Finite Group Databasebeta

← C2×#22550166 #30648481

Subgroups — C2×#22550166 #30648481

Order 16256.

480 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 181 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×#16861209C211yesmaximal, characteristic, complemented
2#22550166C218yesmaximal, direct product factor
3C2×#16861140C212yesmaximal, complemented
4C2×C4⋊(C127⋊C8)C211yesmaximal, characteristic, complemented
5C2×#16861130C212yesmaximal
6C2×C4×D4×C127C211yesmaximal, characteristic
7C22×Dic254—21no—
8C22×C127⋊C8C2212yes—
9#16861209—22no—
10#16861209C2214yescomplemented
11C2×C4⋊Dic127—21no—
12C2×C4⋊Dic127C2212yes—
13#16861140C2218yescomplemented
14C4⋊(C127⋊C8)C2214yescomplemented
15#16861130C2218yes—
16C22×D4×C127C2211yescharacteristic
17C23×C4×C127—21no—
18C4×D4×C127—22no—
19C4×D4×C127C2214yes—
20C2×C4⋊C4×C127C2211yescharacteristic
21C2×C22⋊C4×C127—21no—
22C2×C42×C127C2211yescharacteristic
23C2×Dic254—24no—
24C2×Dic254—44no—
25C22×Dic127—22no—
26C2×C127⋊C8—24no—
27C2×C127⋊C8C2314yes—
28C4⋊Dic127—22no—
29C4⋊Dic127—24no—
30C4⋊Dic127C2314yes—
31C24×C127—21no—
32C2×D4×C127—21no—
33C2×D4×C127—22no—
34C2×D4×C127Q814yescomplemented
35C2×D4×C127C2312yes—
36C22×C4×C127—21no—
37C22×C4×C127—24no—
38C22×C4×C127—44no—
39C22×C4×C127D411yescharacteristic
40C22×C4×C127D411yescharacteristic
41C22×C4×C127C2311yescharacteristic
42C4⋊C4×C127—21no—
43C4⋊C4×C127C2312yes—
44C22⋊C4×C127—24no—
45C42×C127D412yes—
46C42×C127C2312yes—
47Dic254—48no—
48C2×Dic127—24no—
49C2×Dic127—44no—
50C127⋊C8—24no—
51C23×C127—24no—
52C23×C127—22no—
53C23×C127—42no—
54C23×C127—42no—
55C23×C127C2×D411yescharacteristic
56D4×C127—24no—
57D4×C127—22no—
58D4×C127C2×Q814yescomplemented
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—42no—
62C2×C4×C127—48no—
63C2×C4×C127SD1614yes—
64C2×C4×C127C2×D412yes—
65C2×C4×C127C2×D411yescharacteristic
66C2×C4×C127C2×D412yes—
67C2×C4×C127C2×Q812yes—
68C2×C4×C127#4112yes—
69C2×C4×C127C2411yescharacteristic
70C22×C127—24no—
71C22×C127—42no—
72C22×C127—42no—
73C22×C127—48no—
74C22×C127C22⋊Q812yes—
75C22×C127C2×SD1612yes—
76C22×C127AGL(1,ℤ/8)12yes—
77C22×C127C22×D411yescharacteristic
78C4×C127—22no—
79C4×C127—42no—
80C4×C127C22⋊Q812yes—
81C4×C127C2×SD1614yes—
82C4×C127C22×Q811yescharacteristic
83C4×C127C2×#4111yescharacteristic
84Dic127—44no—
85C2×C127—44no—
86C2×C127#47614yescomplemented
87C2×C127C2×C22⋊Q811yescharacteristic
88C2×C127C22×SD1611yescharacteristic
89C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
90C2×#476—1271nomaximal, Sylow
91C127C2×#47611yescharacteristic, complemented, Sylow
92C2×C4⋊Q8—1271no—
93C2×C4×D4D12711yescharacteristic
94#476—1278no—
95C2×#106—1272no—
96C2×C4⋊C8—1271no—
97C2×#102—1272no—
98C22×Q8—2541no—
99C22×D4C2×D12711yescharacteristic
100C23×C4—21no—
101C22×C8—1272no—
102C4⋊Q8—1274no—
103C4⋊Q8—2542no—
104C4×D4—22no—
105C4×D4C2×D12714yes—
106C2×C4⋊C4—1272no—
107C2×C4⋊C4—2541no—
108C2×C4⋊C4C2×D12711yescharacteristic
109C2×C22⋊C4—21no—
110C2×C42C2×D12711yescharacteristic
111#106—1278no—
112C4⋊C8—1274no—
113#102—1278no—
114C24—21no—
115C2×Q8—5084no—
116C2×Q8—2544no—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4Dic25414yescomplemented
120C2×D4C22×D12712yes—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—44no—
124C22×C4—2542no—
125C22×C4C22⋊D12711yescharacteristic
126C22×C4C22⋊D12711yescharacteristic
127C22×C4C22×D12711yescharacteristic
128C2×C8—1274no—
129C2×C8—2544no—
130C4⋊C4—21no—
131C4⋊C4—1274no—
132C4⋊C4—2542no—
133C4⋊C4—2544no—
134C4⋊C4C22×D12712yes—
135C22⋊C4—24no—
136C42C22⋊D12712yes—
137C42C22×D12712yes—
138C23—24no—
139C23—22no—
140C23—42no—
141C23—42no—
142C23C2×C22⋊D12711yescharacteristic
143Q8—5088no—
144D4—24no—
145D4—22no—
146D4C2×Dic25414yescomplemented
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—42no—
150C2×C4—48no—
151C2×C4—5084no—
152C2×C4—2544no—
153C2×C4#1452323614yes—
154C2×C4C2×C22⋊D12712yes—
155C2×C4C2×C22⋊D12711yescharacteristic
156C2×C4C2×C22⋊D12712yes—
157C2×C4C2×Dic25412yes—
158C2×C4#1452324812yes—
159C2×C4C23×D12711yescharacteristic
160C8—2544no—
161C22—24no—
162C22—42no—
163C22—42no—
164C22—48no—
165C22#1686118812yes—
166C22C2×#1452323612yes—
167C22#1686123412yes—
168C22C22×C22⋊D12711yescharacteristic
169C4—22no—
170C4—42no—
171C4—5084no—
172C4#1686118812yes—
173C4C2×#1452323614yes—
174C4C22×Dic25411yescharacteristic
175C4C2×#1452324811yescharacteristic
176C2—44no—
177C2#2255016614yesdirect product factor
178C2C2×#1686118811yescharacteristic
179C2C22×#1452323611yescharacteristic
180C2C2×#1686123411yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C4×D4×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC2×C4×D4D127
hypocenterC127C2×#476
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16861209C2semidirect
2. #22550166C2direct (N × H)
3. C2×#16861140C2semidirect
4. C2×C4⋊(C127⋊C8)C2semidirect
10. #16861209C22semidirect
13. #16861140C22semidirect
14. C4⋊(C127⋊C8)C22semidirect
34. C2×D4×C127Q8semidirect
58. D4×C127C2×Q8semidirect
86. C2×C127#476semidirect
91. C127C2×#476semidirect
119. C2×D4Dic254semidirect
146. D4C2×Dic254semidirect
177. C2#22550166direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612092
#225501662
C2×#168611402
C2×C4⋊(C127⋊C8)2
C2×#168611302
C2×C4×D4×C1272
C2×#476127