Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
270 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 45 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 | C3×H(3,𝔽4) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5061 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5032 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #4541 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #4536 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #2199 | — | 3 | 1 | no | maximal, Sylow |
| 7 | #1004 | C22 | 1 | 1 | yes | characteristic |
| 8 | C3×#124 | — | 2 | 1 | no | — |
| 9 | C3×#124 | — | 2 | 2 | no | — |
| 10 | C3×#124 | — | 2 | 1 | no | — |
| 11 | C3×#124 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×#120 | — | 2 | 2 | no | — |
| 13 | C3×#120 | C22 | 1 | 2 | yes | — |
| 14 | #953 | — | 2 | 2 | no | — |
| 15 | #942 | — | 2 | 1 | no | — |
| 16 | #834 | C22 | 1 | 2 | yes | — |
| 17 | #833 | C22 | 1 | 1 | yes | characteristic |
| 18 | #806 | — | 2 | 2 | no | — |
| 19 | #805 | — | 2 | 1 | no | — |
| 20 | H(3,𝔽4) | D3 | 1 | 1 | yes | characteristic, complemented |
| 21 | B(4,𝔽2) | — | 3 | 2 | no | — |
| 22 | #455 | — | 3 | 1 | no | — |
| 23 | #354 | — | 3 | 1 | no | — |
| 24 | AΣL(1,𝔽16) | — | 3 | 2 | no | — |
| 25 | C24×C3 | D4 | 1 | 2 | yes | complemented |
| 26 | C2×C3×Q8 | — | 2 | 1 | no | — |
| 27 | C2×C3×Q8 | D4 | 1 | 1 | yes | characteristic, complemented |
| 28 | C2×C3×D4 | — | 2 | 1 | no | — |
| 29 | C2×C3×D4 | — | 2 | 1 | no | — |
| 30 | C2×C3×D4 | — | 2 | 2 | no | — |
| 31 | C2×C3×D4 | C23 | 1 | 1 | yes | characteristic |
| 32 | C2×#67 | — | 2 | 2 | no | — |
| 33 | C2×#67 | — | 4 | 2 | no | — |
| 34 | #237 | — | 4 | 1 | no | — |
| 35 | D3×D4 | — | 4 | 1 | no | — |
| 36 | #235 | — | 2 | 1 | no | — |
| 37 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 38 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 39 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 40 | C3×C22⋊C4 | — | 2 | 2 | no | — |
| 41 | C3×C22⋊C4 | D4 | 1 | 2 | yes | complemented |
| 42 | C3×C42 | — | 2 | 1 | no | — |
| 43 | C3×C42 | D4 | 1 | 1 | yes | characteristic, complemented |
| 44 | #217 | — | 2 | 2 | no | — |
| 45 | #217 | — | 4 | 2 | no | — |
| 46 | #216 | — | 4 | 1 | no | — |
| 47 | #215 | — | 4 | 1 | no | — |
| 48 | #208 | — | 2 | 1 | no | — |
| 49 | H(5,𝔽2) | — | 3 | 1 | no | — |
| 50 | #137 | — | 6 | 1 | no | — |
| 51 | #124 | — | 2 | 1 | no | — |
| 52 | #124 | — | 2 | 2 | no | — |
| 53 | #124 | — | 2 | 1 | no | — |
| 54 | #124 | C2×D3 | 1 | 1 | yes | characteristic |
| 55 | #120 | — | 2 | 2 | no | — |
| 56 | #120 | — | 6 | 2 | no | — |
| 57 | #120 | C2×D3 | 1 | 2 | yes | — |
| 58 | #104 | — | 6 | 1 | no | — |
| 59 | #100 | — | 3 | 1 | no | — |
| 60 | #99 | — | 3 | 2 | no | — |
| 61 | #99 | — | 6 | 2 | no | — |
| 62 | C23×C3 | — | 2 | 2 | no | — |
| 63 | C23×C3 | — | 4 | 2 | no | — |
| 64 | C23×C3 | — | 8 | 2 | no | — |
| 65 | C23×C3 | C2×D4 | 1 | 2 | yes | — |
| 66 | C22×D3 | — | 2 | 2 | no | — |
| 67 | C3×Q8 | — | 4 | 1 | no | — |
| 68 | C3×Q8 | — | 8 | 1 | no | — |
| 69 | C3×D4 | — | 4 | 1 | no | — |
| 70 | C3×D4 | — | 8 | 2 | no | — |
| 71 | C3×D4 | — | 8 | 1 | no | — |
| 72 | C3×D4 | — | 8 | 1 | no | — |
| 73 | C2×C3×C4 | — | 2 | 1 | no | — |
| 74 | C2×C3×C4 | — | 2 | 2 | no | — |
| 75 | C2×C3×C4 | — | 2 | 1 | no | — |
| 76 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 77 | #67 | — | 4 | 2 | no | — |
| 78 | #67 | — | 4 | 1 | no | — |
| 79 | #67 | — | 8 | 4 | no | — |
| 80 | C2×Dic3 | — | 2 | 2 | no | — |
| 81 | C2×Dic3 | — | 4 | 2 | no | — |
| 82 | D12 | — | 2 | 1 | no | — |
| 83 | C4×D3 | — | 4 | 1 | no | — |
| 84 | Dic6 | — | 2 | 1 | no | — |
| 85 | C3⋊C8 | — | 4 | 1 | no | — |
| 86 | C24 | #67 | 1 | 2 | yes | complemented |
| 87 | #41 | — | 6 | 1 | no | — |
| 88 | #41 | — | 12 | 1 | no | — |
| 89 | C2×Q8 | — | 2 | 1 | no | — |
| 90 | C2×Q8 | #67 | 1 | 1 | yes | characteristic, complemented |
| 91 | C2×D4 | — | 2 | 1 | no | — |
| 92 | C2×D4 | — | 2 | 1 | no | — |
| 93 | C2×D4 | — | 2 | 2 | no | — |
| 94 | C2×D4 | — | 6 | 2 | no | — |
| 95 | C2×D4 | — | 12 | 1 | no | — |
| 96 | C2×D4 | — | 12 | 2 | no | — |
| 97 | C2×D4 | C22×D3 | 1 | 1 | yes | characteristic |
| 98 | Q16 | — | 12 | 1 | no | — |
| 99 | SD16 | — | 12 | 1 | no | — |
| 100 | M16 | — | 6 | 1 | no | — |
| 101 | C22⋊C4 | — | 2 | 2 | no | — |
| 102 | C22⋊C4 | — | 2 | 2 | no | — |
| 103 | C22⋊C4 | — | 2 | 2 | no | — |
| 104 | C22⋊C4 | — | 2 | 2 | no | — |
| 105 | C22⋊C4 | — | 6 | 2 | no | — |
| 106 | C22⋊C4 | — | 12 | 2 | no | — |
| 107 | C22⋊C4 | #67 | 1 | 2 | yes | complemented |
| 108 | C42 | — | 2 | 1 | no | — |
| 109 | C42 | #67 | 1 | 1 | yes | characteristic, complemented |
| 110 | C22×C3 | — | 2 | 2 | no | — |
| 111 | C22×C3 | — | 4 | 2 | no | — |
| 112 | C22×C3 | — | 4 | 2 | no | — |
| 113 | C22×C3 | — | 4 | 2 | no | — |
| 114 | C22×C3 | — | 4 | 4 | no | — |
| 115 | C22×C3 | — | 4 | 2 | no | — |
| 116 | C22×C3 | — | 8 | 2 | no | — |
| 117 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 118 | C2×D3 | — | 4 | 1 | no | — |
| 119 | C2×D3 | — | 8 | 2 | no | — |
| 120 | C3×C4 | — | 2 | 1 | no | — |
| 121 | C3×C4 | — | 4 | 2 | no | — |
| 122 | C3×C4 | — | 4 | 1 | no | — |
| 123 | C3×C4 | — | 4 | 1 | no | — |
| 124 | Dic3 | — | 4 | 1 | no | — |
| 125 | Dic3 | — | 8 | 2 | no | — |
| 126 | C23 | — | 2 | 2 | no | — |
| 127 | C23 | — | 4 | 2 | no | — |
| 128 | C23 | — | 6 | 2 | no | — |
| 129 | C23 | — | 8 | 2 | no | — |
| 130 | C23 | C2×#67 | 1 | 2 | yes | — |
| 131 | Q8 | — | 4 | 1 | no | — |
| 132 | Q8 | — | 6 | 1 | no | — |
| 133 | Q8 | — | 8 | 1 | no | — |
| 134 | D4 | — | 4 | 1 | no | — |
| 135 | D4 | — | 6 | 1 | no | — |
| 136 | D4 | — | 8 | 2 | no | — |
| 137 | D4 | — | 8 | 1 | no | — |
| 138 | D4 | — | 8 | 1 | no | — |
| 139 | D4 | — | 12 | 2 | no | — |
| 140 | D4 | — | 12 | 1 | no | — |
| 141 | D4 | — | 24 | 4 | no | — |
| 142 | C2×C4 | — | 2 | 1 | no | — |
| 143 | C2×C4 | — | 2 | 2 | no | — |
| 144 | C2×C4 | — | 2 | 1 | no | — |
| 145 | C2×C4 | — | 6 | 2 | no | — |
| 146 | C2×C4 | — | 12 | 1 | no | — |
| 147 | C2×C4 | — | 12 | 2 | no | — |
| 148 | C2×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 149 | C8 | — | 12 | 1 | no | — |
| 150 | C2×C3 | — | 2 | 1 | no | — |
| 151 | C2×C3 | — | 4 | 2 | no | — |
| 152 | C2×C3 | — | 8 | 2 | no | — |
| 153 | C2×C3 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 154 | D3 | — | 8 | 1 | no | — |
| 155 | C22 | — | 2 | 2 | no | — |
| 156 | C22 | — | 4 | 2 | no | — |
| 157 | C22 | — | 4 | 2 | no | — |
| 158 | C22 | — | 4 | 2 | no | — |
| 159 | C22 | — | 4 | 4 | no | — |
| 160 | C22 | — | 4 | 2 | no | — |
| 161 | C22 | — | 8 | 2 | no | — |
| 162 | C22 | — | 12 | 1 | no | — |
| 163 | C22 | — | 24 | 2 | no | — |
| 164 | C22 | #953 | 1 | 1 | yes | characteristic |
| 165 | C4 | — | 2 | 1 | no | — |
| 166 | C4 | — | 4 | 2 | no | — |
| 167 | C4 | — | 4 | 1 | no | — |
| 168 | C4 | — | 4 | 1 | no | — |
| 169 | C4 | — | 12 | 1 | no | — |
| 170 | C4 | — | 24 | 2 | no | — |
| 171 | C3 | #2199 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 172 | C2 | — | 2 | 1 | no | — |
| 173 | C2 | — | 4 | 2 | no | — |
| 174 | C2 | — | 8 | 2 | no | — |
| 175 | C2 | — | 24 | 1 | no | — |
| 176 | C2 | #5244 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×H(3,𝔽4) | C2 | semidirect |
| 2. #5061 | C2 | semidirect |
| 3. #5032 | C2 | semidirect |
| 4. #4541 | C2 | semidirect |
| 5. #4536 | C2 | semidirect |
| 20. H(3,𝔽4) | D3 | semidirect |
| 25. C24×C3 | D4 | semidirect |
| 27. C2×C3×Q8 | D4 | semidirect |
| 41. C3×C22⋊C4 | D4 | semidirect |
| 43. C3×C42 | D4 | semidirect |
| 86. C24 | #67 | semidirect |
| 90. C2×Q8 | #67 | semidirect |
| 107. C22⋊C4 | #67 | semidirect |
| 109. C42 | #67 | semidirect |
| 171. C3 | #2199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×H(3,𝔽4) | 2 |
| #5061 | 2 |
| #5032 | 2 |
| #4541 | 2 |
| #4536 | 2 |
| #2199 | 3 |