Finite Group Databasebeta

← #35223262

Subgroups — #35223262

Order 20864.

502 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104743C211yesmaximal, characteristic, complemented
2C22×#14579682C211yesmaximal, characteristic, complemented
3#23782174C211yesmaximal, characteristic
4#23781742C211yesmaximal, characteristic, complemented
5#23781741C211yesmaximal, characteristic, complemented
6#23781738C211yesmaximal, characteristic, complemented
7#23781733C211yesmaximal, characteristic, complemented
8#23781733C211yesmaximal, characteristic, complemented
9D4×C163⋊C8C211yesmaximal, characteristic, complemented
10C8×C22⋊D163C211yesmaximal, characteristic, complemented
11#23781690C211yesmaximal, characteristic, complemented
12#23781630C211yesmaximal, characteristic, complemented
13C22⋊C8×D163C211yesmaximal, characteristic, complemented
14#23781622C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18#19104743—24no—
19C2×#14579682—24no—
20C2×#14579682C2211yescharacteristic
21C2×#14579682C2211yescharacteristic
22C2×#14579682C2211yescharacteristic
23C2×#14579682C2211yescharacteristic
24C2×#14579682C2211yescharacteristic, complemented
25C22×C163⋊C8C2211yescharacteristic
26C22×C163⋊C8C2211yescharacteristic, complemented
27C2×C8×D163C2211yescharacteristic
28C2×C8×D163C2211yescharacteristic
29C2×C8×D163C2211yescharacteristic, complemented
30#19104720C411yescharacteristic
31#19104717C411yescharacteristic
32#19104713C411yescharacteristic
33#19104712C411yescharacteristic
34#19104708C411yescharacteristic
35#19104707C411yescharacteristic
36#19104705C411yescharacteristic
37#19104703C411yescharacteristic
38C4×D652C2211yescharacteristic
39C4×C22⋊D163C2211yescharacteristic
40C4×C22⋊D163C2211yescharacteristic
41#19104692C2211yescharacteristic
42C4⋊(C163⋊C8)C2211yescharacteristic, complemented
43Dic163⋊C8C2211yescharacteristic, complemented
44Dic163⋊C8C2211yescharacteristic, complemented
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic, complemented
47(C2×D163)⋊C8C2211yescharacteristic
48(C2×D163)⋊C8C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C4×C163⋊C8C2211yescharacteristic, complemented
53C8×Dic163C2211yescharacteristic, complemented
54C2×M16×C163C2211yescharacteristic, complemented
55C22×C8×C163C2211yescharacteristic
56C4×D4×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic
59C22⋊C8×C163C2211yescharacteristic, complemented
60#97×C163C2211yescharacteristic, complemented
61C23×D163C2×C411yescharacteristic
62#14579699—22no—
63#14579699—22no—
64C2×Dic326C2×C411yescharacteristic
65C2×D652C2×C411yescharacteristic
66C2×D652C2×C411yescharacteristic
67C2×C22⋊D163C2×C411yescharacteristic
68C2×C22⋊D163C2×C411yescharacteristic
69C2×C22⋊D163C2×C411yescharacteristic
70C2×C22⋊D163C2×C411yescharacteristic
71C22×Dic163C2×C411yescharacteristic
72C2×C4×D163—22no—
73C2×C4×D163—22no—
74C2×C4×D163C2×C411yescharacteristic
75C2×C4×D163C2×C411yescharacteristic
76C2×C4×D163C2311yescharacteristic
77C2×C4×D163C2311yescharacteristic
78C2×C4×D163C2311yescharacteristic
79#14579682—22no—
80#14579682—22no—
81#14579682—22no—
82#14579682—22no—
83#14579682D414yescomplemented
84C2×C163⋊C8—22no—
85C2×C163⋊C8—22no—
86C2×C163⋊C8C2311yescharacteristic
87C2×C163⋊C8C2311yescharacteristic
88C2×C163⋊C8C2311yescharacteristic
89C2×C163⋊C8C2311yescharacteristic
90C8×D163—22no—
91C8×D163—22no—
92C8×D163—22no—
93C4⋊Dic163C2×C411yescharacteristic
94Dic163⋊C4C2×C411yescharacteristic
95Dic163⋊C4C2×C411yescharacteristic
96Dic163⋊C4C2×C411yescharacteristic
97Dic163⋊C4C2×C411yescharacteristic
98C22⋊Dic163C2×C411yescharacteristic
99C22⋊Dic163C2×C411yescharacteristic
100(C2×D163)⋊C4C2×C411yescharacteristic
101(C2×D163)⋊C4C2×C411yescharacteristic
102(C2×D163)⋊C4C2×C411yescharacteristic
103(C2×D163)⋊C4C2×C411yescharacteristic
104(C2×D163)⋊C4C2×C411yescharacteristic
105(C2×D163)⋊C4C2×C411yescharacteristic
106C4×Dic163C2311yescharacteristic
107C2×D4×C163C2×C411yescharacteristic
108C22×C4×C163C2311yescharacteristic
109C22×C4×C163C2311yescharacteristic
110M16×C163—22no—
111C2×C8×C163—22no—
112C2×C8×C163C2311yescharacteristic
113C2×C8×C163C2311yescharacteristic
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C4⋊C4×C163C2×C411yescharacteristic
117C22⋊C4×C163C2×C411yescharacteristic
118C22⋊C4×C163C2×C411yescharacteristic
119C42×C163C2311yescharacteristic
120C22×D163—22no—
121C22×D163—22no—
122C22×D163—41no—
123C22×D163C22×C411yescharacteristic
124C22×D163C22×C411yescharacteristic
125C22×D163C22×C411yescharacteristic
126Dic326—22no—
127D652—22no—
128D652—41no—
129C22⋊D163—22no—
130C22⋊D163—22no—
131C22⋊D163—22no—
132C22⋊D163—41no—
133C2×Dic163—22no—
134C2×Dic163C22×C411yescharacteristic
135C2×Dic163C22×C411yescharacteristic
136C2×Dic163C22×C411yescharacteristic
137C2×Dic163C22×C411yescharacteristic
138C2×Dic163C22×C411yescharacteristic
139C4×D163—22no—
140C4×D163—21no—
141C4×D163—21no—
142C4×D163—21no—
143C4×D163—21no—
144C4×D163—22no—
145C4×D163C2×D412yes—
146C4×D163#4112yes—
147C163⋊C8—21no—
148C163⋊C8—21no—
149C163⋊C8—21no—
150C163⋊C8C2×D412yes—
151C23×C163C22×C411yescharacteristic
152C23×C163C22×C411yescharacteristic
153D4×C163—22no—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
156C2×C4×C163—21no—
157C2×C4×C163—21no—
158C2×C4×C163C22×C411yescharacteristic
159C2×C4×C163C22×C411yescharacteristic
160C2×C4×C163C22×C411yescharacteristic
161C2×C4×C163C22×C411yescharacteristic
162C2×C4×C163C2411yescharacteristic
163C8×C163—21no—
164C8×C163—21no—
165C8×C163—21no—
166C8×C163C2×D412yes—
167C22×C163—21no—
168C22×C163—21no—
169C22×C163—21no—
170C22×C163—21no—
171C22×C163—21no—
172C22×C163#13112yescomplemented
173C22×C163C23×C411yescharacteristic
174C2×D163—21no—
175C2×D163—21no—
176C2×D163—21no—
177C2×D163—21no—
178C2×D163—21no—
179C2×D163—21no—
180C2×D163—21no—
181C2×D163—21no—
182C2×D163—44no—
183C2×D163C4×D412yes—
184C4×C163—21no—
185C4×C163—21no—
186C4×C163—21no—
187C4×C163—21no—
188C4×C163C22×D411yescharacteristic
189C4×C163C2×#4111yescharacteristic
190Dic163—21no—
191Dic163—21no—
192Dic163—21no—
193Dic163—21no—
194Dic163C4×D412yes—
195C2×C163—22no—
196C2×C163—41no—
197C2×C163C2×C4×D411yescharacteristic
198C2×C163C2×#13111yescharacteristic
199C2×C163#56811yescharacteristic
200D163—22no—
201D163—41no—
202D163—41no—
203C163#293311yescharacteristic, complemented, Sylow
204#2933—1631nomaximal, Sylow
205C2×#131—1631no—
206C22×M16—1631no—
207#525—1631no—
208#436—1631no—
209#436—1631no—
210#435—1631no—
211#435—1631no—
212#435—1631no—
213#435D16311yescharacteristic, complemented
214C8×D4—1631no—
215C8×D4—1631no—
216#424—1631no—
217#408—1631no—
218C2×C22⋊C8—1631no—
219#405—1631no—
220C2×#41—1631no—
221C23×C4—1631no—
222#131—3264no—
223C2×M16—3264no—
224C2×M16—1631no—
225C2×M16—1631no—
226C2×M16—1631no—
227C2×M16—1631no—
228C2×M16—1631no—
229C2×M16C2×D16311yescharacteristic, complemented
230C22×C8—1631no—
231C22×C8—1631no—
232C22×C8—1631no—
233C22×C8—1631no—
234C22×C8—1631no—
235C22×C8C2×D16311yescharacteristic
236#123—1631no—
237#123—1631no—
238C22⋊Q8—1631no—
239C22⋊Q8—1631no—
240#121—1631no—
241#121—1631no—
242#120—1631no—
243#120—1631no—
244C4×D4—1631no—
245C4×D4—1631no—
246C4×D4—1631no—
247C4×D4C2×D16311yescharacteristic
248#117—1631no—
249C4⋊C8—1631no—
250C4⋊C8—1631no—
251C4⋊C8—1631no—
252C4⋊C8C2×D16311yescharacteristic, complemented
253C22⋊C8—1631no—
254C22⋊C8—1631no—
255C22⋊C8—1631no—
256C22⋊C8—1631no—
257C22⋊C8—1631no—
258C22⋊C8—1631no—
259C22⋊C8C2×D16311yescharacteristic
260C22⋊C8C2×D16311yescharacteristic, complemented
261#97—1631no—
262#97C2×D16311yescharacteristic, complemented
263C4×C8—1631no—
264C4×C8—1631no—
265C24—1631no—
266#41—3262no—
267#41—3262no—
268C2×Q8—1631no—
269C2×D4—1631no—
270C2×D4—1631no—
271C2×D4—1631no—
272C2×D4—1631no—
273C2×D4—1631no—
274C2×D4—1631no—
275C2×D4C4×D16311yescharacteristic
276C22×C4—3262no—
277C22×C4—3262no—
278C22×C4—1631no—
279C22×C4—1631no—
280C22×C4—1631no—
281C22×C4—1631no—
282C22×C4—1631no—
283C22×C4—1631no—
284C22×C4C22×D16311yescharacteristic
285C22×C4C22×D16311yescharacteristic
286M16—22no—
287M16—3262no—
288M16—3262no—
289M16—3262no—
290M16—3262no—
291M16—1634no—
292C2×C8—22no—
293C2×C8—3262no—
294C2×C8—3262no—
295C2×C8—3262no—
296C2×C8—3262no—
297C2×C8—3262no—
298C2×C8—1631no—
299C2×C8—1631no—
300C2×C8—1631no—
301C2×C8—1631no—
302C2×C8C22×D16311yescharacteristic
303C2×C8C22×D16311yescharacteristic
304C2×C8C22×D16311yescharacteristic
305C2×C8C22×D16311yescharacteristic
306C4⋊C4—1631no—
307C4⋊C4—1631no—
308C4⋊C4—1631no—
309C4⋊C4—1631no—
310C4⋊C4—1631no—
311C4⋊C4C4×D16311yescharacteristic
312C22⋊C4—1631no—
313C22⋊C4—1631no—
314C22⋊C4—1631no—
315C22⋊C4—1631no—
316C22⋊C4—1631no—
317C22⋊C4—1631no—
318C22⋊C4—1631no—
319C22⋊C4—1631no—
320C22⋊C4C4×D16311yescharacteristic
321C22⋊C4C4×D16311yescharacteristic
322C42—1631no—
323C42C22×D16311yescharacteristic
324C23—3262no—
325C23—3262no—
326C23—6521no—
327C23—1631no—
328C23—1631no—
329C23—1631no—
330C23C2×C4×D16311yescharacteristic
331C23C2×C4×D16311yescharacteristic
332Q8—3262no—
333D4—22no—
334D4—3262no—
335D4—3262no—
336D4—3262no—
337D4—3262no—
338D4—6521no—
339D4—6521no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—3262no—
345C2×C4—3262no—
346C2×C4—3262no—
347C2×C4—3261no—
348C2×C4—3261no—
349C2×C4—3261no—
350C2×C4—3261no—
351C2×C4—1631no—
352C2×C4—1631no—
353C2×C4—1631no—
354C2×C4—1631no—
355C2×C4—1631no—
356C2×C4—1632no—
357C2×C4—1632no—
358C2×C4C2×C4×D16311yescharacteristic
359C2×C4C2×C4×D16311yescharacteristic
360C2×C4C2×C4×D16311yescharacteristic
361C2×C4C2×C4×D16311yescharacteristic
362C2×C4C23×D16311yescharacteristic
363C8—21no—
364C8—21no—
365C8—21no—
366C8—3261no—
367C8—3261no—
368C8—3261no—
369C8—1632no—
370C8D4×D16312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—3261no—
377C22—3261no—
378C22—3261no—
379C22—3261no—
380C22—3261no—
381C22—3261no—
382C22—3261no—
383C22—3261no—
384C22—6524no—
385C22—1632no—
386C22#1910474312yescomplemented
387C22C22×C4×D16311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—3261no—
393C4—3261no—
394C4—3261no—
395C4—3261no—
396C4—1632no—
397C4C2×D4×D16311yescharacteristic
398C4#41×D16311yescharacteristic
399C2—22no—
400C2—41no—
401C2—3262no—
402C2—6521no—
403C2—6521no—
404C2C4×D4×D16311yescharacteristic
405C2C2×#1910474311yescharacteristic
406C2#2378243011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D163
commutator, socleC22×C163C23×C4
Fitting#435×C163C2
prime radical#23782174C2
cosocleC2×C4×C163C24
hypercenter#435D163
hypocenterC163#2933
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#19104743C2semidirect
2. C22×#14579682C2semidirect
4. #23781742C2semidirect
5. #23781741C2semidirect
6. #23781738C2semidirect
7. #23781733C2semidirect
8. #23781733C2semidirect
9. D4×C163⋊C8C2semidirect
10. C8×C22⋊D163C2semidirect
11. #23781690C2semidirect
12. #23781630C2semidirect
13. C22⋊C8×D163C2semidirect
14. #23781622C2semidirect
15. #435×C163C2semidirect
24. C2×#14579682C22semidirect
26. C22×C163⋊C8C22semidirect
29. C2×C8×D163C22semidirect
42. C4⋊(C163⋊C8)C22semidirect
43. Dic163⋊C8C22semidirect
44. Dic163⋊C8C22semidirect
46. C22⋊(C163⋊C8)C22semidirect
49. (C2×D163)⋊C8C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
52. C4×C163⋊C8C22semidirect
53. C8×Dic163C22semidirect
54. C2×M16×C163C22semidirect
57. C4⋊C8×C163C22semidirect
59. C22⋊C8×C163C22semidirect
60. #97×C163C22semidirect
83. #14579682D4semidirect
172. C22×C163#131semidirect
203. C163#2933semidirect
213. #435D163semidirect
229. C2×M16C2×D163semidirect
252. C4⋊C8C2×D163semidirect
260. C22⋊C8C2×D163semidirect
262. #97C2×D163semidirect
386. C22#19104743semidirect

Maximal subgroups

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

subgroupindex
C2×#191047432
C22×#145796822
#237821742
#237817422
#237817412
#237817382
#237817332
D4×C163⋊C82
C8×C22⋊D1632
#237816902
#237816302
C22⋊C8×D1632
#237816222
#435×C1632
#2933163