes | characteristic |
| 37 | C4×A4 | C23 | 1 | 3 | yes | — |
| 38 | SL(2,ℤ/4) | C23 | 1 | 2 | yes | — |
| 39 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 40 | H(5,𝔽2) | — | 6 | 3 | no | — |
| 41 | H(5,𝔽2) | — | 6 | 3 | no | — |
| 42 | H(5,𝔽2) | — | 6 | 1 | no | — |
| 43 | C2×#41 | — | 3 | 1 | no | — |
| 44 | C2×#41 | — | 3 | 3 | no | — |
| 45 | C2×#41 | — | 3 | 3 | no | — |
| 46 | C2×#41 | — | 3 | 1 | no | — |
| 47 | C2×#41 | — | 3 | 3 | no | — |
| 48 | C2×#41 | — | 3 | 2 | no | — |
| 49 | C22×Q8 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 50 | C22×D4 | — | 3 | 3 | no | — |
| 51 | C22×D4 | — | 3 | 6 | no | — |
| 52 | C22×D4 | — | 3 | 6 | no | — |
| 53 | C22×D4 | C2×D3 | 1 | 3 | yes | complemented |
| 54 | C23×C4 | C2×D3 | 1 | 3 | yes | complemented |
| 55 | #127 | — | 3 | 6 | no | — |
| 56 | #124 | — | 3 | 6 | no | — |
| 57 | #123 | — | 3 | 6 | no | — |
| 58 | C22⋊Q8 | — | 3 | 3 | no | — |
| 59 | C22⋊Q8 | — | 3 | 6 | no | — |
| 60 | C22⋊Q8 | — | 3 | 1 | no | — |
| 61 | #121 | — | 3 | 6 | no | — |
| 62 | #121 | — | 3 | 6 | no | — |
| 63 | #121 | — | 3 | 3 | no | — |
| 64 | #121 | — | 3 | 12 | no | — |
| 65 | #121 | — | 3 | 6 | no | — |
| 66 | #121 | — | 3 | 3 | no | — |
| 67 | #121 | — | 3 | 6 | no | — |
| 68 | #120 | — | 3 | 12 | no | — |
| 69 | #120 | — | 3 | 6 | no | — |
| 70 | C4×Q8 | — | 3 | 2 | no | — |
| 71 | C4×D4 | — | 3 | 6 | no | — |
| 72 | C4×D4 | — | 3 | 6 | no | — |
| 73 | C4×D4 | — | 3 | 6 | no | — |
| 74 | #117 | — | 3 | 3 | no | — |
| 75 | C2×C4⋊C4 | — | 3 | 3 | no | — |
| 76 | C2×C22⋊C4 | — | 3 | 6 | no | — |
| 77 | C22×D3 | — | 4 | 6 | no | — |
| 78 | C2×A4 | — | 2 | 3 | no | — |
| 79 | C2×A4 | C24 | 1 | 1 | yes | characteristic |
| 80 | S4 | — | 2 | 6 | no | — |
| 81 | C3×Q8 | — | 4 | 1 | no | — |
| 82 | C3×D4 | — | 4 | 3 | no | — |
| 83 | C2×C3×C4 | — | 4 | 3 | no | — |
| 84 | #67 | — | 4 | 6 | no | — |
| 85 | D12 | — | 4 | 3 | no | — |
| 86 | D12 | — | 4 | 6 | no | — |
| 87 | C4×D3 | — | 4 | 6 | no | — |
| 88 | Dic6 | — | 4 | 1 | no | — |
| 89 | C24 | — | 3 | 6 | no | — |
| 90 | C24 | C22×D3 | 1 | 3 | yes | — |
| 91 | #41 | — | 3 | 1 | no | — |
| 92 | #41 | — | 3 | 3 | no | — |
| 93 | #41 | — | 3 | 3 | no | — |
| 94 | #41 | — | 6 | 1 | no | — |
| 95 | #41 | — | 6 | 3 | no | — |
| 96 | #41 | — | 6 | 3 | no | — |
| 97 | #41 | — | 6 | 3 | no | — |
| 98 | #41 | — | 6 | 3 | no | — |
| 99 | #41 | — | 6 | 3 | no | — |
| 100 | #41 | — | 6 | 6 | no | — |
| 101 | #41 | — | 6 | 2 | no | — |
| 102 | #41 | — | 6 | 3 | no | — |
| 103 | #41 | S4 | 1 | 1 | yes | characteristic, complemented |
| 104 | C2×Q8 | — | 3 | 1 | no | — |
| 105 | C2×Q8 | — | 3 | 3 | no | — |
| 106 | C2×Q8 | — | 3 | 1 | no | — |
| 107 | C2×D4 | — | 3 | 3 | no | — |
| 108 | C2×D4 | — | 3 | 6 | no | — |
| 109 | C2×D4 | — | 3 | 3 | no | — |
| 110 | C2×D4 | — | 3 | 6 | no | — |
| 111 | C2×D4 | — | 3 | 3 | no | — |
| 112 | C2×D4 | — | 3 | 6 | no | — |
| 113 | C2×D4 | — | 3 | 6 | no | — |
| 114 | C2×D4 | — | 3 | 6 | no | — |
| 115 | C2×D4 | — | 3 | 12 | no | — |
| 116 | C2×D4 | — | 6 | 3 | no | — |
| 117 | C2×D4 | — | 6 | 3 | no | — |
| 118 | C2×D4 | — | 6 | 3 | no | — |
| 119 | C2×D4 | — | 6 | 12 | no | — |
| 120 | C2×D4 | — | 6 | 6 | no | — |
| 121 | C2×D4 | — | 6 | 6 | no | — |
| 122 | C2×D4 | — | 6 | 3 | no | — |
| 123 | C2×D4 | — | 12 | 6 | no | — |
| 124 | C2×D4 | — | 12 | 6 | no | — |
| 125 | C22×C4 | — | 3 | 3 | no | — |
| 126 | C22×C4 | — | 3 | 3 | no | — |
| 127 | C22×C4 | — | 3 | 3 | no | — |
| 128 | C22×C4 | — | 3 | 3 | no | — |
| 129 | C22×C4 | — | 3 | 6 | no | — |
| 130 | C22×C4 | — | 3 | 6 | no | — |
| 131 | C22×C4 | C22×D3 | 1 | 3 | yes | — |
| 132 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 133 | C4⋊C4 | — | 3 | 3 | no | — |
| 134 | C4⋊C4 | — | 3 | 6 | no | — |
| 135 | C4⋊C4 | — | 3 | 1 | no | — |
| 136 | C4⋊C4 | — | 3 | 2 | no | — |
| 137 | C4⋊C4 | — | 3 | 6 | no | — |
| 138 | C22⋊C4 | — | 3 | 6 | no | — |
| 139 | C22⋊C4 | — | 3 | 12 | no | — |
| 140 | C22⋊C4 | — | 3 | 6 | no | — |
| 141 | C22⋊C4 | — | 3 | 6 | no | — |
| 142 | C22⋊C4 | — | 3 | 6 | no | — |
| 143 | C22⋊C4 | — | 3 | 2 | no | — |
| 144 | C42 | — | 3 | 6 | no | — |
| 145 | C22×C3 | — | 4 | 3 | no | — |
| 146 | C2×D3 | — | 4 | 6 | no | — |
| 147 | C2×D3 | — | 16 | 6 | no | — |
| 148 | A4 | H(5,𝔽2) | 1 | 1 | yes | characteristic, complemented |
| 149 | C3×C4 | — | 4 | 1 | no | — |
| 150 | C3×C4 | — | 4 | 3 | no | — |
| 151 | Dic3 | — | 4 | 2 | no | — |
| 152 | C23 | — | 2 | 3 | no | — |
| 153 | C23 | — | 3 | 3 | no | — |
| 154 | C23 | — | 3 | 3 | no | — |
| 155 | C23 | — | 3 | 6 | no | — |
| 156 | C23 | — | 6 | 3 | no | — |
| 157 | C23 | — | 6 | 6 | no | — |
| 158 | C23 | — | 6 | 6 | no | — |
| 159 | C23 | — | 12 | 6 | no | — |
| 160 | C23 | — | 12 | 6 | no | — |
| 161 | C23 | C23×D3 | 1 | 1 | yes | characteristic |
| 162 | Q8 | — | 3 | 3 | no | — |
| 163 | Q8 | — | 6 | 1 | no | — |
| 164 | Q8 | — | 6 | 1 | no | — |
| 165 | Q8 | — | 6 | 1 | no | — |
| 166 | Q8 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 167 | D4 | — | 3 | 6 | no | — |
| 168 | D4 | — | 3 | 3 | no | — |
| 169 | D4 | — | 6 | 3 | no | — |
| 170 | D4 | — | 6 | 3 | no | — |
| 171 | D4 | — | 6 | 6 | no | — |
| 172 | D4 | — | 6 | 6 | no | — |
| 173 | D4 | — | 6 | 6 | no | — |
| 174 | D4 | — | 6 | 6 | no | — |
| 175 | D4 | — | 6 | 6 | no | — |
| 176 | D4 | — | 6 | 3 | no | — |
| 177 | D4 | — | 6 | 6 | no | — |
| 178 | D4 | — | 12 | 6 | no | — |
| 179 | D4 | — | 12 | 12 | no | — |
| 180 | D4 | C2×S4 | 1 | 3 | yes | complemented |
| 181 | C2×C4 | — | 3 | 3 | no | — |
| 182 | C2×C4 | — | 3 | 3 | no | — |
| 183 | C2×C4 | — | 3 | 1 | no | — |
| 184 | C2×C4 | — | 3 | 3 | no | — |
| 185 | C2×C4 | — | 3 | 3 | no | — |
| 186 | C2×C4 | — | 3 | 1 | no | — |
| 187 | C2×C4 | — | 3 | 3 | no | — |
| 188 | C2×C4 | — | 3 | 2 | no | — |
| 189 | C2×C4 | — | 3 | 6 | no | — |
| 190 | C2×C4 | — | 3 | 2 | no | — |
| 191 | C2×C4 | — | 6 | 3 | no | — |
| 192 | C2×C4 | — | 6 | 6 | no | — |
| 193 | C2×C4 | — | 6 | 6 | no | — |
| 194 | C2×C4 | — | 12 | 6 | no | — |
| 195 | C2×C4 | C2×S4 | 1 | 3 | yes | complemented |
| 196 | C2×C3 | — | 4 | 1 | no | — |
| 197 | C2×C3 | — | 8 | 3 | no | — |
| 198 | D3 | — | 8 | 6 | no | — |
| 199 | C22 | — | 3 | 1 | no | — |
| 200 | C22 | — | 3 | 1 | no | — |
| 201 | C22 | — | 3 | 3 | no | — |
| 202 | C22 | — | 6 | 3 | no | — |
| 203 | C22 | — | 6 | 3 | no | — |
| 204 | C22 | — | 6 | 3 | no | — |
| 205 | C22 | — | 6 | 3 | no | — |
| 206 | C22 | — | 6 | 6 | no | — |
| 207 | C22 | — | 6 | 6 | no | — |
| 208 | C22 | — | 6 | 6 | no | — |
| 209 | C22 | — | 24 | 6 | no | — |
| 210 | C22 | — | 24 | 6 | no | — |
| 211 | C22 | #1009 | 1 | 1 | yes | characteristic, complemented |
| 212 | C22 | C22×S4 | 1 | 3 | yes | — |
| 213 | C4 | — | 3 | 1 | no | — |
| 214 | C4 | — | 3 | 3 | no | — |
| 215 | C4 | — | 6 | 6 | no | — |
| 216 | C4 | — | 6 | 2 | no | — |
| 217 | C4 | — | 6 | 2 | no | — |
| 218 | C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 219 | C4 | C22×S4 | 1 | 3 | yes | — |
| 220 | C3 | — | 4 | 1 | no | Sylow |
| 221 | C2 | — | 2 | 3 | no | — |
| 222 | C2 | — | 3 | 1 | no | — |
| 223 | C2 | — | 3 | 1 | no | — |
| 224 | C2 | — | 6 | 3 | no | — |
| 225 | C2 | — | 12 | 6 | no | — |
| 226 | C2 | C23×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Fitting | C22×#41 | D3 |
| socle | C23 | C23×D3 |
| hypercenter | #41 | S4 |
| hypocenter | A4 | H(5,𝔽2) |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ #41 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. A4×#41 | C2 | semidirect |
| 2. #5919 | C2 | semidirect |
| 3. D4×S4 | C2 | semidirect |
| 4. #5912 | C2 | semidirect |
| 5. C2×C4⋊S4 | C2 | semidirect |
| 7. C22×S4 | C22 | semidirect |
| 9. Q8×A4 | C22 | semidirect |
| 10. D4×A4 | C22 | semidirect |
| 11. C2×C4×A4 | C22 | semidirect |
| 12. GL(2,ℤ/4) | C22 | semidirect |
| 13. C4⋊S4 | C22 | semidirect |
| 14. C4⋊S4 | C22 | semidirect |
| 15. C4×S4 | C22 | semidirect |
| 16. A4⋊Q8 | C22 | semidirect |
| 18. C22×#41 | D3 | semidirect |
| 49. C22×Q8 | C2×D3 | semidirect |
| 53. C22×D4 | C2×D3 | semidirect |
| 54. C23×C4 | C2×D3 | semidirect |
| 103. #41 | S4 | semidirect |
| 148. A4 | H(5,𝔽2) | semidirect |
| 166. Q8 | C2×S4 | semidirect |
| 180. D4 | C2×S4 | semidirect |
| 195. C2×C4 | C2×S4 | semidirect |
| 211. C22 | #1009 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| A4×#41 | 2 |
| #5919 | 2 |
| D4×S4 | 2 |
| #5912 | 2 |
| C2×C4⋊S4 | 2 |
| #3488 | 3 |
| #1009 | 4 |