Finite Group Databasebeta

← D4×#14581939 #36335836

Subgroups — D4×#14581939 #36335836

Order 21376.

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

normal subgroups only

This group has 546 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×#14581939C212yesmaximal, complemented
2D4×Dic334C211yesmaximal, characteristic, complemented
3D4×D668C211yesmaximal, characteristic
4#24923642C211yesmaximal, characteristic, complemented
5#24923498C212yesmaximal, complemented
6#24923475C211yesmaximal, characteristic, complemented
7#24923470C211yesmaximal, characteristic, complemented
8#24923422C212yesmaximal, complemented
9#24923410C212yesmaximal, complemented
10C4×#14581939C211yesmaximal, characteristic, complemented
11C8×D4×C167C211yesmaximal, characteristic, complemented
12C22×Dic334C2212yescomplemented
13C22×D668C2212yes—
14C2×D4×D167—21no—
15C2×#14581939—28no—
16C2×#14581939C2212yes—
17C2×#14581939C2211yescharacteristic
18C2×#14581939C2212yescomplemented
19C2×#14581939C2212yescomplemented
20C2×#14581939C2212yescomplemented
21#19128832C2211yescharacteristic, complemented
22#19128831—21no—
23#19128829C2211yescharacteristic
24#19128811C2212yescomplemented
25#19128809—22no—
26#19128807—21no—
27#19128806C2212yes—
28#19128802—22no—
29C4×Dic334C2211yescharacteristic, complemented
30C4×D668C2211yescharacteristic
31D4×Dic167—21no—
32#19128764C2211yescharacteristic, complemented
33#19128757C2211yescharacteristic, complemented
34#19128757C2212yescomplemented
35#19128754C2211yescharacteristic
36#19128754C2212yescomplemented
37C22×C8×C167C2212yescomplemented
38C4×D4×C167C2211yescharacteristic
39C4⋊C8×C167C2211yescharacteristic, complemented
40C22⋊C8×C167C2212yescomplemented
41C4×C8×C167C2211yescharacteristic, complemented
42C23×D167—22no—
43C2×Dic334—24no—
44C2×Dic334—22no—
45C2×Dic334—42no—
46C2×Dic334C2312yes—
47C2×Dic334C2311yescharacteristic
48C2×D668—24no—
49C2×D668—21no—
50C2×D668—22no—
51C2×D668—21no—
52C2×D668—42no—
53C2×D668C2312yes—
54C2×D668C2311yescharacteristic
55C2×C22⋊D167—22no—
56D4×D167—44no—
57C22×Dic167—22no—
58C2×C4×D167—21no—
59#14581939—24no—
60#14581939—24no—
61#14581939—24no—
62#14581939—42no—
63#14581939D414yesdirect product factor
64C4⋊Dic167—21no—
65C4⋊Dic167—21no—
66C4⋊Dic167C2311yescharacteristic
67C4⋊Dic167C2311yescharacteristic
68Dic167⋊C4—21no—
69Dic167⋊C4—22no—
70C22⋊Dic167—22no—
71(C2×D167)⋊C4—21no—
72(C2×D167)⋊C4—22no—
73C4×Dic167—21no—
74C2×D4×C167D411yescharacteristic
75C22×C4×C167C2312yes—
76C2×C8×C167—24no—
77C2×C8×C167C2312yes—
78C2×C8×C167C2311yescharacteristic
79C2×C8×C167C2311yescharacteristic
80C4⋊C4×C167D411yescharacteristic
81C22⋊C4×C167D412yes—
82C42×C167C2311yescharacteristic
83C22×D167—21no—
84C22×D167—22no—
85C22×D167—44no—
86C22×D167—44no—
87C22×D167—82no—
88Dic334—21no—
89Dic334—22no—
90Dic334—22no—
91Dic334—44no—
92Dic334C2×D412yescomplemented
93D668—22no—
94D668—21no—
95D668—22no—
96D668—44no—
97D668—42no—
98D668—81no—
99D668C2×D412yes—
100C22⋊D167—44no—
101C2×Dic167—21no—
102C2×Dic167—21no—
103C2×Dic167—21no—
104C2×Dic167—22no—
105C2×Dic167—44no—
106C4×D167—42no—
107C23×C167C2×D412yes—
108D4×C167SD1614yescomplemented
109C2×C4×C167—22no—
110C2×C4×C167—22no—
111C2×C4×C167C2×D411yescharacteristic
112C2×C4×C167C2×D412yes—
113C2×C4×C167C2×D411yescharacteristic
114C2×C4×C167C2411yescharacteristic
115C8×C167—22no—
116C8×C167—21no—
117C8×C167C2×D412yescomplemented
118C22×C167—22no—
119C22×C167—22no—
120C22×C167C2×SD1614yescomplemented
121C22×C167C22×D411yescharacteristic
122C2×D167—22no—
123C2×D167—41no—
124C2×D167—41no—
125C2×D167—42no—
126C2×D167—42no—
127C2×D167—42no—
128C2×D167—88no—
129C4×C167—22no—
130C4×C167—21no—
131C4×C167C2×SD1612yescomplemented
132C4×C167C22×D411yescharacteristic
133C4×C167H(5,𝔽2)11yescharacteristic
134Dic167—22no—
135Dic167—42no—
136Dic167—41no—
137Dic167—41no—
138C2×C167—24no—
139C2×C167D4211yescharacteristic
140C2×C167C22×SD1611yescharacteristic
141C2×C167#57711yescharacteristic
142D167—42no—
143D167—82no—
144C167D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1671nomaximal, Sylow
146C22×SD16—1672no—
147D4×Q8—1671no—
148D42—1671no—
149#492—1671no—
150#465—1672no—
151#461—1671no—
152#460—1671no—
153#450—1672no—
154#448—1672no—
155C4×SD16—1671no—
156C8×D4D16711yescharacteristic, complemented
157C22×Q8—1672no—
158C22×D4—3341no—
159C22×D4—1672no—
160C2×SD16—3348no—
161C2×SD16—1672no—
162C2×SD16—1672no—
163C2×SD16—1672no—
164C2×SD16—1672no—
165C2×SD16—1671no—
166C22×C8C2×D16712yescomplemented
167C4⋊Q8—3341no—
168C4⋊Q8—1671no—
169#127—1671no—
170C22⋊Q8—3342no—
171C22⋊Q8—1672no—
172#121—3341no—
173#121—1672no—
174#120—3342no—
175C4×Q8—1671no—
176C4×D4—3341no—
177C4×D4—1671no—
178C4×D4C2×D16711yescharacteristic
179#106—1671no—
180C4⋊C8C2×D16711yescharacteristic, complemented
181#103—1671no—
182#103—1672no—
183#102—1671no—
184#102—1672no—
185C22⋊C8C2×D16712yescomplemented
186C4×C8C2×D16711yescharacteristic, complemented
187C24—3342no—
188C2×Q8—3344no—
189C2×Q8—3342no—
190C2×Q8—6682no—
191C2×Q8—1672no—
192C2×Q8—1671no—
193C2×D4—3344no—
194C2×D4—3342no—
195C2×D4—3341no—
196C2×D4—3342no—
197C2×D4—3341no—
198C2×D4—6682no—
199C2×D4—6684no—
200C2×D4—1672no—
201C2×D4—1671no—
202C2×D4D66811yescharacteristic
203C22×C4—3342no—
204C22×C4—3341no—
205C22×C4C22×D16712yes—
206SD16—3344no—
207SD16—3344no—
208SD16—3344no—
209SD16—6682no—
210SD16—1674no—
211C2×C8—24no—
212C2×C8C22×D16711yescharacteristic
213C2×C8C22×D16712yes—
214C2×C8C22×D16711yescharacteristic
215C4⋊C4—3341no—
216C4⋊C4—3341no—
217C4⋊C4—3341no—
218C4⋊C4—3342no—
219C4⋊C4—1671no—
220C4⋊C4—1671no—
221C4⋊C4D66811yescharacteristic
222C22⋊C4—3342no—
223C22⋊C4—3341no—
224C22⋊C4—3342no—
225C22⋊C4D66812yes—
226C42—3341no—
227C42C22×D16711yescharacteristic
228C23—13362no—
229C23—3341no—
230C23—3342no—
231C23—6684no—
232C23—6684no—
233C23C2×D66812yes—
234Q8—3341no—
235Q8—3342no—
236Q8—3342no—
237Q8—6684no—
238Q8—1672no—
239D4—13361no—
240D4—3342no—
241D4—3341no—
242D4—3342no—
243D4—6684no—
244D4—6684no—
245D4—6682no—
246D4—1672no—
247D4#1458193914yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—3341no—
251C2×C4—3341no—
252C2×C4—3341no—
253C2×C4—3342no—
254C2×C4—6684no—
255C2×C4—6682no—
256C2×C4C2×D66811yescharacteristic
257C2×C4C2×D66812yes—
258C2×C4C2×D66811yescharacteristic
259C2×C4C23×D16711yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D16712yescomplemented
263C22—22no—
264C22—22no—
265C22—13368no—
266C22—3342no—
267C22—6682no—
268C22—6682no—
269C22—6682no—
270C22—6681no—
271C22—6681no—
272C22C2×#1458193914yescomplemented
273C22C22×D66811yescharacteristic
274C4—22no—
275C4—21no—
276C4—3342no—
277C4—6682no—
278C4—6681no—
279C4—6681no—
280C4C2×#1458193912yescomplemented
281C4C2×D4×D16711yescharacteristic
282C4#1912887811yescharacteristic
283C2—24no—
284C2—13362no—
285C2—6682no—
286C2D4×D66811yescharacteristic
287C2C22×#1458193911yescharacteristic
288C2#2492408111yescharacteristic
subgroupG/H
centerC22C22×D668
commutator, cosocleC2×C4×C167C24
FrattiniC2×C4C23×D167
FittingC8×D4×C167C2
socleC22×C167C22×D4
prime radicalD4×D668C2
hypercenterC8×D4D167
hypocenterC167D4×SD16
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×#14581939C2semidirect
2. D4×Dic334C2semidirect
4. #24923642C2semidirect
5. #24923498C2semidirect
6. #24923475C2semidirect
7. #24923470C2semidirect
8. #24923422C2semidirect
9. #24923410C2semidirect
10. C4×#14581939C2semidirect
11. C8×D4×C167C2semidirect
12. C22×Dic334C22semidirect
18. C2×#14581939C22semidirect
19. C2×#14581939C22semidirect
20. C2×#14581939C22semidirect
21. #19128832C22semidirect
24. #19128811C22semidirect
29. C4×Dic334C22semidirect
32. #19128764C22semidirect
33. #19128757C22semidirect
34. #19128757C22semidirect
36. #19128754C22semidirect
37. C22×C8×C167C22semidirect
39. C4⋊C8×C167C22semidirect
40. C22⋊C8×C167C22semidirect
41. C4×C8×C167C22semidirect
63. #14581939D4direct (N × H)
92. Dic334C2×D4semidirect
108. D4×C167SD16semidirect
117. C8×C167C2×D4semidirect
120. C22×C167C2×SD16semidirect
131. C4×C167C2×SD16semidirect
144. C167D4×SD16semidirect
156. C8×D4D167semidirect
166. C22×C8C2×D167semidirect
180. C4⋊C8C2×D167semidirect
185. C22⋊C8C2×D167semidirect
186. C4×C8C2×D167semidirect
247. D4#14581939direct (N × H)
262. C8D4×D167semidirect
272. C22C2×#14581939semidirect
280. C4C2×#14581939semidirect

Maximal subgroups

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

subgroupindex
C22×#145819392
D4×Dic3342
D4×D6682
#249236422
#249234982
#249234752
#249234702
#249234222
#249234102
C4×#145819392
C8×D4×C1672
D4×SD16167