Finite Group Databasebeta

← D8×D668 #36335814

Subgroups — D8×D668 #36335814

Order 21376.

620 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 141 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×D167C212yesmaximal, complemented
2D4×D668C212yesmaximal, complemented
3#24923646C211yesmaximal, characteristic, complemented
4#24923506C212yesmaximal, complemented
5#24923464C212yesmaximal, complemented
6#24923404C214yesmaximal, complemented
7C8×D668C211yesmaximal, characteristic, complemented
8C4×D8×C167C211yesmaximal, characteristic, complemented
9C22×D668—22no—
10C2×D4×D167C2214yescomplemented
11C2×D1336C2212yescomplemented
12C2×D1336C2212yescomplemented
13C2×#14581935C2214yescomplemented
14D8×D167—28no—
15C2×C8×D167C2212yescomplemented
16#19128829C2212yescomplemented
17#19128807C2214yescomplemented
18#19128806—22no—
19#19128802—24no—
20C4×D668C2211yescharacteristic
21C4⋊(C167⋊C8)C2211yescharacteristic, complemented
22#19128752C2214yescomplemented
23(C2×D167)⋊C8C2212yescomplemented
24C2×D8×C167C2211yescharacteristic
25C4×D4×C167C2212yescomplemented
26#107×C167C2211yescharacteristic, complemented
27#102×C167C2212yescomplemented
28C4×C8×C167C2211yescharacteristic, complemented
29C23×D167—24no—
30C2×D668—24no—
31C2×D668—22no—
32C2×D668—22no—
33C2×D668—48no—
34C2×D668D411yescharacteristic
35C2×D668C2314yes—
36C2×C22⋊D167—24no—
37D4×D167—28no—
38D4×D167—44no—
39C2×C4×D167C2312yes—
40D1336—24no—
41D1336—42no—
42#14581935—28no—
43C2×C167⋊C8C2312yes—
44C8×D167—24no—
45C4⋊Dic167D411yescharacteristic
46(C2×D167)⋊C4—24no—
47(C2×D167)⋊C4D412yes—
48C2×D4×C167C2312yes—
49C22×C4×C167—22no—
50D8×C167D414yescomplemented
51C2×C8×C167C2311yescharacteristic
52C2×C8×C167C2311yescharacteristic
53C4⋊C4×C167C2312yes—
54C22⋊C4×C167—22no—
55C42×C167C2311yescharacteristic
56C22×D167—24no—
57C22×D167—48no—
58C22×D167—48no—
59C22×D167—84no—
60C22×D167C2×D412yes—
61D668—24no—
62D668—24no—
63D668—48no—
64D668—44no—
65D668—82no—
66D668D814yesdirect product factor
67C22⋊D167—48no—
68C2×Dic167C2×D412yes—
69C4×D167—22no—
70C4×D167—22no—
71C167⋊C8—22no—
72C23×C167—22no—
73D4×C167—22no—
74D4×C167C2×D414yescomplemented
75C2×C4×C167—22no—
76C2×C4×C167—44no—
77C2×C4×C167C2×D411yescharacteristic
78C2×C4×C167C2×D411yescharacteristic
79C2×C4×C167C2411yescharacteristic
80C8×C167—21no—
81C8×C167C2×D412yescomplemented
82C22×C167—24no—
83C22×C167—42no—
84C22×C167—42no—
85C22×C167C22×D411yescharacteristic
86C2×D167—22no—
87C2×D167—22no—
88C2×D167—44no—
89C2×D167—44no—
90C2×D167—44no—
91C2×D167—816no—
92C2×D167C2×D814yescomplemented
93C4×C167—21no—
94C4×C167—42no—
95C4×C167C2×D812yescomplemented
96C4×C167C22×D411yescharacteristic
97C4×C167H(5,𝔽2)11yescharacteristic
98Dic167—22no—
99C2×C167—44no—
100C2×C167D4211yescharacteristic
101C2×C167C22×D811yescharacteristic
102C2×C167#57611yescharacteristic
103D167—24no—
104D167—84no—
105C167D4×D811yescharacteristic, complemented, Sylow
106D4×D8—1671nomaximal, Sylow
107C22×D8—1672no—
108D42—1672no—
109#493—1671no—
110#466—1672no—
111#459—1672no—
112#447—1674no—
113C4×D8D16711yescharacteristic, complemented
114C8×D4—1671no—
115C22×D4—3342no—
116C22×D4—1674no—
117C2×D8—3348no—
118C2×D8—1674no—
119C2×D8—1672no—
120C2×D8—1672no—
121C2×D8C2×D16711yescharacteristic
122C22×C8—1672no—
123#127—1672no—
124#121—3342no—
125#121—1674no—
126#120—3344no—
127C4×D4—1671no—
128C4×D4C2×D16712yescomplemented
129#107C2×D16711yescharacteristic, complemented
130C4⋊C8—1671no—
131#102—1674no—
132#102C2×D16712yescomplemented
133C22⋊C8—1672no—
134C4×C8C2×D16711yescharacteristic, complemented
135C24—3344no—
136C2×D4—3344no—
137C2×D4—3348no—
138C2×D4—3344no—
139C2×D4—3342no—
140C2×D4—3342no—
141C2×D4—6688no—
142C2×D4—6684no—
143C2×D4—1671no—
144C2×D4—1674no—
145C2×D4C22×D16712yes—
146C22×C4—22no—
147C22×C4—1672no—
148D8—3348no—
149D8—3344no—
150D8—6682no—
151D8D66814yesdirect product factor
152C2×C8—3344no—
153C2×C8—1672no—
154C2×C8C22×D16711yescharacteristic
155C2×C8C22×D16711yescharacteristic
156C4⋊C4—1671no—
157C4⋊C4C22×D16712yes—
158C22⋊C4—22no—
159C22⋊C4—3344no—
160C22⋊C4—1672no—
161C42C22×D16711yescharacteristic
162C23—22no—
163C23—13364no—
164C23—3344no—
165C23—6688no—
166C23—6688no—
167C23—1672no—
168D4—22no—
169D4—13362no—
170D4—3344no—
171D4—3344no—
172D4—6688no—
173D4—6688no—
174D4—6684no—
175D4—1674no—
176D4C2×D66814yescomplemented
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—3342no—
180C2×C4—3342no—
181C2×C4—1672no—
182C2×C4D4×D16711yescharacteristic
183C2×C4D4×D16711yescharacteristic
184C2×C4C23×D16711yescharacteristic
185C8—21no—
186C8—3342no—
187C8C2×D66812yescomplemented
188C22—24no—
189C22—42no—
190C22—42no—
191C22—133616no—
192C22—3342no—
193C22—3342no—
194C22—6684no—
195C22—6684no—
196C22—6684no—
197C22—1674no—
198C22C2×D4×D16711yescharacteristic
199C4—21no—
200C4—42no—
201C4—3342no—
202C4D8×D16712yescomplemented
203C4C22×D66811yescharacteristic
204C4#1912887811yescharacteristic
205C2—44no—
206C2—13364no—
207C2—3344no—
208C2D4×D66811yescharacteristic
209C2C2×D8×D16711yescharacteristic
210C2#2492407111yescharacteristic
subgroupG/H
centerC22C2×D4×D167
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC4×D8×C167C2
socleC22×C167C22×D4
hypercenterC4×D8D167
hypocenterC167D4×D8
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×D8×D167C2semidirect
2. D4×D668C2semidirect
3. #24923646C2semidirect
4. #24923506C2semidirect
5. #24923464C2semidirect
6. #24923404C2semidirect
7. C8×D668C2semidirect
8. C4×D8×C167C2semidirect
10. C2×D4×D167C22semidirect
11. C2×D1336C22semidirect
12. C2×D1336C22semidirect
13. C2×#14581935C22semidirect
15. C2×C8×D167C22semidirect
16. #19128829C22semidirect
17. #19128807C22semidirect
21. C4⋊(C167⋊C8)C22semidirect
22. #19128752C22semidirect
23. (C2×D167)⋊C8C22semidirect
25. C4×D4×C167C22semidirect
26. #107×C167C22semidirect
27. #102×C167C22semidirect
28. C4×C8×C167C22semidirect
50. D8×C167D4semidirect
66. D668D8direct (N × H)
74. D4×C167C2×D4semidirect
81. C8×C167C2×D4semidirect
92. C2×D167C2×D8semidirect
95. C4×C167C2×D8semidirect
105. C167D4×D8semidirect
113. C4×D8D167semidirect
128. C4×D4C2×D167semidirect
129. #107C2×D167semidirect
132. #102C2×D167semidirect
134. C4×C8C2×D167semidirect
151. D8D668direct (N × H)
176. D4C2×D668semidirect
187. C8C2×D668semidirect
202. C4D8×D167semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1672
D4×D6682
#249236462
#249235062
#249234642
#249234042
C8×D6682
C4×D8×C1672
D4×D8167