Finite Group Databasebeta

← D4×D1336 #36335815

Subgroups — D4×D1336 #36335815

Order 21376.

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×D1336C212yesmaximal, complemented
2D4×D668C212yesmaximal, complemented
3#24923646C211yesmaximal, characteristic, complemented
4#24923505C212yesmaximal, complemented
5#24923465C212yesmaximal, complemented
6#24923405C214yesmaximal, complemented
7C4×D1336C211yesmaximal, characteristic, complemented
8C8×D4×C167C211yesmaximal, characteristic, complemented
9C22×D668C2214yescomplemented
10C2×D4×D167—22no—
11C2×D1336—28no—
12C2×D1336C2211yescharacteristic
13C2×D1336C2214yescomplemented
14C2×D1336C2212yescomplemented
15C2×D1336C2212yescomplemented
16#19128829C2212yescomplemented
17#19128807—22no—
18#19128806C2214yescomplemented
19#19128802—24no—
20C4×D668C2212yescomplemented
21#19128766C2211yescharacteristic, complemented
22#19128754C2212yescomplemented
23#19128754C2214yescomplemented
24C22×C8×C167C2212yescomplemented
25C4×D4×C167C2211yescharacteristic
26C4⋊C8×C167C2211yescharacteristic, complemented
27C22⋊C8×C167C2212yescomplemented
28C4×C8×C167C2211yescharacteristic, complemented
29C23×D167—24no—
30C2×D668—28no—
31C2×D668—22no—
32C2×D668—24no—
33C2×D668—22no—
34C2×D668—44no—
35C2×D668C2314yes—
36C2×D668C2312yes—
37C2×C22⋊D167—24no—
38D4×D167—48no—
39C2×C4×D167—22no—
40D1336—28no—
41D1336—24no—
42D1336—42no—
43D1336D414yesdirect product factor
44C4⋊Dic167C2312yes—
45(C2×D167)⋊C4—22no—
46(C2×D167)⋊C4—24no—
47C2×D4×C167D411yescharacteristic
48C22×C4×C167C2312yes—
49C2×C8×C167—24no—
50C2×C8×C167C2311yescharacteristic
51C2×C8×C167C2312yes—
52C2×C8×C167C2311yescharacteristic
53C4⋊C4×C167D411yescharacteristic
54C22⋊C4×C167D412yes—
55C42×C167C2311yescharacteristic
56C22×D167—22no—
57C22×D167—24no—
58C22×D167—48no—
59C22×D167—48no—
60C22×D167—84no—
61D668—24no—
62D668—22no—
63D668—24no—
64D668—48no—
65D668—44no—
66D668—82no—
67D668C2×D414yescomplemented
68C22⋊D167—48no—
69C2×Dic167—22no—
70C4×D167—44no—
71C23×C167C2×D412yes—
72D4×C167D814yescomplemented
73C2×C4×C167—22no—
74C2×C4×C167—22no—
75C2×C4×C167C2×D411yescharacteristic
76C2×C4×C167C2×D412yes—
77C2×C4×C167C2×D411yescharacteristic
78C2×C4×C167C2411yescharacteristic
79C8×C167—22no—
80C8×C167—21no—
81C8×C167C2×D412yescomplemented
82C22×C167—22no—
83C22×C167—22no—
84C22×C167C2×D814yescomplemented
85C22×C167C22×D411yescharacteristic
86C2×D167—24no—
87C2×D167—42no—
88C2×D167—42no—
89C2×D167—44no—
90C2×D167—44no—
91C2×D167—44no—
92C2×D167—816no—
93C4×C167—22no—
94C4×C167—21no—
95C4×C167C2×D812yescomplemented
96C4×C167C22×D411yescharacteristic
97C4×C167H(5,𝔽2)11yescharacteristic
98Dic167—42no—
99C2×C167—24no—
100C2×C167D4211yescharacteristic
101C2×C167C22×D811yescharacteristic
102C2×C167#57611yescharacteristic
103D167—44no—
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×D8—1671no—
114C8×D4D16711yescharacteristic, complemented
115C22×D4—3342no—
116C22×D4—1674no—
117C2×D8—3348no—
118C2×D8—1674no—
119C2×D8—1672no—
120C2×D8—1672no—
121C2×D8—1671no—
122C22×C8C2×D16712yescomplemented
123#127—1672no—
124#121—3342no—
125#121—1674no—
126#120—3344no—
127C4×D4—1672no—
128C4×D4C2×D16711yescharacteristic
129#107—1671no—
130C4⋊C8C2×D16711yescharacteristic, complemented
131#102—1672no—
132#102—1674no—
133C22⋊C8C2×D16712yescomplemented
134C4×C8C2×D16711yescharacteristic, complemented
135C24—3344no—
136C2×D4—3348no—
137C2×D4—3344no—
138C2×D4—3342no—
139C2×D4—3344no—
140C2×D4—3342no—
141C2×D4—6684no—
142C2×D4—6688no—
143C2×D4—1674no—
144C2×D4—1672no—
145C2×D4D66811yescharacteristic
146C22×C4—3342no—
147C22×C4C22×D16712yes—
148D8—3348no—
149D8—3344no—
150D8—6682no—
151D8—1674no—
152C2×C8—24no—
153C2×C8C22×D16711yescharacteristic
154C2×C8C22×D16712yes—
155C2×C8C22×D16711yescharacteristic
156C4⋊C4—1672no—
157C4⋊C4D66811yescharacteristic
158C22⋊C4—3342no—
159C22⋊C4—3344no—
160C22⋊C4D66812yes—
161C42C22×D16711yescharacteristic
162C23—13364no—
163C23—3342no—
164C23—3344no—
165C23—6688no—
166C23—6688no—
167C23C2×D66812yes—
168D4—13362no—
169D4—3344no—
170D4—3342no—
171D4—3344no—
172D4—6688no—
173D4—6688no—
174D4—6684no—
175D4—1674no—
176D4D133614yesdirect product factor
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—3342no—
180C2×C4—6684no—
181C2×C4C2×D66811yescharacteristic
182C2×C4C2×D66812yes—
183C2×C4C2×D66811yescharacteristic
184C2×C4C23×D16711yescharacteristic
185C8—22no—
186C8—21no—
187C8D4×D16712yescomplemented
188C22—22no—
189C22—22no—
190C22—133616no—
191C22—3344no—
192C22—6684no—
193C22—6684no—
194C22—6684no—
195C22—6682no—
196C22—6682no—
197C22C2×D133614yescomplemented
198C22C22×D66811yescharacteristic
199C4—22no—
200C4—21no—
201C4—6682no—
202C4C2×D133612yescomplemented
203C4C2×D4×D16711yescharacteristic
204C4#1912887811yescharacteristic
205C2—24no—
206C2—13364no—
207C2—6684no—
208C2D4×D66811yescharacteristic
209C2C22×D133611yescharacteristic
210C2#2492407411yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC8×D4×C167C2
socleC22×C167C22×D4
hypercenterC8×D4D167
hypocenterC167D4×D8
LCSG ▷ C2×C4×C167 ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×D1336C2semidirect
2. D4×D668C2semidirect
3. #24923646C2semidirect
4. #24923505C2semidirect
5. #24923465C2semidirect
6. #24923405C2semidirect
7. C4×D1336C2semidirect
8. C8×D4×C167C2semidirect
9. C22×D668C22semidirect
13. C2×D1336C22semidirect
14. C2×D1336C22semidirect
15. C2×D1336C22semidirect
16. #19128829C22semidirect
18. #19128806C22semidirect
20. C4×D668C22semidirect
21. #19128766C22semidirect
22. #19128754C22semidirect
23. #19128754C22semidirect
24. C22×C8×C167C22semidirect
26. C4⋊C8×C167C22semidirect
27. C22⋊C8×C167C22semidirect
28. C4×C8×C167C22semidirect
43. D1336D4direct (N × H)
67. D668C2×D4semidirect
72. D4×C167D8semidirect
81. C8×C167C2×D4semidirect
84. C22×C167C2×D8semidirect
95. C4×C167C2×D8semidirect
105. C167D4×D8semidirect
114. C8×D4D167semidirect
122. C22×C8C2×D167semidirect
130. C4⋊C8C2×D167semidirect
133. C22⋊C8C2×D167semidirect
134. C4×C8C2×D167semidirect
176. D4D1336direct (N × H)
187. C8D4×D167semidirect
197. C22C2×D1336semidirect
202. C4C2×D1336semidirect

Maximal subgroups

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

subgroupindex
C22×D13362
D4×D6682
#249236462
#249235052
#249234652
#249234052
C4×D13362
C8×D4×C1672
D4×D8167