Finite Group Databasebeta

← D4×#14579688 #35226234

Subgroups — D4×#14579688 #35226234

Order 20864.

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×#14579688C212yesmaximal, complemented
2D4×Dic326C211yesmaximal, characteristic, complemented
3D4×D652C211yesmaximal, characteristic
4#23782038C211yesmaximal, characteristic, complemented
5#23781894C212yesmaximal, complemented
6#23781871C211yesmaximal, characteristic, complemented
7#23781866C211yesmaximal, characteristic, complemented
8#23781818C212yesmaximal, complemented
9#23781806C212yesmaximal, complemented
10C4×#14579688C211yesmaximal, characteristic, complemented
11C8×D4×C163C211yesmaximal, characteristic, complemented
12C22×Dic326C2212yescomplemented
13C22×D652C2212yes—
14C2×D4×D163—21no—
15C2×#14579688—28no—
16C2×#14579688C2212yes—
17C2×#14579688C2211yescharacteristic
18C2×#14579688C2212yescomplemented
19C2×#14579688C2212yescomplemented
20C2×#14579688C2212yescomplemented
21#19104735C2211yescharacteristic, complemented
22#19104734—21no—
23#19104732C2211yescharacteristic
24#19104714C2212yescomplemented
25#19104712—22no—
26#19104710—21no—
27#19104709C2212yes—
28#19104705—22no—
29C4×Dic326C2211yescharacteristic, complemented
30C4×D652C2211yescharacteristic
31D4×Dic163—21no—
32#19104667C2211yescharacteristic, complemented
33#19104660C2212yescomplemented
34#19104660C2211yescharacteristic, complemented
35#19104657C2211yescharacteristic
36#19104657C2212yescomplemented
37C22×C8×C163C2212yescomplemented
38C4×D4×C163C2211yescharacteristic
39C4⋊C8×C163C2211yescharacteristic, complemented
40C22⋊C8×C163C2212yescomplemented
41C4×C8×C163C2211yescharacteristic, complemented
42C23×D163—22no—
43C2×Dic326—24no—
44C2×Dic326—22no—
45C2×Dic326—42no—
46C2×Dic326C2312yes—
47C2×Dic326C2311yescharacteristic
48C2×D652—24no—
49C2×D652—21no—
50C2×D652—22no—
51C2×D652—21no—
52C2×D652—42no—
53C2×D652C2312yes—
54C2×D652C2311yescharacteristic
55C2×C22⋊D163—22no—
56D4×D163—44no—
57C22×Dic163—22no—
58C2×C4×D163—21no—
59#14579688—24no—
60#14579688—24no—
61#14579688—24no—
62#14579688—42no—
63#14579688D414yesdirect product factor
64C4⋊Dic163—21no—
65C4⋊Dic163—21no—
66C4⋊Dic163C2311yescharacteristic
67C4⋊Dic163C2311yescharacteristic
68Dic163⋊C4—21no—
69Dic163⋊C4—22no—
70C22⋊Dic163—22no—
71(C2×D163)⋊C4—21no—
72(C2×D163)⋊C4—22no—
73C4×Dic163—21no—
74C2×D4×C163D411yescharacteristic
75C22×C4×C163C2312yes—
76C2×C8×C163—24no—
77C2×C8×C163C2311yescharacteristic
78C2×C8×C163C2312yes—
79C2×C8×C163C2311yescharacteristic
80C4⋊C4×C163D411yescharacteristic
81C22⋊C4×C163D412yes—
82C42×C163C2311yescharacteristic
83C22×D163—21no—
84C22×D163—22no—
85C22×D163—44no—
86C22×D163—44no—
87C22×D163—82no—
88Dic326—21no—
89Dic326—22no—
90Dic326—22no—
91Dic326—44no—
92Dic326C2×D412yescomplemented
93D652—22no—
94D652—21no—
95D652—22no—
96D652—44no—
97D652—42no—
98D652—81no—
99D652C2×D412yes—
100C22⋊D163—44no—
101C2×Dic163—21no—
102C2×Dic163—21no—
103C2×Dic163—21no—
104C2×Dic163—22no—
105C2×Dic163—44no—
106C4×D163—42no—
107C23×C163C2×D412yes—
108D4×C163SD1614yescomplemented
109C2×C4×C163—22no—
110C2×C4×C163—22no—
111C2×C4×C163C2×D411yescharacteristic
112C2×C4×C163C2×D412yes—
113C2×C4×C163C2×D411yescharacteristic
114C2×C4×C163C2411yescharacteristic
115C8×C163—22no—
116C8×C163—21no—
117C8×C163C2×D412yescomplemented
118C22×C163—22no—
119C22×C163—22no—
120C22×C163C2×SD1614yescomplemented
121C22×C163C22×D411yescharacteristic
122C2×D163—22no—
123C2×D163—41no—
124C2×D163—41no—
125C2×D163—42no—
126C2×D163—42no—
127C2×D163—42no—
128C2×D163—88no—
129C4×C163—22no—
130C4×C163—21no—
131C4×C163C2×SD1612yescomplemented
132C4×C163C22×D411yescharacteristic
133C4×C163H(5,𝔽2)11yescharacteristic
134Dic163—22no—
135Dic163—42no—
136Dic163—41no—
137Dic163—41no—
138C2×C163—24no—
139C2×C163D4211yescharacteristic
140C2×C163C22×SD1611yescharacteristic
141C2×C163#57711yescharacteristic
142D163—42no—
143D163—82no—
144C163D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1631nomaximal, Sylow
146C22×SD16—1632no—
147D4×Q8—1631no—
148D42—1631no—
149#492—1631no—
150#465—1632no—
151#461—1631no—
152#460—1631no—
153#450—1632no—
154#448—1632no—
155C4×SD16—1631no—
156C8×D4D16311yescharacteristic, complemented
157C22×Q8—1632no—
158C22×D4—3261no—
159C22×D4—1632no—
160C2×SD16—3268no—
161C2×SD16—1632no—
162C2×SD16—1632no—
163C2×SD16—1632no—
164C2×SD16—1632no—
165C2×SD16—1631no—
166C22×C8C2×D16312yescomplemented
167C4⋊Q8—3261no—
168C4⋊Q8—1631no—
169#127—1631no—
170C22⋊Q8—3262no—
171C22⋊Q8—1632no—
172#121—3261no—
173#121—1632no—
174#120—3262no—
175C4×Q8—1631no—
176C4×D4—3261no—
177C4×D4—1631no—
178C4×D4C2×D16311yescharacteristic
179#106—1631no—
180C4⋊C8C2×D16311yescharacteristic, complemented
181#103—1631no—
182#103—1632no—
183#102—1631no—
184#102—1632no—
185C22⋊C8C2×D16312yescomplemented
186C4×C8C2×D16311yescharacteristic, complemented
187C24—3262no—
188C2×Q8—3264no—
189C2×Q8—3262no—
190C2×Q8—6522no—
191C2×Q8—1632no—
192C2×Q8—1631no—
193C2×D4—3264no—
194C2×D4—3262no—
195C2×D4—3261no—
196C2×D4—3262no—
197C2×D4—3261no—
198C2×D4—6522no—
199C2×D4—6524no—
200C2×D4—1632no—
201C2×D4—1631no—
202C2×D4D65211yescharacteristic
203C22×C4—3262no—
204C22×C4—3261no—
205C22×C4C22×D16312yes—
206SD16—3264no—
207SD16—3264no—
208SD16—3264no—
209SD16—6522no—
210SD16—1634no—
211C2×C8—24no—
212C2×C8C22×D16311yescharacteristic
213C2×C8C22×D16312yes—
214C2×C8C22×D16311yescharacteristic
215C4⋊C4—3261no—
216C4⋊C4—3261no—
217C4⋊C4—3261no—
218C4⋊C4—3262no—
219C4⋊C4—1631no—
220C4⋊C4—1631no—
221C4⋊C4D65211yescharacteristic
222C22⋊C4—3262no—
223C22⋊C4—3261no—
224C22⋊C4—3262no—
225C22⋊C4D65212yes—
226C42—3261no—
227C42C22×D16311yescharacteristic
228C23—13042no—
229C23—3261no—
230C23—3262no—
231C23—6524no—
232C23—6524no—
233C23C2×D65212yes—
234Q8—3261no—
235Q8—3262no—
236Q8—3262no—
237Q8—6524no—
238Q8—1632no—
239D4—13041no—
240D4—3262no—
241D4—3261no—
242D4—3262no—
243D4—6524no—
244D4—6524no—
245D4—6522no—
246D4—1632no—
247D4#1457968814yesdirect product factor
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—3261no—
251C2×C4—3261no—
252C2×C4—3261no—
253C2×C4—3262no—
254C2×C4—6524no—
255C2×C4—6522no—
256C2×C4C2×D65211yescharacteristic
257C2×C4C2×D65212yes—
258C2×C4C2×D65211yescharacteristic
259C2×C4C23×D16311yescharacteristic
260C8—22no—
261C8—21no—
262C8D4×D16312yescomplemented
263C22—22no—
264C22—22no—
265C22—13048no—
266C22—3262no—
267C22—6522no—
268C22—6522no—
269C22—6522no—
270C22—6521no—
271C22—6521no—
272C22C2×#1457968814yescomplemented
273C22C22×D65211yescharacteristic
274C4—21no—
275C4—22no—
276C4—3262no—
277C4—6521no—
278C4—6521no—
279C4—6522no—
280C4C2×#1457968812yescomplemented
281C4C2×D4×D16311yescharacteristic
282C4#1910478111yescharacteristic
283C2—24no—
284C2—13042no—
285C2—6522no—
286C2D4×D65211yescharacteristic
287C2C22×#1457968811yescharacteristic
288C2#2378247711yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC8×D4×C163C2
socleC22×C163C22×D4
prime radicalD4×D652C2
hypercenterC8×D4D163
hypocenterC163D4×SD16
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14579688C2semidirect
2. D4×Dic326C2semidirect
4. #23782038C2semidirect
5. #23781894C2semidirect
6. #23781871C2semidirect
7. #23781866C2semidirect
8. #23781818C2semidirect
9. #23781806C2semidirect
10. C4×#14579688C2semidirect
11. C8×D4×C163C2semidirect
12. C22×Dic326C22semidirect
18. C2×#14579688C22semidirect
19. C2×#14579688C22semidirect
20. C2×#14579688C22semidirect
21. #19104735C22semidirect
24. #19104714C22semidirect
29. C4×Dic326C22semidirect
32. #19104667C22semidirect
33. #19104660C22semidirect
34. #19104660C22semidirect
36. #19104657C22semidirect
37. C22×C8×C163C22semidirect
39. C4⋊C8×C163C22semidirect
40. C22⋊C8×C163C22semidirect
41. C4×C8×C163C22semidirect
63. #14579688D4direct (N × H)
92. Dic326C2×D4semidirect
108. D4×C163SD16semidirect
117. C8×C163C2×D4semidirect
120. C22×C163C2×SD16semidirect
131. C4×C163C2×SD16semidirect
144. C163D4×SD16semidirect
156. C8×D4D163semidirect
166. C22×C8C2×D163semidirect
180. C4⋊C8C2×D163semidirect
185. C22⋊C8C2×D163semidirect
186. C4×C8C2×D163semidirect
247. D4#14579688direct (N × H)
262. C8D4×D163semidirect
272. C22C2×#14579688semidirect
280. C4C2×#14579688semidirect

Maximal subgroups

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

subgroupindex
C22×#145796882
D4×Dic3262
D4×D6522
#237820382
#237818942
#237818712
#237818662
#237818182
#237818062
C4×#145796882
C8×D4×C1632
D4×SD16163