Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1299 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1299 conjugacy classes of subgroups — too many to draw. Only its 487 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×#578 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | #3293 | C2 | 1 | 16 | yes | maximal, direct product factor |
| 3 | C2×#494 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#473 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | C2×#456 | C2 | 1 | 4 | yes | maximal, complemented |
| 6 | C2×#443 | C2 | 1 | 2 | yes | maximal, complemented |
| 7 | C2×2−1+4 | C22 | 1 | 4 | yes | — |
| 8 | #578 | — | 2 | 16 | no | — |
| 9 | #578 | C22 | 1 | 32 | yes | complemented |
| 10 | C2×#137 | — | 2 | 8 | no | — |
| 11 | C2×#137 | C22 | 1 | 8 | yes | — |
| 12 | C2×#135 | — | 2 | 4 | no | — |
| 13 | C2×#135 | C22 | 1 | 2 | yes | — |
| 14 | C22×Q16 | — | 2 | 4 | no | — |
| 15 | C22×Q16 | C22 | 1 | 4 | yes | — |
| 16 | C2×#131 | C22 | 1 | 4 | yes | — |
| 17 | C2×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 18 | #494 | C22 | 1 | 8 | yes | complemented |
| 19 | #473 | C22 | 1 | 32 | yes | complemented |
| 20 | #456 | C22 | 1 | 32 | yes | complemented |
| 21 | #443 | C22 | 1 | 16 | yes | complemented |
| 22 | C2×#108 | C22 | 1 | 2 | yes | — |
| 23 | C2×#104 | C22 | 1 | 4 | yes | — |
| 24 | C2×#101 | C22 | 1 | 4 | yes | — |
| 25 | C2×C4×C8 | C22 | 1 | 1 | yes | characteristic |
| 26 | 2−1+4 | — | 2 | 16 | no | — |
| 27 | 2−1+4 | — | 4 | 4 | no | — |
| 28 | 2−1+4 | C23 | 1 | 16 | yes | — |
| 29 | C2×#41 | — | 2 | 8 | no | — |
| 30 | C2×#41 | — | 4 | 4 | no | — |
| 31 | C2×#41 | C23 | 1 | 4 | yes | — |
| 32 | C22×Q8 | — | 2 | 8 | no | — |
| 33 | C22×Q8 | C23 | 1 | 4 | yes | — |
| 34 | #137 | — | 2 | 16 | no | — |
| 35 | #137 | — | 2 | 64 | no | — |
| 36 | #137 | C23 | 1 | 32 | yes | — |
| 37 | #135 | — | 2 | 32 | no | — |
| 38 | #135 | D4 | 1 | 8 | yes | — |
| 39 | #135 | C23 | 1 | 8 | yes | — |
| 40 | C2×Q16 | — | 2 | 8 | no | — |
| 41 | C2×Q16 | — | 2 | 32 | no | — |
| 42 | C2×Q16 | — | 2 | 4 | no | — |
| 43 | C2×Q16 | — | 2 | 4 | no | — |
| 44 | C2×Q16 | — | 2 | 8 | no | — |
| 45 | C2×Q16 | D4 | 1 | 2 | yes | — |
| 46 | C2×Q16 | C23 | 1 | 16 | yes | — |
| 47 | C2×SD16 | — | 2 | 8 | no | — |
| 48 | C2×SD16 | D4 | 1 | 4 | yes | — |
| 49 | C2×D8 | D4 | 1 | 2 | yes | — |
| 50 | #131 | — | 2 | 8 | no | — |
| 51 | #131 | C23 | 1 | 16 | yes | — |
| 52 | C2×M16 | — | 2 | 4 | no | — |
| 53 | C2×M16 | C23 | 1 | 4 | yes | — |
| 54 | C22×C8 | — | 2 | 4 | no | — |
| 55 | C22×C8 | C23 | 1 | 2 | yes | — |
| 56 | C4⋊Q8 | — | 4 | 2 | no | — |
| 57 | C4⋊Q8 | C23 | 1 | 8 | yes | — |
| 58 | C2×C4⋊C4 | — | 4 | 2 | no | — |
| 59 | C2×C42 | C23 | 1 | 1 | yes | characteristic |
| 60 | #108 | C23 | 1 | 8 | yes | — |
| 61 | #104 | C23 | 1 | 16 | yes | — |
| 62 | #101 | C23 | 1 | 16 | yes | — |
| 63 | C4×C8 | C23 | 1 | 4 | yes | — |
| 64 | #41 | — | 2 | 4 | no | — |
| 65 | #41 | — | 2 | 4 | no | — |
| 66 | #41 | — | 2 | 32 | no | — |
| 67 | #41 | — | 4 | 16 | no | — |
| 68 | #41 | — | 4 | 16 | no | — |
| 69 | #41 | — | 4 | 16 | no | — |
| 70 | #41 | C2×D4 | 1 | 8 | yes | — |
| 71 | #41 | C24 | 1 | 8 | yes | — |
| 72 | C2×Q8 | — | 2 | 4 | no | — |
| 73 | C2×Q8 | — | 2 | 32 | no | — |
| 74 | C2×Q8 | — | 2 | 8 | no | — |
| 75 | C2×Q8 | — | 2 | 8 | no | — |
| 76 | C2×Q8 | — | 4 | 4 | no | — |
| 77 | C2×Q8 | — | 4 | 16 | no | — |
| 78 | C2×Q8 | C2×D4 | 1 | 4 | yes | — |
| 79 | C2×Q8 | C24 | 1 | 8 | yes | — |
| 80 | C2×D4 | — | 2 | 4 | no | — |
| 81 | C2×D4 | — | 4 | 4 | no | — |
| 82 | C2×D4 | C2×D4 | 1 | 4 | yes | — |
| 83 | C22×C4 | — | 2 | 1 | no | — |
| 84 | C22×C4 | — | 2 | 4 | no | — |
| 85 | C22×C4 | — | 2 | 4 | no | — |
| 86 | C22×C4 | — | 4 | 8 | no | — |
| 87 | C22×C4 | C24 | 1 | 1 | yes | characteristic |
| 88 | Q16 | — | 2 | 32 | no | — |
| 89 | Q16 | — | 2 | 16 | no | — |
| 90 | Q16 | — | 2 | 16 | no | — |
| 91 | Q16 | C2×D4 | 1 | 8 | yes | — |
| 92 | SD16 | — | 2 | 32 | no | — |
| 93 | SD16 | C2×D4 | 1 | 16 | yes | — |
| 94 | D8 | C2×D4 | 1 | 8 | yes | — |
| 95 | M16 | — | 2 | 4 | no | — |
| 96 | M16 | — | 2 | 16 | no | — |
| 97 | M16 | C24 | 1 | 8 | yes | — |
| 98 | C2×C8 | — | 2 | 4 | no | — |
| 99 | C2×C8 | — | 2 | 16 | no | — |
| 100 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 101 | C2×C8 | C2×D4 | 1 | 2 | yes | — |
| 102 | C2×C8 | C2×D4 | 1 | 4 | yes | — |
| 103 | C2×C8 | C24 | 1 | 4 | yes | — |
| 104 | C4⋊C4 | — | 4 | 8 | no | — |
| 105 | C42 | — | 2 | 1 | no | — |
| 106 | C42 | C24 | 1 | 2 | yes | — |
| 107 | C23 | — | 2 | 4 | no | — |
| 108 | C23 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 109 | Q8 | — | 2 | 4 | no | — |
| 110 | Q8 | — | 2 | 16 | no | — |
| 111 | Q8 | — | 4 | 16 | no | — |
| 112 | Q8 | — | 4 | 8 | no | — |
| 113 | Q8 | C22×D4 | 1 | 8 | yes | — |
| 114 | D4 | — | 2 | 8 | no | — |
| 115 | D4 | — | 2 | 8 | no | — |
| 116 | D4 | — | 2 | 4 | no | — |
| 117 | D4 | — | 4 | 16 | no | — |
| 118 | D4 | C22×D4 | 1 | 8 | yes | — |
| 119 | C2×C4 | — | 2 | 4 | no | — |
| 120 | C2×C4 | — | 2 | 4 | no | — |
| 121 | C2×C4 | — | 2 | 2 | no | — |
| 122 | C2×C4 | — | 2 | 8 | no | — |
| 123 | C2×C4 | — | 2 | 8 | no | — |
| 124 | C2×C4 | — | 2 | 8 | no | — |
| 125 | C2×C4 | — | 4 | 32 | no | — |
| 126 | C2×C4 | — | 4 | 4 | no | — |
| 127 | C2×C4 | — | 4 | 4 | no | — |
| 128 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 129 | C2×C4 | C22×D4 | 1 | 2 | yes | — |
| 130 | C2×C4 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 131 | C2×C4 | C25 | 1 | 1 | yes | characteristic |
| 132 | C8 | — | 2 | 8 | no | — |
| 133 | C8 | C22×D4 | 1 | 4 | yes | — |
| 134 | C8 | C22×D4 | 1 | 4 | yes | — |
| 135 | C22 | — | 2 | 2 | no | — |
| 136 | C22 | — | 2 | 8 | no | — |
| 137 | C22 | — | 4 | 8 | no | — |
| 138 | C22 | D42 | 1 | 1 | yes | characteristic |
| 139 | C22 | D42 | 1 | 1 | yes | characteristic |
| 140 | C22 | C2×H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 141 | C4 | — | 2 | 4 | no | — |
| 142 | C4 | — | 2 | 8 | no | — |
| 143 | C4 | — | 4 | 8 | no | — |
| 144 | C4 | D42 | 1 | 2 | yes | — |
| 145 | C4 | C23×D4 | 1 | 2 | yes | — |
| 146 | C2 | — | 2 | 1 | no | — |
| 147 | C2 | — | 2 | 1 | no | — |
| 148 | C2 | — | 4 | 8 | no | — |
| 149 | C2 | #3293 | 1 | 2 | yes | direct product factor |
| 150 | C2 | C2×D42 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#578 | C2 | semidirect |
| 2. #3293 | C2 | direct (N × H) |
| 3. C2×#494 | C2 | semidirect |
| 4. C2×#473 | C2 | semidirect |
| 5. C2×#456 | C2 | semidirect |
| 6. C2×#443 | C2 | semidirect |
| 9. #578 | C22 | semidirect |
| 18. #494 | C22 | semidirect |
| 19. #473 | C22 | semidirect |
| 20. #456 | C22 | semidirect |
| 21. #443 | C22 | semidirect |
| 149. C2 | #3293 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#578 | 2 |
| #3293 | 2 |
| C2×#494 | 2 |
| C2×#473 | 2 |
| C2×#456 | 2 |
| C2×#443 | 2 |