Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
751 conjugacy classes of subgroups; only the 191 normal subgroups are shown.
This group has 751 conjugacy classes of subgroups — too many to draw. Only its 191 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×#497 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C2×#496 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#495 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C2×#492 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | #2658 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #2657 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #1726 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | C22×Q16 | C22 | 1 | 3 | yes | — |
| 9 | C22×SD16 | C22 | 1 | 3 | yes | — |
| 10 | C22×SD16 | C22 | 1 | 6 | yes | — |
| 11 | C22×SD16 | C22 | 1 | 1 | yes | characteristic |
| 12 | C22×D8 | C22 | 1 | 3 | yes | — |
| 13 | C2×C4⋊Q8 | C22 | 1 | 3 | yes | — |
| 14 | C2×#127 | C22 | 1 | 3 | yes | — |
| 15 | C2×#124 | C22 | 1 | 3 | yes | — |
| 16 | C2×#124 | C22 | 1 | 3 | yes | — |
| 17 | #497 | — | 2 | 12 | no | — |
| 18 | #496 | — | 2 | 12 | no | — |
| 19 | #495 | — | 2 | 12 | no | — |
| 20 | #492 | — | 2 | 12 | no | — |
| 21 | C2×#97 | C22 | 1 | 3 | yes | — |
| 22 | C2×C4×C8 | C22 | 1 | 3 | yes | — |
| 23 | #399 | — | 2 | 4 | no | — |
| 24 | #392 | — | 2 | 4 | no | — |
| 25 | #376 | C22 | 1 | 1 | yes | characteristic |
| 26 | C22×Q8 | C23 | 1 | 1 | yes | characteristic |
| 27 | C22×Q8 | C23 | 1 | 3 | yes | — |
| 28 | C22×D4 | — | 2 | 6 | no | — |
| 29 | C22×D4 | C23 | 1 | 1 | yes | characteristic |
| 30 | C22×D4 | C23 | 1 | 3 | yes | — |
| 31 | C2×Q16 | — | 2 | 6 | no | — |
| 32 | C2×Q16 | — | 2 | 12 | no | — |
| 33 | C2×SD16 | — | 2 | 12 | no | — |
| 34 | C2×SD16 | — | 2 | 12 | no | — |
| 35 | C2×SD16 | — | 2 | 6 | no | — |
| 36 | C2×SD16 | — | 2 | 6 | no | — |
| 37 | C2×SD16 | — | 2 | 12 | no | — |
| 38 | C2×SD16 | — | 2 | 12 | no | — |
| 39 | C2×D8 | — | 2 | 6 | no | — |
| 40 | C2×D8 | — | 2 | 12 | no | — |
| 41 | C22×C8 | C23 | 1 | 3 | yes | — |
| 42 | C22×C8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C4⋊Q8 | — | 2 | 6 | no | — |
| 44 | C4⋊Q8 | — | 2 | 6 | no | — |
| 45 | #127 | — | 2 | 6 | no | — |
| 46 | #127 | — | 2 | 6 | no | — |
| 47 | #124 | — | 2 | 6 | no | — |
| 48 | #124 | — | 2 | 6 | no | — |
| 49 | #124 | — | 2 | 6 | no | — |
| 50 | #124 | — | 2 | 6 | no | — |
| 51 | C2×C4⋊C4 | — | 2 | 6 | no | — |
| 52 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 53 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 54 | C2×C42 | C23 | 1 | 3 | yes | — |
| 55 | #97 | D4 | 1 | 12 | yes | complemented |
| 56 | C4×C8 | D4 | 1 | 12 | yes | complemented |
| 57 | #95 | D4 | 1 | 4 | yes | — |
| 58 | C24 | — | 2 | 1 | no | — |
| 59 | C24 | — | 2 | 3 | no | — |
| 60 | C2×Q8 | — | 2 | 3 | no | — |
| 61 | C2×Q8 | — | 2 | 3 | no | — |
| 62 | C2×Q8 | — | 2 | 3 | no | — |
| 63 | C2×Q8 | — | 2 | 3 | no | — |
| 64 | C2×Q8 | — | 2 | 6 | no | — |
| 65 | C2×Q8 | — | 2 | 6 | no | — |
| 66 | C2×D4 | — | 2 | 3 | no | — |
| 67 | C2×D4 | — | 2 | 3 | no | — |
| 68 | C2×D4 | — | 2 | 3 | no | — |
| 69 | C2×D4 | — | 2 | 3 | no | — |
| 70 | C2×D4 | — | 2 | 6 | no | — |
| 71 | C2×D4 | — | 2 | 6 | no | — |
| 72 | C2×D4 | — | 4 | 12 | no | — |
| 73 | C2×D4 | — | 8 | 12 | no | — |
| 74 | C22×C4 | — | 2 | 1 | no | — |
| 75 | C22×C4 | — | 2 | 3 | no | — |
| 76 | C22×C4 | C2×D4 | 1 | 6 | yes | — |
| 77 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 78 | Q16 | — | 4 | 12 | no | — |
| 79 | SD16 | — | 4 | 24 | no | — |
| 80 | SD16 | — | 4 | 12 | no | — |
| 81 | SD16 | — | 4 | 4 | no | — |
| 82 | D8 | — | 4 | 12 | no | — |
| 83 | C2×C8 | C2×D4 | 1 | 6 | yes | — |
| 84 | C2×C8 | C2×D4 | 1 | 6 | yes | — |
| 85 | C2×C8 | C2×D4 | 1 | 12 | yes | — |
| 86 | C4⋊C4 | — | 4 | 12 | no | — |
| 87 | C22⋊C4 | — | 4 | 12 | no | — |
| 88 | C22⋊C4 | — | 4 | 12 | no | — |
| 89 | C42 | C2×D4 | 1 | 6 | yes | — |
| 90 | C42 | C2×D4 | 1 | 6 | yes | — |
| 91 | C23 | — | 4 | 3 | no | — |
| 92 | C23 | — | 4 | 3 | no | — |
| 93 | C23 | — | 4 | 6 | no | — |
| 94 | C23 | — | 4 | 12 | no | — |
| 95 | C23 | — | 4 | 4 | no | — |
| 96 | C23 | C22×D4 | 1 | 1 | yes | characteristic |
| 97 | Q8 | — | 4 | 6 | no | — |
| 98 | Q8 | — | 4 | 1 | no | — |
| 99 | Q8 | — | 4 | 3 | no | — |
| 100 | Q8 | — | 4 | 3 | no | — |
| 101 | Q8 | — | 4 | 3 | no | — |
| 102 | D4 | — | 4 | 3 | no | — |
| 103 | D4 | — | 4 | 3 | no | — |
| 104 | D4 | — | 4 | 1 | no | — |
| 105 | D4 | — | 4 | 3 | no | — |
| 106 | D4 | — | 4 | 6 | no | — |
| 107 | D4 | — | 8 | 24 | no | — |
| 108 | C2×C4 | — | 2 | 12 | no | — |
| 109 | C2×C4 | — | 4 | 3 | no | — |
| 110 | C2×C4 | — | 4 | 3 | no | — |
| 111 | C2×C4 | — | 4 | 6 | no | — |
| 112 | C2×C4 | #127 | 1 | 12 | yes | — |
| 113 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 114 | C2×C4 | C22×D4 | 1 | 3 | yes | — |
| 115 | C8 | #127 | 1 | 12 | yes | — |
| 116 | C8 | #127 | 1 | 4 | yes | — |
| 117 | C22 | — | 8 | 1 | no | — |
| 118 | C22 | — | 8 | 3 | no | — |
| 119 | C22 | — | 8 | 12 | no | — |
| 120 | C22 | — | 8 | 6 | no | — |
| 121 | C22 | — | 8 | 6 | no | — |
| 122 | C22 | C2×#127 | 1 | 3 | yes | — |
| 123 | C22 | #577 | 1 | 4 | yes | — |
| 124 | C4 | — | 2 | 12 | no | — |
| 125 | C4 | — | 8 | 1 | no | — |
| 126 | C4 | — | 8 | 3 | no | — |
| 127 | C4 | C2×#127 | 1 | 3 | yes | — |
| 128 | C4 | C2×#127 | 1 | 1 | yes | characteristic |
| 129 | C2 | — | 16 | 3 | no | — |
| 130 | C2 | — | 16 | 1 | no | — |
| 131 | C2 | #2867 | 1 | 1 | yes | characteristic |
| 132 | C2 | #3154 | 1 | 6 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#497 | C2 | semidirect |
| 2. C2×#496 | C2 | semidirect |
| 3. C2×#495 | C2 | semidirect |
| 4. C2×#492 | C2 | semidirect |
| 5. #2658 | C2 | semidirect |
| 6. #2657 | C2 | semidirect |
| 7. #1726 | C2 | semidirect |
| 55. #97 | D4 | semidirect |
| 56. C4×C8 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#497 | 2 |
| C2×#496 | 2 |
| C2×#495 | 2 |
| C2×#492 | 2 |
| #2658 | 2 |
| #2657 | 2 |
| #1726 | 2 |