Finite Group Databasebeta

← #30640806

Subgroups — #30640806

Order 16256.

170 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22550311C212yesmaximal, complemented
2#22550253C211yesmaximal, characteristic, complemented
3#22549697C211yesmaximal, characteristic, complemented
4#22549688C212yesmaximal, complemented
5#443×C127C211yesmaximal, characteristic, complemented
6C2×D1016—21no—
7C2×D1016C2212yes—
8#16861206—21no—
9#16861152—22no—
10#16861150—22no—
11#16861149C2212yes—
12#135×C127C2211yescharacteristic
13#131×C127—21no—
14#108×C127C2211yescharacteristic
15#104×C127—21no—
16C4×C8×C127C2211yescharacteristic
17C2×D508—22no—
18C2×D508—42no—
19D1016—22no—
20D1016—22no—
21D1016—41no—
22D1016—41no—
23C127⋊C16—22no—
24#41×C127—21no—
25Q16×C127D411yescharacteristic, complemented
26SD16×C127—21no—
27D8×C127D411yescharacteristic, complemented
28M16×C127—21no—
29M16×C127—41no—
30C2×C8×C127—41no—
31C2×C8×C127D411yescharacteristic
32C2×C8×C127C2311yescharacteristic
33C42×C127D411yescharacteristic
34C22×D127—42no—
35D508—42no—
36D508—42no—
37D508—84no—
38Q8×C127—21no—
39D4×C127—21no—
40D4×C127—41no—
41C2×C4×C127—21no—
42C2×C4×C127—41no—
43C2×C4×C127C2×D411yescharacteristic
44C8×C127—41no—
45C8×C127D811yescharacteristic
46C8×C127D811yescharacteristic
47C8×C127C2×D411yescharacteristic
48C8×C127#4111yescharacteristic
49C22×C127—41no—
50C22×C127AGL(1,ℤ/8)11yescharacteristic
51C2×D127—84no—
52C2×D127—82no—
53C4×C127—22no—
54C4×C127—41no—
55C4×C127#12111yescharacteristic
56C4×C127C2×D811yescharacteristic
57C2×C127—21no—
58C2×C127—81no—
59C2×C127#45911yescharacteristic
60D127—162no—
61#2213—1271nomaximal, Sylow
62C127#221311yescharacteristic, complemented, Sylow
63#509—1272no—
64#493—1271no—
65#443D12711yescharacteristic, complemented
66#364—1271no—
67#361—1272no—
68#135C2×D12711yescharacteristic
69C2×D8—1272no—
70C2×D8—2541no—
71#131—21no—
72#127—2541no—
73SD32—2542no—
74D16—2542no—
75M32—1272no—
76#108C2×D12711yescharacteristic
77#104—21no—
78C4×C8C2×D12711yescharacteristic
79#41—21no—
80C2×D4—5082no—
81C2×D4—2542no—
82Q16D50811yescharacteristic, complemented
83SD16—21no—
84D8—5081no—
85D8—5081no—
86D8—2542no—
87D8—2542no—
88D8D50811yescharacteristic, complemented
89M16—21no—
90M16—41no—
91C2×C8—41no—
92C2×C8C22⋊D12711yescharacteristic
93C2×C8C22×D12711yescharacteristic
94C42C22⋊D12711yescharacteristic
95C16—2542no—
96C23—5082no—
97Q8—21no—
98D4—21no—
99D4—41no—
100D4—10164no—
101D4—5082no—
102D4—5082no—
103C2×C4—21no—
104C2×C4—41no—
105C2×C4C2×C22⋊D12711yescharacteristic
106C8—41no—
107C8#1452323311yescharacteristic
108C8#1452323311yescharacteristic
109C8C2×D50811yescharacteristic
110C8#1452324811yescharacteristic
111C22—41no—
112C22—10164no—
113C22—10162no—
114C22#1686123411yescharacteristic
115C4—22no—
116C4—41no—
117C4#1686118311yescharacteristic
118C4C2×#1452323311yescharacteristic
119C2—21no—
120C2—81no—
121C2—20322no—
122C2#2255007111yescharacteristic
subgroupG/H
centerC2#22550071
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
Fitting#443×C127C2
socleC2×C127#459
hypercenter#443D127
hypocenterC127#2213
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #443

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #22550311C2semidirect
2. #22550253C2semidirect
3. #22549697C2semidirect
4. #22549688C2semidirect
5. #443×C127C2semidirect
25. Q16×C127D4semidirect
27. D8×C127D4semidirect
62. C127#2213semidirect
65. #443D127semidirect
82. Q16D508semidirect
88. D8D508semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225503112
#225502532
#225496972
#225496882
#443×C1272
#2213127