Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
1917 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 1917 conjugacy classes of subgroups — too many to draw. Only its 78 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×D4×AGL(1,𝔽8) | C2 | 1 | 6 | yes | maximal, complemented |
| 2 | C42×AGL(1,𝔽8) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C23×AGL(1,𝔽8) | C22 | 1 | 4 | yes | — |
| 4 | D4×AGL(1,𝔽8) | — | 2 | 12 | no | — |
| 5 | C2×C4×AGL(1,𝔽8) | C22 | 1 | 3 | yes | — |
| 6 | C23×#127 | C7 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 7 | C22×AGL(1,𝔽8) | — | 2 | 12 | no | — |
| 8 | C22×AGL(1,𝔽8) | C23 | 1 | 1 | yes | characteristic |
| 9 | C4×AGL(1,𝔽8) | D4 | 1 | 6 | yes | complemented |
| 10 | C7×#127 | — | 8 | 1 | no | maximal |
| 11 | C24×D4 | C2×C7 | 1 | 6 | yes | complemented |
| 12 | C22×#127 | — | 7 | 1 | no | — |
| 13 | C22×#127 | — | 7 | 6 | no | — |
| 14 | C22×#127 | — | 7 | 1 | no | — |
| 15 | C23×C42 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×AGL(1,𝔽8) | — | 4 | 4 | no | — |
| 17 | C2×AGL(1,𝔽8) | C2×D4 | 1 | 3 | yes | — |
| 18 | C2×C7×D4 | — | 8 | 6 | no | — |
| 19 | C42×C7 | — | 8 | 1 | no | — |
| 20 | C26 | C22×C7 | 1 | 4 | yes | — |
| 21 | C23×D4 | — | 2 | 12 | no | — |
| 22 | C23×D4 | — | 7 | 6 | no | — |
| 23 | C23×D4 | — | 7 | 6 | no | — |
| 24 | C23×D4 | — | 7 | 12 | no | — |
| 25 | C23×D4 | — | 14 | 12 | no | — |
| 26 | C24×C4 | C22×C7 | 1 | 3 | yes | — |
| 27 | C2×#127 | — | 7 | 24 | no | — |
| 28 | C2×#127 | — | 7 | 18 | no | — |
| 29 | C2×#127 | — | 7 | 1 | no | — |
| 30 | C2×#127 | — | 7 | 3 | no | — |
| 31 | C2×#127 | — | 7 | 18 | no | — |
| 32 | C22×C42 | — | 7 | 1 | no | — |
| 33 | C22×C42 | — | 7 | 3 | no | — |
| 34 | C23×C7 | — | 8 | 4 | no | — |
| 35 | AGL(1,𝔽8) | #127 | 1 | 1 | yes | characteristic, direct product factor |
| 36 | C7×D4 | — | 16 | 12 | no | — |
| 37 | C2×C4×C7 | — | 8 | 3 | no | — |
| 38 | C25 | — | 2 | 12 | no | — |
| 39 | C25 | — | 7 | 4 | no | — |
| 40 | C25 | — | 7 | 4 | no | — |
| 41 | C25 | — | 14 | 12 | no | — |
| 42 | C25 | C23×C7 | 1 | 1 | yes | characteristic |
| 43 | C22×D4 | — | 7 | 36 | no | — |
| 44 | C22×D4 | — | 7 | 6 | no | — |
| 45 | C22×D4 | — | 7 | 18 | no | — |
| 46 | C22×D4 | — | 7 | 36 | no | — |
| 47 | C22×D4 | — | 14 | 72 | no | — |
| 48 | C22×D4 | — | 14 | 36 | no | — |
| 49 | C22×D4 | — | 14 | 36 | no | — |
| 50 | C22×D4 | — | 14 | 12 | no | — |
| 51 | C22×D4 | — | 14 | 12 | no | — |
| 52 | C22×D4 | — | 14 | 24 | no | — |
| 53 | C23×C4 | — | 7 | 3 | no | — |
| 54 | C23×C4 | — | 7 | 3 | no | — |
| 55 | C23×C4 | — | 7 | 6 | no | — |
| 56 | C23×C4 | C7×D4 | 1 | 6 | yes | complemented |
| 57 | #127 | — | 7 | 24 | no | — |
| 58 | #127 | — | 7 | 24 | no | — |
| 59 | #127 | — | 7 | 18 | no | — |
| 60 | #127 | — | 7 | 6 | no | — |
| 61 | #127 | — | 7 | 1 | no | — |
| 62 | #127 | AGL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 63 | C2×C42 | — | 7 | 6 | no | — |
| 64 | C2×C42 | — | 7 | 1 | no | — |
| 65 | C2×C42 | — | 7 | 9 | no | — |
| 66 | C22×C7 | — | 8 | 1 | no | — |
| 67 | C22×C7 | — | 16 | 12 | no | — |
| 68 | C4×C7 | — | 8 | 6 | no | — |
| 69 | C24 | — | 4 | 4 | no | — |
| 70 | C24 | — | 7 | 3 | no | — |
| 71 | C24 | — | 7 | 1 | no | — |
| 72 | C24 | — | 7 | 12 | no | — |
| 73 | C24 | — | 7 | 4 | no | — |
| 74 | C24 | — | 14 | 36 | no | — |
| 75 | C24 | — | 14 | 36 | no | — |
| 76 | C24 | — | 14 | 12 | no | — |
| 77 | C24 | — | 14 | 12 | no | — |
| 78 | C24 | — | 28 | 12 | no | — |
| 79 | C24 | — | 28 | 24 | no | — |
| 80 | C24 | C2×C7×D4 | 1 | 3 | yes | — |
| 81 | C2×D4 | — | 7 | 36 | no | — |
| 82 | C2×D4 | — | 7 | 12 | no | — |
| 83 | C2×D4 | — | 7 | 6 | no | — |
| 84 | C2×D4 | — | 14 | 12 | no | — |
| 85 | C2×D4 | — | 14 | 12 | no | — |
| 86 | C2×D4 | — | 14 | 72 | no | — |
| 87 | C2×D4 | — | 14 | 72 | no | — |
| 88 | C2×D4 | — | 14 | 72 | no | — |
| 89 | C2×D4 | — | 14 | 36 | no | — |
| 90 | C2×D4 | — | 14 | 36 | no | — |
| 91 | C2×D4 | — | 14 | 72 | no | — |
| 92 | C2×D4 | C2×AGL(1,𝔽8) | 1 | 6 | yes | complemented |
| 93 | C22×C4 | — | 7 | 18 | no | — |
| 94 | C22×C4 | — | 7 | 3 | no | — |
| 95 | C22×C4 | — | 7 | 9 | no | — |
| 96 | C22×C4 | — | 7 | 18 | no | — |
| 97 | C22×C4 | — | 7 | 6 | no | — |
| 98 | C22×C4 | — | 7 | 6 | no | — |
| 99 | C42 | — | 7 | 6 | no | — |
| 100 | C42 | — | 7 | 3 | no | — |
| 101 | C42 | C2×AGL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 102 | C2×C7 | — | 8 | 3 | no | — |
| 103 | C2×C7 | — | 32 | 4 | no | — |
| 104 | C23 | — | 7 | 3 | no | — |
| 105 | C23 | — | 7 | 3 | no | — |
| 106 | C23 | — | 7 | 9 | no | — |
| 107 | C23 | — | 7 | 6 | no | — |
| 108 | C23 | — | 7 | 1 | no | — |
| 109 | C23 | — | 7 | 4 | no | — |
| 110 | C23 | — | 14 | 12 | no | — |
| 111 | C23 | — | 14 | 12 | no | — |
| 112 | C23 | — | 14 | 36 | no | — |
| 113 | C23 | — | 14 | 36 | no | — |
| 114 | C23 | — | 28 | 4 | no | — |
| 115 | C23 | — | 28 | 4 | no | — |
| 116 | C23 | — | 28 | 36 | no | — |
| 117 | C23 | — | 28 | 36 | no | — |
| 118 | C23 | — | 28 | 24 | no | — |
| 119 | C23 | — | 28 | 24 | no | — |
| 120 | C23 | C7×#127 | 1 | 1 | yes | characteristic, complemented |
| 121 | C23 | C22×AGL(1,𝔽8) | 1 | 4 | yes | — |
| 122 | D4 | — | 2 | 12 | no | — |
| 123 | D4 | — | 14 | 24 | no | — |
| 124 | D4 | — | 14 | 72 | no | — |
| 125 | D4 | — | 14 | 12 | no | — |
| 126 | C2×C4 | — | 7 | 18 | no | — |
| 127 | C2×C4 | — | 7 | 3 | no | — |
| 128 | C2×C4 | — | 7 | 6 | no | — |
| 129 | C2×C4 | — | 7 | 6 | no | — |
| 130 | C2×C4 | — | 7 | 18 | no | — |
| 131 | C2×C4 | C22×AGL(1,𝔽8) | 1 | 3 | yes | — |
| 132 | C7 | — | 8 | 1 | no | Sylow |
| 133 | C22 | — | 2 | 12 | no | — |
| 134 | C22 | — | 7 | 1 | no | — |
| 135 | C22 | — | 7 | 9 | no | — |
| 136 | C22 | — | 7 | 6 | no | — |
| 137 | C22 | — | 7 | 3 | no | — |
| 138 | C22 | — | 7 | 3 | no | — |
| 139 | C22 | — | 14 | 12 | no | — |
| 140 | C22 | — | 28 | 12 | no | — |
| 141 | C22 | — | 28 | 4 | no | — |
| 142 | C22 | — | 28 | 12 | no | — |
| 143 | C22 | — | 28 | 36 | no | — |
| 144 | C22 | C23×AGL(1,𝔽8) | 1 | 1 | yes | characteristic |
| 145 | C4 | — | 7 | 6 | no | — |
| 146 | C4 | D4×AGL(1,𝔽8) | 1 | 6 | yes | complemented |
| 147 | C2 | — | 4 | 4 | no | — |
| 148 | C2 | — | 7 | 1 | no | — |
| 149 | C2 | — | 7 | 3 | no | — |
| 150 | C2 | — | 28 | 4 | no | — |
| 151 | C2 | C2×D4×AGL(1,𝔽8) | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽8) |
| commutator, socle, cosocle | C25 | C23×C7 |
| Fitting | C23×#127 | C7 |
| hypercenter | #127 | AGL(1,𝔽8) |
| hypocenter | C23 | C7×#127 |
| LCS | G ▷ C25 ▷ C23 | |
| UCS | C1 ◁ C22 ◁ #127 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D4×AGL(1,𝔽8) | C2 | semidirect |
| 2. C42×AGL(1,𝔽8) | C2 | semidirect |
| 6. C23×#127 | C7 | semidirect |
| 9. C4×AGL(1,𝔽8) | D4 | semidirect |
| 11. C24×D4 | C2×C7 | semidirect |
| 15. C23×C42 | C2×C7 | semidirect |
| 35. AGL(1,𝔽8) | #127 | direct (N × H) |
| 56. C23×C4 | C7×D4 | semidirect |
| 62. #127 | AGL(1,𝔽8) | direct (N × H) |
| 92. C2×D4 | C2×AGL(1,𝔽8) | semidirect |
| 101. C42 | C2×AGL(1,𝔽8) | semidirect |
| 120. C23 | C7×#127 | semidirect |
| 146. C4 | D4×AGL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D4×AGL(1,𝔽8) | 2 |
| C42×AGL(1,𝔽8) | 2 |
| C23×#127 | 7 |
| C7×#127 | 8 |