Finite Group Databasebeta

← D4×#14579687 #35226228

Subgroups — D4×#14579687 #35226228

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×#14579687C212yesmaximal, complemented
2D4×Dic326C211yesmaximal, characteristic, complemented
3#23782037C211yesmaximal, characteristic, complemented
4#23781892C212yesmaximal, complemented
5#23781870C211yesmaximal, characteristic, complemented
6#23781862C211yesmaximal, characteristic, complemented
7#23781816C212yesmaximal, complemented
8#23781803C212yesmaximal, complemented
9C4×#14579687C211yesmaximal, characteristic, complemented
10D4×C163⋊C8C211yesmaximal, characteristic, complemented
11D42×C163C211yesmaximal, characteristic
12C22×Dic326C2212yescomplemented
13C2×#14579687—28no—
14C2×#14579687C2212yes—
15C2×#14579687C2211yescharacteristic
16C2×#14579687C2212yescomplemented
17C2×#14579687C2212yescomplemented
18C2×#14579687C2212yescomplemented
19C22×C163⋊C8C2212yescomplemented
20#19104735C2211yescharacteristic, complemented
21#19104734—21no—
22#19104714C2212yescomplemented
23#19104712—22no—
24C4×Dic326C2211yescharacteristic, complemented
25D4×Dic163—21no—
26#19104666C2211yescharacteristic, complemented
27C4⋊(C163⋊C8)C2211yescharacteristic, complemented
28#19104658C2212yescomplemented
29#19104658C2211yescharacteristic, complemented
30#19104656C2211yescharacteristic
31#19104656C2212yescomplemented
32C22⋊(C163⋊C8)C2212yescomplemented
33C4×C163⋊C8C2211yescharacteristic, complemented
34C22×D4×C163—21no—
35C22×D4×C163C2212yes—
36#127×C163C2211yescharacteristic
37#121×C163—21no—
38#121×C163C2212yes—
39#120×C163—22no—
40C4×D4×C163C2211yescharacteristic
41C4×D4×C163C2211yescharacteristic
42C2×Dic326—24no—
43C2×Dic326—22no—
44C2×Dic326—42no—
45C2×Dic326C2312yes—
46C2×Dic326C2311yescharacteristic
47C22×Dic163—22no—
48#14579687—24no—
49#14579687—24no—
50#14579687—24no—
51#14579687—42no—
52#14579687D414yesdirect product factor
53C2×C163⋊C8—24no—
54C2×C163⋊C8C2311yescharacteristic
55C2×C163⋊C8C2312yes—
56C2×C163⋊C8C2311yescharacteristic
57C4⋊Dic163—21no—
58C4⋊Dic163—21no—
59C4⋊Dic163C2311yescharacteristic
60Dic163⋊C4—21no—
61Dic163⋊C4—22no—
62C22⋊Dic163—22no—
63C4×Dic163—21no—
64C24×C163—22no—
65C2×D4×C163—24no—
66C2×D4×C163—22no—
67C2×D4×C163—22no—
68C2×D4×C163—21no—
69C2×D4×C163—21no—
70C2×D4×C163—42no—
71C2×D4×C163—44no—
72C2×D4×C163D411yescharacteristic
73C2×D4×C163C2312yes—
74C2×D4×C163C2311yescharacteristic
75C22×C4×C163—21no—
76C22×C4×C163C2312yes—
77C4⋊C4×C163D411yescharacteristic
78C4⋊C4×C163C2311yescharacteristic
79C22⋊C4×C163—21no—
80C22⋊C4×C163—22no—
81C22⋊C4×C163D412yes—
82C42×C163C2311yescharacteristic
83Dic326—21no—
84Dic326—22no—
85Dic326—22no—
86Dic326—44no—
87Dic326C2×D412yescomplemented
88C2×Dic163—21no—
89C2×Dic163—21no—
90C2×Dic163—22no—
91C2×Dic163—44no—
92C163⋊C8—22no—
93C163⋊C8—21no—
94C163⋊C8C2×D412yescomplemented
95C23×C163—21no—
96C23×C163—22no—
97C23×C163—44no—
98C23×C163—44no—
99C23×C163—82no—
100C23×C163C2×D412yes—
101D4×C163—22no—
102D4×C163—21no—
103D4×C163—22no—
104D4×C163—44no—
105D4×C163—44no—
106D4×C163—42no—
107D4×C163—81no—
108D4×C163SD1614yescomplemented
109D4×C163C2×D412yes—
110C2×C4×C163—22no—
111C2×C4×C163—22no—
112C2×C4×C163—21no—
113C2×C4×C163—42no—
114C2×C4×C163C2×D411yescharacteristic
115C2×C4×C163C2×D412yes—
116C2×C4×C163C2×D411yescharacteristic
117C2×C4×C163C2411yescharacteristic
118C22×C163—22no—
119C22×C163—22no—
120C22×C163—22no—
121C22×C163—41no—
122C22×C163—41no—
123C22×C163—42no—
124C22×C163—42no—
125C22×C163—42no—
126C22×C163—88no—
127C22×C163C2×SD1614yescomplemented
128C22×C163C22×D411yescharacteristic
129C4×C163—22no—
130C4×C163—21no—
131C4×C163—41no—
132C4×C163C2×SD1612yescomplemented
133C4×C163C22×D411yescharacteristic
134C4×C163H(5,𝔽2)11yescharacteristic
135Dic163—22no—
136Dic163—42no—
137Dic163—41no—
138C2×C163—24no—
139C2×C163—42no—
140C2×C163—82no—
141C2×C163D4211yescharacteristic
142C2×C163C22×SD1611yescharacteristic
143C2×C163#57711yescharacteristic
144C163D4×SD1611yescharacteristic, complemented, Sylow
145D4×SD16—1631nomaximal, Sylow
146C22×SD16—1632no—
147D4×Q8—1631no—
148D42D16311yescharacteristic
149#492—1631no—
150#465—1632no—
151#461—1631no—
152#460—1631no—
153#450—1632no—
154#448—1632no—
155C4×SD16—1631no—
156C8×D4—1631no—
157C22×Q8—1632no—
158C22×D4—21no—
159C22×D4C2×D16312yes—
160C2×SD16—3268no—
161C2×SD16—1632no—
162C2×SD16—1632no—
163C2×SD16—1632no—
164C2×SD16—1632no—
165C2×SD16—1631no—
166C22×C8—1632no—
167C4⋊Q8—3261no—
168C4⋊Q8—1631no—
169#127C2×D16311yescharacteristic
170C22⋊Q8—3262no—
171C22⋊Q8—1632no—
172#121—21no—
173#121C2×D16312yes—
174#120—22no—
175C4×Q8—1631no—
176C4×D4—3261no—
177C4×D4C2×D16311yescharacteristic
178C4×D4C2×D16311yescharacteristic
179#106—1631no—
180C4⋊C8—1631no—
181#103—1631no—
182#103—1632no—
183#102—1631no—
184#102—1632no—
185C22⋊C8—1632no—
186C4×C8—1631no—
187C24—22no—
188C2×Q8—3264no—
189C2×Q8—3262no—
190C2×Q8—6522no—
191C2×Q8—1632no—
192C2×Q8—1631no—
193C2×D4—24no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—42no—
199C2×D4—44no—
200C2×D4C22⋊D16311yescharacteristic
201C2×D4C22×D16312yes—
202C2×D4C22×D16311yescharacteristic
203C22×C4—21no—
204C22×C4—3262no—
205C22×C4C22×D16312yes—
206SD16—3264no—
207SD16—3264no—
208SD16—3264no—
209SD16—6522no—
210SD16—1634no—
211C2×C8—3264no—
212C2×C8—1631no—
213C2×C8—1632no—
214C2×C8—1631no—
215C4⋊C4—3261no—
216C4⋊C4—3261no—
217C4⋊C4—3261no—
218C4⋊C4—3262no—
219C4⋊C4—1631no—
220C4⋊C4C22⋊D16311yescharacteristic
221C4⋊C4C22×D16311yescharacteristic
222C22⋊C4—21no—
223C22⋊C4—22no—
224C22⋊C4—3262no—
225C22⋊C4C22⋊D16312yes—
226C42—3261no—
227C42C22×D16311yescharacteristic
228C23—21no—
229C23—22no—
230C23—44no—
231C23—44no—
232C23—82no—
233C23C2×C22⋊D16312yes—
234Q8—3261no—
235Q8—3262no—
236Q8—3262no—
237Q8—6524no—
238Q8—1632no—
239D4—22no—
240D4—21no—
241D4—22no—
242D4—44no—
243D4—44no—
244D4—42no—
245D4—81no—
246D4#1457968714yesdirect product factor
247D4D4×D16312yes—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—3261no—
253C2×C4—3261no—
254C2×C4—3262no—
255C2×C4—6524no—
256C2×C4C2×C22⋊D16311yescharacteristic
257C2×C4C2×C22⋊D16312yes—
258C2×C4C2×C22⋊D16311yescharacteristic
259C2×C4C23×D16311yescharacteristic
260C8—3262no—
261C8—3261no—
262C8—1632no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—88no—
272C22C2×#1457968714yescomplemented
273C22C22×C22⋊D16311yescharacteristic
274C4—21no—
275C4—22no—
276C4—41no—
277C4—3262no—
278C4—6521no—
279C4—6522no—
280C4C2×#1457968712yescomplemented
281C4C2×D4×D16311yescharacteristic
282C4#1910478011yescharacteristic
283C2—24no—
284C2—42no—
285C2—82no—
286C2D4×C22⋊D16311yescharacteristic
287C2C22×#1457968711yescharacteristic
288C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting, prime radicalD42×C163C2
socleC22×C163C22×D4
hypercenterD42D163
hypocenterC163D4×SD16
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C22×#14579687C2semidirect
2. D4×Dic326C2semidirect
3. #23782037C2semidirect
4. #23781892C2semidirect
5. #23781870C2semidirect
6. #23781862C2semidirect
7. #23781816C2semidirect
8. #23781803C2semidirect
9. C4×#14579687C2semidirect
10. D4×C163⋊C8C2semidirect
12. C22×Dic326C22semidirect
16. C2×#14579687C22semidirect
17. C2×#14579687C22semidirect
18. C2×#14579687C22semidirect
19. C22×C163⋊C8C22semidirect
20. #19104735C22semidirect
22. #19104714C22semidirect
24. C4×Dic326C22semidirect
26. #19104666C22semidirect
27. C4⋊(C163⋊C8)C22semidirect
28. #19104658C22semidirect
29. #19104658C22semidirect
31. #19104656C22semidirect
32. C22⋊(C163⋊C8)C22semidirect
33. C4×C163⋊C8C22semidirect
52. #14579687D4direct (N × H)
87. Dic326C2×D4semidirect
94. C163⋊C8C2×D4semidirect
108. D4×C163SD16semidirect
127. C22×C163C2×SD16semidirect
132. C4×C163C2×SD16semidirect
144. C163D4×SD16semidirect
246. D4#14579687direct (N × H)
272. C22C2×#14579687semidirect
280. C4C2×#14579687semidirect

Maximal subgroups

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

subgroupindex
C22×#145796872
D4×Dic3262
#237820372
#237818922
#237818702
#237818622
#237818162
#237818032
C4×#145796872
D4×C163⋊C82
D42×C1632
D4×SD16163