Finite Group Databasebeta

← D4×C22⋊D367 #37554819

Subgroups — D4×C22⋊D367 #37554819

Order 23488.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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×C22⋊D367C212yesmaximal, complemented
2C2×D4×D367C211yesmaximal, characteristic, complemented
3#26044257C211yesmaximal, characteristic, complemented
4#26044235C211yesmaximal, characteristic, complemented
5#26044233C212yesmaximal, complemented
6#26044232C211yesmaximal, characteristic, complemented
7#26044230C212yesmaximal, complemented
8#26044229C212yesmaximal, complemented
9D4×Dic367C211yesmaximal, characteristic, complemented
10C4×C22⋊D367C211yesmaximal, characteristic, complemented
11C22×D4×C367C211yesmaximal, characteristic, complemented
12C23×D367C2212yescomplemented
13C2×D1468C2211yescharacteristic, complemented
14C2×C22⋊D367—28no—
15C2×C22⋊D367C2211yescharacteristic
16C2×C22⋊D367C2212yescomplemented
17C2×C22⋊D367C2212yescomplemented
18C2×C22⋊D367C2212yescomplemented
19C2×C22⋊D367C2212yescomplemented
20C2×C22⋊D367C2212yescomplemented
21D4×D367—24no—
22C22×Dic367C2212yescomplemented
23C2×C4×D367C2211yescharacteristic, complemented
24C4⋊Dic367C2211yescharacteristic, complemented
25Dic367⋊C4C2211yescharacteristic, complemented
26C22⋊Dic367C2211yescharacteristic, complemented
27C22⋊Dic367C2212yescomplemented
28C22⋊Dic367C2212yescomplemented
29(C2×D367)⋊C4C2211yescharacteristic, complemented
30(C2×D367)⋊C4C2212yescomplemented
31C4×Dic367C2211yescharacteristic, complemented
32C24×C367C2212yescomplemented
33C2×D4×C367—24no—
34C2×D4×C367C2211yescharacteristic
35C2×D4×C367C2212yescomplemented
36C2×D4×C367C2211yescharacteristic, complemented
37C22×C4×C367C2211yescharacteristic, complemented
38C22×D367—24no—
39C22×D367—24no—
40C22×D367—42no—
41C22×D367C2312yes—
42C22×D367C2311yescharacteristic
43D1468—22no—
44C22⋊D367—24no—
45C22⋊D367—24no—
46C22⋊D367—24no—
47C22⋊D367—24no—
48C22⋊D367—42no—
49C22⋊D367D414yesdirect product factor
50C2×Dic367—24no—
51C2×Dic367C2311yescharacteristic
52C2×Dic367C2312yes—
53C2×Dic367C2311yescharacteristic
54C2×Dic367C2311yescharacteristic
55C4×D367—22no—
56C23×C367—24no—
57C23×C367—24no—
58C23×C367—42no—
59C23×C367C2312yes—
60C23×C367C2311yescharacteristic
61C23×C367C2312yes—
62D4×C367—24no—
63D4×C367—22no—
64D4×C367D414yescomplemented
65C2×C4×C367—22no—
66C2×C4×C367C2311yescharacteristic
67C2×C4×C367C2311yescharacteristic
68C22×C367—21no—
69C22×C367—22no—
70C22×C367—21no—
71C22×C367—22no—
72C22×C367—22no—
73C22×C367—22no—
74C22×C367—22no—
75C22×C367—48no—
76C22×C367C2×D412yescomplemented
77C22×C367C2×D414yescomplemented
78C22×C367C2411yescharacteristic
79C2×D367—21no—
80C2×D367—21no—
81C2×D367—22no—
82C2×D367—22no—
83C2×D367—22no—
84C2×D367—48no—
85C2×D367C2×D412yescomplemented
86C4×C367—21no—
87C4×C367C2×D412yescomplemented
88Dic367—21no—
89Dic367—22no—
90Dic367—21no—
91Dic367C2×D412yescomplemented
92C2×C367—24no—
93C2×C367—22no—
94C2×C367—42no—
95C2×C367C22×D411yescharacteristic
96C2×C367C22×D411yescharacteristic
97C2×C367H(5,𝔽2)11yescharacteristic
98D367—22no—
99D367—42no—
100C367D4211yescharacteristic, complemented, Sylow
101D42—3671nomaximal, Sylow
102C22×D4—3672no—
103C22×D4—3671no—
104C22×D4D36711yescharacteristic, complemented
105#127—3671no—
106#121—3671no—
107#121—3672no—
108#121—3671no—
109#120—3672no—
110#120—3672no—
111C4×D4—3671no—
112C4×D4—3671no—
113C24—3672no—
114C24C2×D36712yescomplemented
115C2×D4—24no—
116C2×D4—3671no—
117C2×D4—3672no—
118C2×D4—3672no—
119C2×D4—3672no—
120C2×D4—3672no—
121C2×D4—3672no—
122C2×D4—3671no—
123C2×D4—7348no—
124C2×D4—7344no—
125C2×D4C2×D36711yescharacteristic
126C2×D4C2×D36712yescomplemented
127C2×D4C2×D36711yescharacteristic, complemented
128C22×C4—3672no—
129C22×C4—3671no—
130C22×C4C2×D36711yescharacteristic, complemented
131C4⋊C4—3671no—
132C4⋊C4—3671no—
133C22⋊C4—3671no—
134C22⋊C4—3672no—
135C22⋊C4—3672no—
136C22⋊C4—3671no—
137C22⋊C4—3672no—
138C42—3671no—
139C23—24no—
140C23—24no—
141C23—42no—
142C23—3671no—
143C23—3672no—
144C23—14682no—
145C23—7344no—
146C23—7344no—
147C23C22×D36712yes—
148C23C22×D36711yescharacteristic
149C23C22×D36712yes—
150D4—24no—
151D4—22no—
152D4—3674no—
153D4—14682no—
154D4—7344no—
155D4—7344no—
156D4—7344no—
157D4—7344no—
158D4—7342no—
159D4C22⋊D36714yesdirect product factor
160C2×C4—22no—
161C2×C4—3671no—
162C2×C4—3671no—
163C2×C4—3672no—
164C2×C4—3671no—
165C2×C4—7344no—
166C2×C4—7342no—
167C2×C4C22×D36711yescharacteristic
168C2×C4C22×D36711yescharacteristic
169C22—22no—
170C22—21no—
171C22—21no—
172C22—22no—
173C22—22no—
174C22—22no—
175C22—22no—
176C22—48no—
177C22—3672no—
178C22—14688no—
179C22—7342no—
180C22—7342no—
181C22—7342no—
182C22—7341no—
183C22—7341no—
184C22D4×D36712yescomplemented
185C22C2×C22⋊D36714yescomplemented
186C22C23×D36711yescharacteristic
187C4—21no—
188C4—3672no—
189C4—7342no—
190C4—7341no—
191C4—7341no—
192C4C2×C22⋊D36712yescomplemented
193C2—22no—
194C2—24no—
195C2—42no—
196C2—14682no—
197C2—7342no—
198C2C2×D4×D36711yescharacteristic
199C2C22×C22⋊D36711yescharacteristic
200C2#2604430611yescharacteristic
subgroupG/H
center, FrattiniC22C23×D367
commutator, socle, cosocleC22×C367C24
FittingC22×D4×C367C2
hypercenterC22×D4D367
hypocenterC367D42
LCSG ▷ C22×C367 ▷ C367
UCSC1 ◁ C22 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C22×C22⋊D367C2semidirect
2. C2×D4×D367C2semidirect
3. #26044257C2semidirect
4. #26044235C2semidirect
5. #26044233C2semidirect
6. #26044232C2semidirect
7. #26044230C2semidirect
8. #26044229C2semidirect
9. D4×Dic367C2semidirect
10. C4×C22⋊D367C2semidirect
11. C22×D4×C367C2semidirect
12. C23×D367C22semidirect
13. C2×D1468C22semidirect
16. C2×C22⋊D367C22semidirect
17. C2×C22⋊D367C22semidirect
18. C2×C22⋊D367C22semidirect
19. C2×C22⋊D367C22semidirect
20. C2×C22⋊D367C22semidirect
22. C22×Dic367C22semidirect
23. C2×C4×D367C22semidirect
24. C4⋊Dic367C22semidirect
25. Dic367⋊C4C22semidirect
26. C22⋊Dic367C22semidirect
27. C22⋊Dic367C22semidirect
28. C22⋊Dic367C22semidirect
29. (C2×D367)⋊C4C22semidirect
30. (C2×D367)⋊C4C22semidirect
31. C4×Dic367C22semidirect
32. C24×C367C22semidirect
35. C2×D4×C367C22semidirect
36. C2×D4×C367C22semidirect
37. C22×C4×C367C22semidirect
49. C22⋊D367D4direct (N × H)
64. D4×C367D4semidirect
76. C22×C367C2×D4semidirect
77. C22×C367C2×D4semidirect
85. C2×D367C2×D4semidirect
87. C4×C367C2×D4semidirect
91. Dic367C2×D4semidirect
100. C367D42semidirect
104. C22×D4D367semidirect
114. C24C2×D367semidirect
126. C2×D4C2×D367semidirect
127. C2×D4C2×D367semidirect
130. C22×C4C2×D367semidirect
159. D4C22⋊D367direct (N × H)
184. C22D4×D367semidirect
185. C22C2×C22⋊D367semidirect
192. C4C2×C22⋊D367semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊D3672
C2×D4×D3672
#260442572
#260442352
#260442332
#260442322
#260442302
#260442292
D4×Dic3672
C4×C22⋊D3672
C22×D4×C3672
D42367