Finite Group Databasebeta

← #30640640

Subgroups — #30640640

Order 16256.

266 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 53 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
1C2×D8×D127C211yesmaximal, characteristic, complemented
2C2×#16861150C211yesmaximal, characteristic, complemented
3#22550109C211yesmaximal, characteristic, complemented
4#22549678C211yesmaximal, characteristic, complemented
5#22549677C211yesmaximal, characteristic, complemented
6(C2×D127)⋊C16C211yesmaximal, characteristic, complemented
7C2×D16×C127C211yesmaximal, characteristic, complemented
8C2×D4×D127—21no—
9C2×D1016C2211yescharacteristic
10C2×#14523233—21no—
11D8×D127—22no—
12D8×D127—22no—
13C2×C8×D127C2211yescharacteristic
14#16861180—21no—
15#16861150—22no—
16C2×C127⋊C16C2211yescharacteristic, complemented
17#16861143C2211yescharacteristic, complemented
18#16861130—21no—
19C2×D8×C127C2211yescharacteristic
20C2×D8×C127C2211yescharacteristic, complemented
21D16×C127—22no—
22C2×C16×C127C2211yescharacteristic, complemented
23C23×D127—41no—
24C2×D508—21no—
25C2×C22⋊D127—41no—
26C2×C22⋊D127—41no—
27D4×D127—24no—
28D4×D127—42no—
29C2×C4×D127D411yescharacteristic
30D1016—21no—
31D1016D412yescomplemented
32#14523233—24no—
33C2×C127⋊C8D411yescharacteristic
34C8×D127—21no—
35C8×D127—21no—
36C4⋊Dic127—21no—
37C22⋊Dic127—41no—
38C127⋊C16—21no—
39C2×D4×C127—21no—
40C2×D4×C127—21no—
41D8×C127—21no—
42D8×C127—21no—
43D8×C127—21no—
44D8×C127D412yescomplemented
45C2×C8×C127C2311yescharacteristic
46C16×C127—21no—
47C22×D127—44no—
48C22×D127—41no—
49C22×D127—82no—
50C22×D127—82no—
51C22×D127D811yescharacteristic
52D508—22no—
53D508—41no—
54C22⋊D127—44no—
55C22⋊D127—82no—
56C2×Dic127—41no—
57C2×Dic127D811yescharacteristic
58C4×D127—21no—
59C4×D127—21no—
60C127⋊C8—21no—
61C23×C127—41no—
62C23×C127—41no—
63D4×C127—22no—
64D4×C127—41no—
65D4×C127—41no—
66D4×C127—41no—
67C2×C4×C127C2×D411yescharacteristic
68C8×C127C2×D411yescharacteristic
69C8×C127C2×D411yescharacteristic
70C22×C127—42no—
71C22×C127—81no—
72C22×C127—81no—
73C22×C127—81no—
74C22×C127—81no—
75C22×C127—81no—
76C22×C127C2×D811yescharacteristic
77C2×D127—21no—
78C2×D127—21no—
79C2×D127—42no—
80C2×D127—88no—
81C2×D127—81no—
82C2×D127—81no—
83C2×D127D1612yescomplemented
84C4×C127#12011yescharacteristic
85C4×C127AGL(1,ℤ/8)11yescharacteristic
86Dic127—21no—
87Dic127—81no—
88C2×C127—82no—
89C2×C127—161no—
90C2×C127#44711yescharacteristic
91C2×C127C2×D1611yescharacteristic
92C2×C127#50911yescharacteristic
93D127—22no—
94D127—82no—
95#2184—1271nomaximal, Sylow
96C127#218411yescharacteristic, complemented, Sylow
97C22×D8—1271no—
98C2×D16—1271no—
99C2×D16D12711yescharacteristic, complemented
100#466—1271no—
101#357—1271no—
102#357—1271no—
103C22⋊C16—1271no—
104C22×D4—2541no—
105C2×D8—1271no—
106C2×D8—2541no—
107C2×D8—2542no—
108C2×D8—2542no—
109C2×D8C2×D12711yescharacteristic
110C2×D8C2×D12711yescharacteristic, complemented
111C22×C8—1271no—
112#121—2541no—
113D16—22no—
114D16—2542no—
115C2×C16—1271no—
116C2×C16C2×D12711yescharacteristic, complemented
117#107—1271no—
118#102—2541no—
119C24—5081no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—5081no—
123C2×D4—5081no—
124C2×D4—5082no—
125C2×D4—2544no—
126C2×D4—2541no—
127C22×C4—1271no—
128D8—21no—
129D8—21no—
130D8—21no—
131D8—1272no—
132D8—2544no—
133D8—2541no—
134D8C22⋊D12712yescomplemented
135C2×C8—1271no—
136C2×C8—2541no—
137C2×C8—2541no—
138C2×C8C22×D12711yescharacteristic
139C4⋊C4—2541no—
140C22⋊C4—5081no—
141C16—21no—
142C16—2541no—
143C23—41no—
144C23—41no—
145C23—1271no—
146C23—10162no—
147C23—10162no—
148C23—5084no—
149C23—5081no—
150D4—22no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—10162no—
155D4—5084no—
156D4—5081no—
157D4—2542no—
158C2×C4—1271no—
159C2×C4—5081no—
160C2×C4—2541no—
161C2×C4—2541no—
162C2×C4D4×D12711yescharacteristic
163C8—2541no—
164C8D4×D12711yescharacteristic
165C8C2×C22⋊D12711yescharacteristic
166C22—42no—
167C22—81no—
168C22—81no—
169C22—81no—
170C22—81no—
171C22—81no—
172C22—1272no—
173C22—10168no—
174C22—10161no—
175C22—10161no—
176C22—5082no—
177C22—2541no—
178C22—2541no—
179C22D8×D12711yescharacteristic
180C4—10161no—
181C4—2541no—
182C4#1686117711yescharacteristic
183C4#1686123311yescharacteristic
184C2—82no—
185C2—161no—
186C2—10162no—
187C2—2542no—
188C2#2255000811yescharacteristic
189C2D16×D12711yescharacteristic
190C2#2255031011yescharacteristic
subgroupG/H
centerC22D8×D127
commutator, cosocleC2×C8×C127C23
FrattiniC2×C8C22×D127
FittingC2×D16×C127C2
socleC22×C127C2×D8
hypercenterC2×D16D127
hypocenterC127#2184
LCSG ▷ C2×C8×C127 ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×D16

Complements

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

NH ≅ G/Ntype
1. C2×D8×D127C2semidirect
2. C2×#16861150C2semidirect
3. #22550109C2semidirect
4. #22549678C2semidirect
5. #22549677C2semidirect
6. (C2×D127)⋊C16C2semidirect
7. C2×D16×C127C2semidirect
16. C2×C127⋊C16C22semidirect
17. #16861143C22semidirect
20. C2×D8×C127C22semidirect
22. C2×C16×C127C22semidirect
31. D1016D4semidirect
44. D8×C127D4semidirect
83. C2×D127D16semidirect
96. C127#2184semidirect
99. C2×D16D127semidirect
110. C2×D8C2×D127semidirect
116. C2×C16C2×D127semidirect
134. D8C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1272
C2×#168611502
#225501092
#225496782
#225496772
(C2×D127)⋊C162
C2×D16×C1272
#2184127