Finite Group Databasebeta

← #35226111

Subgroups — #35226111

Order 20864.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104768C211yesmaximal, characteristic, complemented
2C22×Dic652C211yesmaximal, characteristic, complemented
3D4×Dic326C211yesmaximal, characteristic, complemented
4#23782324C211yesmaximal, characteristic
5#23782059C211yesmaximal, characteristic, complemented
6#23781923C211yesmaximal, characteristic, complemented
7#23781894C211yesmaximal, characteristic, complemented
8#23781888C211yesmaximal, characteristic, complemented
9#23781876C211yesmaximal, characteristic, complemented
10#23781829C211yesmaximal, characteristic, complemented
11#23781823C211yesmaximal, characteristic, complemented
12#23781823C211yesmaximal, characteristic, complemented
13#23781806C211yesmaximal, characteristic, complemented
14#23781773C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×Dic326C2211yescharacteristic
18C22×Dic326C2211yescharacteristic
19C22×Dic326C2211yescharacteristic
20#19104768—24no—
21C2×Dic652—24no—
22C2×Dic652C2211yescharacteristic
23C2×Dic652C2211yescharacteristic
24C2×Dic652C2211yescharacteristic
25C2×Dic652C2211yescharacteristic
26C2×Dic652C2211yescharacteristic
27C2×Dic652C2211yescharacteristic
28C2×#14579688C2211yescharacteristic
29C2×#14579688C2211yescharacteristic
30C2×#14579688C2211yescharacteristic
31#19104735C2211yescharacteristic
32#19104734—21no—
33#19104723C2211yescharacteristic
34#19104721—21no—
35#19104714C2211yescharacteristic
36#19104714C2211yescharacteristic
37#19104714C2211yescharacteristic
38#19104713—21no—
39#19104712—21no—
40#19104712—21no—
41#19104709C2211yescharacteristic
42#19104707—21no—
43C4×Dic326C2211yescharacteristic
44C4×Dic326C2211yescharacteristic
45D4×Dic163—21no—
46(C22⋊D163)⋊C4—21no—
47#19104667C2211yescharacteristic
48#19104660C2211yescharacteristic
49#19104660C2211yescharacteristic
50#19104660C2211yescharacteristic
51#19104660C2211yescharacteristic
52#19104660C2211yescharacteristic
53#19104657C2211yescharacteristic
54C2×M16×C163C2211yescharacteristic
55C22×C8×C163C2211yescharacteristic
56C4×D4×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C22⋊C8×C163C2211yescharacteristic
60#97×C163C2211yescharacteristic
61#14579699—22no—
62#14579699—41no—
63C2×Dic326—22no—
64C2×Dic326—22no—
65C2×Dic326—22no—
66C2×Dic326—21no—
67C2×Dic326—21no—
68C2×Dic326—21no—
69C2×Dic326—41no—
70C2×Dic326—41no—
71C2×Dic326—41no—
72C2×Dic326C2311yescharacteristic
73C2×Dic326C2311yescharacteristic
74C2×Dic326C2311yescharacteristic
75C2×Dic326C2311yescharacteristic
76C2×Dic326C2311yescharacteristic
77C2×D652C2311yescharacteristic
78C2×C22⋊D163—21no—
79C2×C22⋊D163—21no—
80C22×Dic163—21no—
81C22×Dic163—21no—
82C22×Dic163—21no—
83C2×C4×D163—21no—
84Dic652—22no—
85Dic652—22no—
86Dic652—22no—
87Dic652—22no—
88Dic652—41no—
89Dic652D414yescomplemented
90#14579688—22no—
91#14579688—22no—
92#14579688—41no—
93C4⋊Dic163—21no—
94C4⋊Dic163—21no—
95C4⋊Dic163C2311yescharacteristic
96C4⋊Dic163C2311yescharacteristic
97Dic163⋊C4—21no—
98Dic163⋊C4—21no—
99Dic163⋊C4—21no—
100Dic163⋊C4—21no—
101Dic163⋊C4—21no—
102C22⋊Dic163—21no—
103C22⋊Dic163—21no—
104C22⋊Dic163—21no—
105(C2×D163)⋊C4—21no—
106(C2×D163)⋊C4—21no—
107(C2×D163)⋊C4—21no—
108C4×Dic163—21no—
109C4×Dic163—21no—
110C2×D4×C163D411yescharacteristic
111C22×C4×C163C2311yescharacteristic
112C22×C4×C163C2311yescharacteristic
113M16×C163—22no—
114C2×C8×C163—22no—
115C2×C8×C163C2311yescharacteristic
116C2×C8×C163C2311yescharacteristic
117C2×C8×C163C2311yescharacteristic
118C2×C8×C163C2311yescharacteristic
119C4⋊C4×C163D411yescharacteristic
120C22⋊C4×C163D411yescharacteristic
121C22⋊C4×C163D411yescharacteristic
122C42×C163C2311yescharacteristic
123C22×D163—21no—
124Dic326—21no—
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128Dic326—21no—
129Dic326—21no—
130Dic326—21no—
131Dic326—21no—
132Dic326—42no—
133Dic326—42no—
134Dic326—42no—
135Dic326C2×D412yes—
136Dic326C2×D412yes—
137D652—21no—
138D652—21no—
139C22⋊D163—42no—
140C22⋊D163—42no—
141C2×Dic163—21no—
142C2×Dic163—21no—
143C2×Dic163—21no—
144C2×Dic163—21no—
145C2×Dic163—21no—
146C2×Dic163—21no—
147C2×Dic163—21no—
148C2×Dic163—42no—
149C2×Dic163—42no—
150C2×Dic163—42no—
151C4×D163—42no—
152C23×C163C2×D411yescharacteristic
153C23×C163C2×D411yescharacteristic
154D4×C163—22no—
155C2×C4×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163C2×D411yescharacteristic
160C2×C4×C163C2×D411yescharacteristic
161C2×C4×C163C2×D411yescharacteristic
162C2×C4×C163C2×D411yescharacteristic
163C2×C4×C163C2411yescharacteristic
164C8×C163—21no—
165C8×C163—21no—
166C8×C163—21no—
167C8×C163C2×D412yes—
168C22×C163—21no—
169C22×C163—21no—
170C22×C163—21no—
171C22×C163—21no—
172C22×C163—21no—
173C22×C163#13712yescomplemented
174C22×C163C22×D411yescharacteristic
175C2×D163—41no—
176C2×D163—41no—
177C2×D163—41no—
178C4×C163—21no—
179C4×C163—21no—
180C4×C163—21no—
181C4×C163—21no—
182C4×C163C22×D411yescharacteristic
183C4×C163H(5,𝔽2)11yescharacteristic
184Dic163—22no—
185Dic163—22no—
186Dic163—41no—
187Dic163—41no—
188Dic163—41no—
189Dic163—41no—
190Dic163—41no—
191C2×C163—22no—
192C2×C163—41no—
193C2×C163D4211yescharacteristic
194C2×C163C2×#13711yescharacteristic
195C2×C163#57811yescharacteristic
196D163—81no—
197C163#327011yescharacteristic, complemented, Sylow
198#3270—1631nomaximal, Sylow
199C2×#137—1631no—
200C22×Q16—1631no—
201D4×Q8—1631no—
202#548—1631no—
203#497—1631no—
204#470—1631no—
205#465—1631no—
206#464—1631no—
207#462—1631no—
208#452—1631no—
209#451—1631no—
210#451—1631no—
211#448—1631no—
212#441—1631no—
213#435D16311yescharacteristic, complemented
214C2×#41—1631no—
215C22×Q8—1631no—
216C22×Q8—1631no—
217C22×Q8—1631no—
218#137—3264no—
219C2×Q16—3264no—
220C2×Q16—1631no—
221C2×Q16—1631no—
222C2×Q16—1631no—
223C2×Q16—1631no—
224C2×Q16—1631no—
225C2×Q16—1631no—
226C2×SD16—1631no—
227C2×SD16—1631no—
228C2×SD16—1631no—
229C2×M16C2×D16311yescharacteristic
230C22×C8C2×D16311yescharacteristic
231C4⋊Q8—3261no—
232C4⋊Q8—1631no—
233#124—3261no—
234#124—1631no—
235C22⋊Q8—3261no—
236C22⋊Q8—3261no—
237C22⋊Q8—3261no—
238C22⋊Q8—1631no—
239C22⋊Q8—1631no—
240C22⋊Q8—1631no—
241#121—3261no—
242#121—1631no—
243C4×Q8—1631no—
244C4×Q8—1631no—
245C4×D4—3261no—
246C4×D4—3261no—
247C4×D4C2×D16311yescharacteristic
248#106—1631no—
249C4⋊C8C2×D16311yescharacteristic
250#103—1631no—
251#103—1631no—
252#103—1631no—
253#103—1631no—
254#103—1631no—
255#102—1631no—
256C22⋊C8C2×D16311yescharacteristic
257C22⋊C8C2×D16311yescharacteristic
258#97C2×D16311yescharacteristic
259#41—3262no—
260#41—6521no—
261C2×Q8—3262no—
262C2×Q8—3262no—
263C2×Q8—3262no—
264C2×Q8—3261no—
265C2×Q8—3261no—
266C2×Q8—3261no—
267C2×Q8—6521no—
268C2×Q8—6521no—
269C2×Q8—6521no—
270C2×Q8—1631no—
271C2×Q8—1631no—
272C2×Q8—1631no—
273C2×Q8—1631no—
274C2×Q8—1631no—
275C2×D4—3261no—
276C2×D4—3261no—
277C2×D4—1631no—
278C2×D4D65211yescharacteristic
279C22×C4—3261no—
280C22×C4—3261no—
281C22×C4—3261no—
282C22×C4—3261no—
283C22×C4C22×D16311yescharacteristic
284C22×C4C22×D16311yescharacteristic
285Q16—3262no—
286Q16—3262no—
287Q16—3262no—
288Q16—3262no—
289Q16—6521no—
290Q16—1634no—
291SD16—3262no—
292SD16—3262no—
293SD16—6521no—
294M16—22no—
295C2×C8—22no—
296C2×C8C22×D16311yescharacteristic
297C2×C8C22×D16311yescharacteristic
298C2×C8C22×D16311yescharacteristic
299C2×C8C22×D16311yescharacteristic
300C4⋊C4—3261no—
301C4⋊C4—3261no—
302C4⋊C4—3261no—
303C4⋊C4—3261no—
304C4⋊C4—3261no—
305C4⋊C4—3261no—
306C4⋊C4—3261no—
307C4⋊C4—1631no—
308C4⋊C4—1631no—
309C4⋊C4D65211yescharacteristic
310C22⋊C4—3261no—
311C22⋊C4—3261no—
312C22⋊C4—3261no—
313C22⋊C4—3261no—
314C22⋊C4—3261no—
315C22⋊C4—3261no—
316C22⋊C4D65211yescharacteristic
317C22⋊C4D65211yescharacteristic
318C42—3261no—
319C42—3261no—
320C42C22×D16311yescharacteristic
321C23—3261no—
322C23C2×D65211yescharacteristic
323C23C2×D65211yescharacteristic
324Q8—3261no—
325Q8—3261no—
326Q8—3261no—
327Q8—3261no—
328Q8—3261no—
329Q8—3261no—
330Q8—3261no—
331Q8—3261no—
332Q8—6522no—
333Q8—6522no—
334Q8—6522no—
335Q8—1632no—
336Q8—1632no—
337D4—22no—
338D4—3261no—
339D4—3261no—
340D4—6522no—
341D4—6522no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—3261no—
351C2×C4—3261no—
352C2×C4—3261no—
353C2×C4—6522no—
354C2×C4—6522no—
355C2×C4—6522no—
356C2×C4—6522no—
357C2×C4C2×D65211yescharacteristic
358C2×C4C2×D65211yescharacteristic
359C2×C4C2×D65211yescharacteristic
360C2×C4C2×D65211yescharacteristic
361C2×C4C23×D16311yescharacteristic
362C8—21no—
363C8—21no—
364C8—21no—
365C8D4×D16312yes—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—6521no—
372C22—6521no—
373C22—6521no—
374C22#1910476812yescomplemented
375C22C22×D65211yescharacteristic
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—3262no—
381C4—3262no—
382C4—6521no—
383C4—6521no—
384C4—6521no—
385C4—6521no—
386C4—6521no—
387C4C2×D4×D16311yescharacteristic
388C4#1910478111yescharacteristic
389C2—22no—
390C2—41no—
391C2—13041no—
392C2D4×D65211yescharacteristic
393C2C2×#1910476811yescharacteristic
394C2#2378248211yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#435×C163C2
socleC22×C163C22×D4
prime radical#23782324C2
hypercenter#435D163
hypocenterC163#3270
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19104768C2semidirect
2. C22×Dic652C2semidirect
3. D4×Dic326C2semidirect
5. #23782059C2semidirect
6. #23781923C2semidirect
7. #23781894C2semidirect
8. #23781888C2semidirect
9. #23781876C2semidirect
10. #23781829C2semidirect
11. #23781823C2semidirect
12. #23781823C2semidirect
13. #23781806C2semidirect
14. #23781773C2semidirect
15. #435×C163C2semidirect
89. Dic652D4semidirect
173. C22×C163#137semidirect
197. C163#3270semidirect
213. #435D163semidirect
374. C22#19104768semidirect

Maximal subgroups

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

subgroupindex
C2×#191047682
C22×Dic6522
D4×Dic3262
#237823242
#237820592
#237819232
#237818942
#237818882
#237818762
#237818292
#237818232
#237818062
#237817732
#435×C1632
#3270163