Finite Group Databasebeta

← #30638229

Subgroups — #30638229

Order 16256.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550274C211yesmaximal, characteristic, complemented
2#22550166C211yesmaximal, characteristic
3#22550081C211yesmaximal, characteristic
4#22549629C211yesmaximal, characteristic, complemented
5#22549624C211yesmaximal, characteristic, complemented
6#22549621C211yesmaximal, characteristic, complemented
7#436×C127C211yesmaximal, characteristic
8C2×#14523236—21no—
9#16861209C2211yescharacteristic
10#16861209C2211yescharacteristic
11#16861141—21no—
12#16861141—21no—
13#16861140—21no—
14C4⋊(C127⋊C8)C2211yescharacteristic
15C4⋊(C127⋊C8)C2211yescharacteristic
16#16861132—21no—
17#16861130—21no—
18C2×M16×C127—21no—
19C4×D4×C127C2211yescharacteristic
20C4⋊C8×C127C2211yescharacteristic
21C22⋊C8×C127—21no—
22C4×C8×C127C2211yescharacteristic
23C2×Dic254—21no—
24C2×Dic254—21no—
25#14523236—42no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32C2×D4×C127D411yescharacteristic
33C22×C4×C127—21no—
34M16×C127—42no—
35C2×C8×C127—21no—
36C2×C8×C127D411yescharacteristic
37C2×C8×C127D411yescharacteristic
38C4⋊C4×C127D411yescharacteristic
39C22⋊C4×C127—21no—
40C42×C127C2311yescharacteristic
41Dic254—41no—
42Dic254—41no—
43Dic254—81no—
44C2×Dic127—41no—
45C2×Dic127—41no—
46C127⋊C8—41no—
47C127⋊C8—41no—
48C23×C127—21no—
49D4×C127—21no—
50D4×C127—21no—
51C2×C4×C127—21no—
52C2×C4×C127—41no—
53C2×C4×C127—41no—
54C2×C4×C127C2×D411yescharacteristic
55C2×C4×C127C2×D411yescharacteristic
56C2×C4×C127#4111yescharacteristic
57C8×C127—21no—
58C8×C127—41no—
59C8×C127SD1612yescomplemented
60C22×C127—41no—
61C22×C127—41no—
62C22×C127—41no—
63C22×C127#12111yescharacteristic
64C4×C127—21no—
65C4×C127—41no—
66C4×C127C2×SD1611yescharacteristic
67C4×C127AGL(1,ℤ/8)11yescharacteristic
68C4×C127#13711yescharacteristic
69C4×C127#13711yescharacteristic
70Dic127—81no—
71Dic127—81no—
72C2×C127—81no—
73C2×C127#46111yescharacteristic
74C2×C127#46811yescharacteristic
75C2×C127#47111yescharacteristic
76#1691—1271nomaximal, Sylow
77C127#169111yescharacteristic, complemented, Sylow
78#498—1271no—
79#476—1271no—
80#461—1271no—
81#436D12711yescharacteristic
82#335—1271no—
83#333—1271no—
84#332—1271no—
85C2×SD16—2541no—
86C2×M16—21no—
87C4⋊Q8—1271no—
88C4⋊Q8—1271no—
89C4×D4C2×D12711yescharacteristic
90#106—2541no—
91#106—2541no—
92#106—2541no—
93C4⋊C8—1271no—
94C4⋊C8—1271no—
95C4⋊C8C2×D12711yescharacteristic
96#103—2541no—
97#102—2541no—
98C22⋊C8—21no—
99C4×C8C2×D12711yescharacteristic
100C2×Q8—2541no—
101C2×Q8—2541no—
102C2×D4D50811yescharacteristic
103C22×C4—21no—
104SD16—5082no—
105M16—42no—
106C2×C8—21no—
107C2×C8—2541no—
108C2×C8—2541no—
109C2×C8C22⋊D12711yescharacteristic
110C2×C8C22⋊D12711yescharacteristic
111C4⋊C4—2541no—
112C4⋊C4—2541no—
113C4⋊C4—2541no—
114C4⋊C4—2541no—
115C4⋊C4D50811yescharacteristic
116C22⋊C4—21no—
117C42C22×D12711yescharacteristic
118C23—21no—
119Q8—10161no—
120Q8—5081no—
121Q8—5081no—
122D4—21no—
123D4—21no—
124C2×C4—21no—
125C2×C4—41no—
126C2×C4—41no—
127C2×C4—5081no—
128C2×C4—5081no—
129C2×C4C2×C22⋊D12711yescharacteristic
130C2×C4C2×D50811yescharacteristic
131C2×C4#1452324811yescharacteristic
132C8—21no—
133C8—41no—
134C8—5081no—
135C8—5081no—
136C8#1452323612yescomplemented
137C22—41no—
138C22—41no—
139C22—41no—
140C22#1686118311yescharacteristic
141C4—21no—
142C4—41no—
143C4—10161no—
144C4—10161no—
145C4C2×#1452323611yescharacteristic
146C4#1686123711yescharacteristic
147C4#1686123911yescharacteristic
148C4#1686124211yescharacteristic
149C2—81no—
150C2#2255008111yescharacteristic
151C2#2255012211yescharacteristic
152C2#2255013911yescharacteristic
subgroupG/H
centerC22#16861183
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#436×C127C2
socleC22×C127#121
prime radicalC2×D4×C127D4
hypercenter#436D127
hypocenterC127#1691
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #436

Complements

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

NH ≅ G/Ntype
1. #22550274C2semidirect
4. #22549629C2semidirect
5. #22549624C2semidirect
6. #22549621C2semidirect
59. C8×C127SD16semidirect
77. C127#1691semidirect
136. C8#14523236semidirect

Maximal subgroups

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

subgroupindex
#225502742
#225501662
#225500812
#225496292
#225496242
#225496212
#436×C1272
#1691127