Finite Group Databasebeta

← D4×D856 #28348138

Subgroups — D4×D856 #28348138

Order 13696.

620 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×D856C212yesmaximal, complemented
2D4×D428C212yesmaximal, complemented
3#20293004C211yesmaximal, characteristic, complemented
4#20292863C212yesmaximal, complemented
5#20292823C212yesmaximal, complemented
6#20292763C214yesmaximal, complemented
7C4×D856C211yesmaximal, characteristic, complemented
8C8×D4×C107C211yesmaximal, characteristic, complemented
9C22×D428C2214yescomplemented
10C2×D4×D107—22no—
11C2×D856—28no—
12C2×D856C2211yescharacteristic
13C2×D856C2212yescomplemented
14C2×D856C2214yescomplemented
15C2×D856C2212yescomplemented
16#16812106C2212yescomplemented
17#16812084—22no—
18#16812083C2214yescomplemented
19#16812079—24no—
20C4×D428C2212yescomplemented
21#16812043C2211yescharacteristic, complemented
22#16812031C2212yescomplemented
23#16812031C2214yescomplemented
24C22×C8×C107C2212yescomplemented
25C4×D4×C107C2211yescharacteristic
26C4⋊C8×C107C2211yescharacteristic, complemented
27C22⋊C8×C107C2212yescomplemented
28C4×C8×C107C2211yescharacteristic, complemented
29C23×D107—24no—
30C2×D428—28no—
31C2×D428—24no—
32C2×D428—22no—
33C2×D428—22no—
34C2×D428—44no—
35C2×D428C2314yes—
36C2×D428C2312yes—
37C2×C22⋊D107—24no—
38D4×D107—48no—
39C2×C4×D107—22no—
40D856—28no—
41D856—24no—
42D856—42no—
43D856D414yesdirect product factor
44C4⋊Dic107C2312yes—
45(C2×D107)⋊C4—22no—
46(C2×D107)⋊C4—24no—
47C2×D4×C107D411yescharacteristic
48C22×C4×C107C2312yes—
49C2×C8×C107—24no—
50C2×C8×C107C2311yescharacteristic
51C2×C8×C107C2312yes—
52C2×C8×C107C2311yescharacteristic
53C4⋊C4×C107D411yescharacteristic
54C22⋊C4×C107D412yes—
55C42×C107C2311yescharacteristic
56C23×C107C2×D412yes—
57C22×D107—22no—
58C22×D107—24no—
59C22×D107—48no—
60C22×D107—48no—
61C22×D107—84no—
62D4×C107D814yescomplemented
63C2×C4×C107—22no—
64C2×C4×C107—22no—
65C2×C4×C107C2×D411yescharacteristic
66C2×C4×C107C2×D412yes—
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2411yescharacteristic
69C22⋊D107—48no—
70C2×Dic107—22no—
71D428—24no—
72D428—22no—
73D428—24no—
74D428—48no—
75D428—44no—
76D428—82no—
77D428C2×D414yescomplemented
78C4×D107—44no—
79C8×C107—22no—
80C8×C107—21no—
81C8×C107C2×D412yescomplemented
82C22×C107—22no—
83C22×C107—22no—
84C22×C107C2×D814yescomplemented
85C22×C107C22×D411yescharacteristic
86C2×D107—24no—
87C2×D107—42no—
88C2×D107—42no—
89C2×D107—44no—
90C2×D107—44no—
91C2×D107—44no—
92C2×D107—816no—
93C4×C107—22no—
94C4×C107—21no—
95C4×C107C2×D812yescomplemented
96C4×C107C22×D411yescharacteristic
97C4×C107H(5,𝔽2)11yescharacteristic
98Dic107—42no—
99C2×C107—24no—
100C2×C107D4211yescharacteristic
101C2×C107C22×D811yescharacteristic
102C2×C107#57611yescharacteristic
103D107—44no—
104D107—84no—
105D4×D8—1071nomaximal, Sylow
106C107D4×D811yescharacteristic, complemented, Sylow
107C22×D8—1072no—
108D42—1072no—
109#493—1071no—
110#466—1072no—
111#459—1072no—
112#447—1074no—
113C4×D8—1071no—
114C8×D4D10711yescharacteristic, complemented
115C22×D4—1074no—
116C22×D4—2142no—
117C2×D8—1071no—
118C2×D8—1072no—
119C2×D8—1074no—
120C2×D8—1072no—
121C2×D8—2148no—
122C22×C8C2×D10712yescomplemented
123#127—1072no—
124#121—1074no—
125#121—2142no—
126#120—2144no—
127C4×D4—1072no—
128C4×D4C2×D10711yescharacteristic
129#107—1071no—
130C4⋊C8C2×D10711yescharacteristic, complemented
131#102—1072no—
132#102—1074no—
133C22⋊C8C2×D10712yescomplemented
134C4×C8C2×D10711yescharacteristic, complemented
135C24—2144no—
136C2×D4—1072no—
137C2×D4—1074no—
138C2×D4—4284no—
139C2×D4—4288no—
140C2×D4—2148no—
141C2×D4—2144no—
142C2×D4—2142no—
143C2×D4—2144no—
144C2×D4—2142no—
145C2×D4D42811yescharacteristic
146C22×C4—2142no—
147C22×C4C22×D10712yes—
148D8—1074no—
149D8—4282no—
150D8—2148no—
151D8—2144no—
152C2×C8—24no—
153C2×C8C22×D10711yescharacteristic
154C2×C8C22×D10712yes—
155C2×C8C22×D10711yescharacteristic
156C4⋊C4—1072no—
157C4⋊C4D42811yescharacteristic
158C22⋊C4—2142no—
159C22⋊C4—2144no—
160C22⋊C4D42812yes—
161C42C22×D10711yescharacteristic
162C23—8564no—
163C23—4288no—
164C23—4288no—
165C23—2144no—
166C23—2142no—
167C23C2×D42812yes—
168D4—8562no—
169D4—1074no—
170D4—4288no—
171D4—4288no—
172D4—4284no—
173D4—2144no—
174D4—2144no—
175D4—2142no—
176D4D85614yesdirect product factor
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—4284no—
180C2×C4—2142no—
181C2×C4C2×D42811yescharacteristic
182C2×C4C2×D42812yes—
183C2×C4C2×D42811yescharacteristic
184C2×C4C23×D10711yescharacteristic
185C8—22no—
186C8—21no—
187C8D4×D10712yescomplemented
188C22—22no—
189C22—22no—
190C22—85616no—
191C22—4284no—
192C22—4284no—
193C22—4284no—
194C22—4282no—
195C22—4282no—
196C22—2144no—
197C22C2×D85614yescomplemented
198C22C22×D42811yescharacteristic
199C4—21no—
200C4—22no—
201C4—4282no—
202C4C2×D85612yescomplemented
203C4C2×D4×D10711yescharacteristic
204C4#1681215511yescharacteristic
205C2—24no—
206C2—8564no—
207C2—4284no—
208C2D4×D42811yescharacteristic
209C2C22×D85611yescharacteristic
210C2#2029343211yescharacteristic
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC8×D4×C107C2
socleC22×C107C22×D4
hypercenterC8×D4D107
hypocenterC107D4×D8
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×D856C2semidirect
2. D4×D428C2semidirect
3. #20293004C2semidirect
4. #20292863C2semidirect
5. #20292823C2semidirect
6. #20292763C2semidirect
7. C4×D856C2semidirect
8. C8×D4×C107C2semidirect
9. C22×D428C22semidirect
13. C2×D856C22semidirect
14. C2×D856C22semidirect
15. C2×D856C22semidirect
16. #16812106C22semidirect
18. #16812083C22semidirect
20. C4×D428C22semidirect
21. #16812043C22semidirect
22. #16812031C22semidirect
23. #16812031C22semidirect
24. C22×C8×C107C22semidirect
26. C4⋊C8×C107C22semidirect
27. C22⋊C8×C107C22semidirect
28. C4×C8×C107C22semidirect
43. D856D4direct (N × H)
62. D4×C107D8semidirect
77. D428C2×D4semidirect
81. C8×C107C2×D4semidirect
84. C22×C107C2×D8semidirect
95. C4×C107C2×D8semidirect
106. C107D4×D8semidirect
114. C8×D4D107semidirect
122. C22×C8C2×D107semidirect
130. C4⋊C8C2×D107semidirect
133. C22⋊C8C2×D107semidirect
134. C4×C8C2×D107semidirect
176. D4D856direct (N × H)
187. C8D4×D107semidirect
197. C22C2×D856semidirect
202. C4C2×D856semidirect

Maximal subgroups

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

subgroupindex
C22×D8562
D4×D4282
#202930042
#202928632
#202928232
#202927632
C4×D8562
C8×D4×C1072
D4×D8107