Finite Group Databasebeta

← #35223269

Subgroups — #35223269

Order 20864.

544 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579682C211yesmaximal, characteristic, complemented
2C2×M16×D163C211yesmaximal, characteristic, complemented
3C4×D4×D163C211yesmaximal, characteristic
4#23781741C211yesmaximal, characteristic, complemented
5#23781740C211yesmaximal, characteristic, complemented
6#23781738C211yesmaximal, characteristic, complemented
7#23781737C211yesmaximal, characteristic, complemented
8#23781735C211yesmaximal, characteristic, complemented
9C8×C22⋊D163C211yesmaximal, characteristic, complemented
10(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
11#23781687C211yesmaximal, characteristic, complemented
12#23781627C211yesmaximal, characteristic, complemented
13#23781627C211yesmaximal, characteristic, complemented
14#23781617C211yesmaximal, characteristic, complemented
15#435×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C411yescharacteristic
17C22×C4×D163C2211yescharacteristic
18C22×C4×D163C2211yescharacteristic
19C2×#14579683C2211yescharacteristic, complemented
20C2×#14579682—24no—
21C2×#14579682C2211yescharacteristic
22C2×#14579682C2211yescharacteristic
23C2×#14579682C2211yescharacteristic
24C2×#14579682C2211yescharacteristic
25C2×#14579682C2211yescharacteristic, complemented
26C2×#14579682C2211yescharacteristic, complemented
27M16×D163—24no—
28C22×C163⋊C8C2211yescharacteristic, complemented
29C2×C8×D163C2211yescharacteristic, complemented
30C2×C8×D163C2211yescharacteristic, complemented
31C4×D652C2211yescharacteristic
32D4×Dic163C411yescharacteristic
33#19104697C411yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35C4×C22⋊D163C2211yescharacteristic
36(C22⋊D163)⋊C4C411yescharacteristic
37(C22⋊D163)⋊C4C411yescharacteristic
38C4⋊C4×D163C411yescharacteristic
39C22⋊C4×D163C411yescharacteristic
40C22⋊C4×D163C411yescharacteristic
41C42×D163C2211yescharacteristic
42C4⋊(C163⋊C8)C2211yescharacteristic, complemented
43Dic163⋊C8C2211yescharacteristic, complemented
44Dic163⋊C8C2211yescharacteristic, complemented
45C22⋊(C163⋊C8)C2211yescharacteristic, complemented
46C22⋊(C163⋊C8)C2211yescharacteristic, complemented
47(C2×D163)⋊C8C2211yescharacteristic, complemented
48(C2×D163)⋊C8C2211yescharacteristic, complemented
49(C2×D163)⋊C8C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104646C2211yescharacteristic, complemented
52C8×Dic163C2211yescharacteristic, complemented
53C8×Dic163C2211yescharacteristic, complemented
54C2×M16×C163C2211yescharacteristic, complemented
55C22×C8×C163C2211yescharacteristic, complemented
56C4×D4×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C22⋊C8×C163C2211yescharacteristic, complemented
60#97×C163C2211yescharacteristic, complemented
61C23×D163C2×C411yescharacteristic
62C23×D163C2×C411yescharacteristic
63C2×D652C2×C411yescharacteristic
64C2×C22⋊D163C2×C411yescharacteristic
65C2×C22⋊D163C2×C411yescharacteristic
66D4×D163—24no—
67C22×Dic163C2×C411yescharacteristic
68C22×Dic163C2×C411yescharacteristic
69C2×C4×D163—22no—
70C2×C4×D163—22no—
71C2×C4×D163—22no—
72C2×C4×D163—22no—
73C2×C4×D163C2×C411yescharacteristic
74C2×C4×D163C2×C411yescharacteristic
75C2×C4×D163C2×C411yescharacteristic
76C2×C4×D163C2×C411yescharacteristic
77C2×C4×D163C2×C411yescharacteristic
78C2×C4×D163C2×C411yescharacteristic
79C2×C4×D163C2311yescharacteristic
80C2×C4×D163C2311yescharacteristic
81C2×C4×D163C2311yescharacteristic
82#14579683—22no—
83#14579682—22no—
84#14579682—22no—
85#14579682—22no—
86#14579682—22no—
87#14579682—22no—
88#14579682D414yescomplemented
89C2×C163⋊C8—22no—
90C2×C163⋊C8C2311yescharacteristic
91C2×C163⋊C8C2311yescharacteristic
92C2×C163⋊C8C2311yescharacteristic
93C2×C163⋊C8C2311yescharacteristic
94C8×D163—22no—
95C8×D163—22no—
96C4⋊Dic163C2×C411yescharacteristic
97Dic163⋊C4C2×C411yescharacteristic
98Dic163⋊C4C2×C411yescharacteristic
99C22⋊Dic163C2×C411yescharacteristic
100C22⋊Dic163C2×C411yescharacteristic
101(C2×D163)⋊C4C2×C411yescharacteristic
102(C2×D163)⋊C4C2×C411yescharacteristic
103(C2×D163)⋊C4C2×C411yescharacteristic
104(C2×D163)⋊C4C2×C411yescharacteristic
105C4×Dic163C2×C411yescharacteristic
106C4×Dic163C2×C411yescharacteristic
107C4×Dic163C2311yescharacteristic
108C2×D4×C163C2×C411yescharacteristic
109C22×C4×C163C2311yescharacteristic
110C22×C4×C163C2311yescharacteristic
111M16×C163—22no—
112C2×C8×C163—22no—
113C2×C8×C163C2311yescharacteristic
114C2×C8×C163C2311yescharacteristic
115C2×C8×C163C2311yescharacteristic
116C2×C8×C163C2311yescharacteristic
117C4⋊C4×C163C2×C411yescharacteristic
118C22⋊C4×C163C2×C411yescharacteristic
119C22⋊C4×C163C2×C411yescharacteristic
120C42×C163C2311yescharacteristic
121C22×D163—22no—
122C22×D163—22no—
123C22×D163—22no—
124C22×D163—22no—
125C22×D163—41no—
126C22×D163—41no—
127C22×D163C22×C411yescharacteristic
128C22×D163C22×C411yescharacteristic
129C22×D163C22×C411yescharacteristic
130D652—22no—
131C22⋊D163—22no—
132C22⋊D163M1614yescomplemented
133C2×Dic163—22no—
134C2×Dic163—22no—
135C2×Dic163C22×C411yescharacteristic
136C2×Dic163C22×C411yescharacteristic
137C2×Dic163C22×C411yescharacteristic
138C2×Dic163C22×C411yescharacteristic
139C2×Dic163C22×C411yescharacteristic
140C4×D163—22no—
141C4×D163—22no—
142C4×D163—21no—
143C4×D163—21no—
144C4×D163—21no—
145C4×D163—21no—
146C4×D163—22no—
147C4×D163—22no—
148C4×D163—22no—
149C4×D163—41no—
150C4×D163C2×D412yes—
151C4×D163#4112yes—
152C163⋊C8—21no—
153C163⋊C8—21no—
154C163⋊C8—21no—
155C163⋊C8C2×D412yescomplemented
156C23×C163C22×C411yescharacteristic
157C23×C163C22×C411yescharacteristic
158D4×C163—22no—
159C2×C4×C163—21no—
160C2×C4×C163—21no—
161C2×C4×C163—21no—
162C2×C4×C163—21no—
163C2×C4×C163C22×C411yescharacteristic
164C2×C4×C163C22×C411yescharacteristic
165C2×C4×C163C22×C411yescharacteristic
166C2×C4×C163C22×C411yescharacteristic
167C2×C4×C163C2411yescharacteristic
168C8×C163—21no—
169C8×C163—21no—
170C8×C163—21no—
171C8×C163C2×D412yescomplemented
172C22×C163—21no—
173C22×C163—21no—
174C22×C163—21no—
175C22×C163—21no—
176C22×C163—21no—
177C22×C163C2×M1612yescomplemented
178C22×C163C23×C411yescharacteristic
179C2×D163—21no—
180C2×D163—21no—
181C2×D163—21no—
182C2×D163—21no—
183C2×D163—21no—
184C2×D163—21no—
185C2×D163—21no—
186C2×D163—44no—
187C2×D163—44no—
188C2×D163C4×D412yes—
189C2×D163C2×M1612yescomplemented
190C4×C163—21no—
191C4×C163—21no—
192C4×C163—21no—
193C4×C163—21no—
194C4×C163C22×D411yescharacteristic
195C4×C163C2×#4111yescharacteristic
196Dic163—21no—
197Dic163—21no—
198Dic163—21no—
199Dic163C4×D412yes—
200Dic163C2×M1612yescomplemented
201C2×C163—22no—
202C2×C163—41no—
203C2×C163C2×C4×D411yescharacteristic
204C2×C163C22×M1611yescharacteristic
205C2×C163#56811yescharacteristic
206D163—22no—
207D163—22no—
208D163—41no—
209C163D4×M1611yescharacteristic, complemented, Sylow
210D4×M16—1631nomaximal, Sylow
211C22×M16—1631no—
212C22×M16—1631no—
213C2×C4×D4—1631no—
214#436—1631no—
215#436—1631no—
216#435—1631no—
217#435—1631no—
218#435—1631no—
219#435D16311yescharacteristic, complemented
220C8×D4—1631no—
221C8×D4—1631no—
222#423—1631no—
223#407—1631no—
224#407—1631no—
225C4×M16—1631no—
226C22×D4—1631no—
227C23×C4—1631no—
228C23×C4—1631no—
229C2×M16—3264no—
230C2×M16—3264no—
231C2×M16—1631no—
232C2×M16—1631no—
233C2×M16—1631no—
234C2×M16—1631no—
235C2×M16—1631no—
236C2×M16—1631no—
237C2×M16—1631no—
238C2×M16C2×D16311yescharacteristic, complemented
239C22×C8—1631no—
240C22×C8—1631no—
241C22×C8—1631no—
242C22×C8C2×D16311yescharacteristic, complemented
243C4×D4—1631no—
244C4×D4—1631no—
245C4×D4—1631no—
246C4×D4—1631no—
247C4×D4—1631no—
248C4×D4—1631no—
249C4×D4—1631no—
250C4×D4C2×D16311yescharacteristic
251C2×C4⋊C4—1631no—
252C2×C22⋊C4—1631no—
253C2×C22⋊C4—1631no—
254C2×C42—1631no—
255C4⋊C8—1631no—
256C4⋊C8—1631no—
257C4⋊C8—1631no—
258C4⋊C8C2×D16311yescharacteristic, complemented
259C22⋊C8—1631no—
260C22⋊C8—1631no—
261C22⋊C8—1631no—
262C22⋊C8—1631no—
263C22⋊C8—1631no—
264C22⋊C8—1631no—
265C22⋊C8C2×D16311yescharacteristic, complemented
266C22⋊C8C2×D16311yescharacteristic, complemented
267#97—1631no—
268#97C2×D16311yescharacteristic, complemented
269C4×C8—1631no—
270C4×C8—1631no—
271C24—1631no—
272C24—1631no—
273C2×D4—3264no—
274C2×D4—1631no—
275C2×D4—1631no—
276C2×D4—1631no—
277C2×D4C4×D16311yescharacteristic
278C22×C4—3262no—
279C22×C4—3262no—
280C22×C4—3262no—
281C22×C4—3262no—
282C22×C4—1631no—
283C22×C4—1631no—
284C22×C4—1631no—
285C22×C4—1631no—
286C22×C4—1631no—
287C22×C4—1631no—
288C22×C4—1631no—
289C22×C4—1631no—
290C22×C4—1631no—
291C22×C4—1631no—
292C22×C4—1631no—
293C22×C4C22×D16311yescharacteristic
294C22×C4C22×D16311yescharacteristic
295M16—22no—
296M16—3262no—
297M16—3262no—
298M16—3262no—
299M16—3262no—
300M16—3262no—
301M16—3262no—
302M16—1634no—
303C2×C8—22no—
304C2×C8—3262no—
305C2×C8—3262no—
306C2×C8—3262no—
307C2×C8—1631no—
308C2×C8—1631no—
309C2×C8—1631no—
310C2×C8—1631no—
311C2×C8C22×D16311yescharacteristic
312C2×C8C22×D16311yescharacteristic
313C2×C8C22×D16311yescharacteristic
314C2×C8C22×D16311yescharacteristic
315C4⋊C4—1631no—
316C4⋊C4—1631no—
317C4⋊C4—1631no—
318C4⋊C4C4×D16311yescharacteristic
319C22⋊C4—1631no—
320C22⋊C4—1631no—
321C22⋊C4—1631no—
322C22⋊C4—1631no—
323C22⋊C4—1631no—
324C22⋊C4—1631no—
325C22⋊C4C4×D16311yescharacteristic
326C22⋊C4C4×D16311yescharacteristic
327C42—1631no—
328C42—1631no—
329C42—1631no—
330C42C22×D16311yescharacteristic
331C23—3262no—
332C23—3262no—
333C23—3262no—
334C23—3262no—
335C23—6521no—
336C23—6521no—
337C23—1631no—
338C23—1631no—
339C23—1631no—
340C23C2×C4×D16311yescharacteristic
341C23C2×C4×D16311yescharacteristic
342D4—22no—
343D4—3262no—
344D4—3262no—
345D4—1634no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—3262no—
351C2×C4—3262no—
352C2×C4—3262no—
353C2×C4—3262no—
354C2×C4—3262no—
355C2×C4—3262no—
356C2×C4—3262no—
357C2×C4—3261no—
358C2×C4—3261no—
359C2×C4—3261no—
360C2×C4—3261no—
361C2×C4—6521no—
362C2×C4—1631no—
363C2×C4—1631no—
364C2×C4—1631no—
365C2×C4—1631no—
366C2×C4—1631no—
367C2×C4—1632no—
368C2×C4—1632no—
369C2×C4C2×C4×D16311yescharacteristic
370C2×C4C2×C4×D16311yescharacteristic
371C2×C4C2×C4×D16311yescharacteristic
372C2×C4C2×C4×D16311yescharacteristic
373C2×C4C23×D16311yescharacteristic
374C8—21no—
375C8—21no—
376C8—21no—
377C8—3261no—
378C8—3261no—
379C8—3261no—
380C8—1632no—
381C8D4×D16312yescomplemented
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—3261no—
388C22—3261no—
389C22—3261no—
390C22—3261no—
391C22—3261no—
392C22—3261no—
393C22—3261no—
394C22—6524no—
395C22—6524no—
396C22—1632no—
397C22—1632no—
398C22M16×D16312yescomplemented
399C22C22×C4×D16311yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—3261no—
405C4—3261no—
406C4—3261no—
407C4—1632no—
408C4—1632no—
409C4C2×D4×D16311yescharacteristic
410C4#41×D16311yescharacteristic
411C2—22no—
412C2—41no—
413C2—3262no—
414C2—3262no—
415C2—6521no—
416C2C4×D4×D16311yescharacteristic
417C2C2×M16×D16311yescharacteristic
418C2#2378243011yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D163
commutator, socleC22×C163C23×C4
Fitting#435×C163C2
prime radicalC2×D4×D163C4
cosocleC2×C4×C163C24
hypercenter#435D163
hypocenterC163D4×M16
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C22×#14579682C2semidirect
2. C2×M16×D163C2semidirect
4. #23781741C2semidirect
5. #23781740C2semidirect
6. #23781738C2semidirect
7. #23781737C2semidirect
8. #23781735C2semidirect
9. C8×C22⋊D163C2semidirect
10. (C22⋊D163)⋊C8C2semidirect
11. #23781687C2semidirect
12. #23781627C2semidirect
13. #23781627C2semidirect
14. #23781617C2semidirect
15. #435×C163C2semidirect
19. C2×#14579683C22semidirect
25. C2×#14579682C22semidirect
26. C2×#14579682C22semidirect
28. C22×C163⋊C8C22semidirect
29. C2×C8×D163C22semidirect
30. C2×C8×D163C22semidirect
42. C4⋊(C163⋊C8)C22semidirect
43. Dic163⋊C8C22semidirect
44. Dic163⋊C8C22semidirect
45. C22⋊(C163⋊C8)C22semidirect
46. C22⋊(C163⋊C8)C22semidirect
47. (C2×D163)⋊C8C22semidirect
48. (C2×D163)⋊C8C22semidirect
49. (C2×D163)⋊C8C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104646C22semidirect
52. C8×Dic163C22semidirect
53. C8×Dic163C22semidirect
54. C2×M16×C163C22semidirect
55. C22×C8×C163C22semidirect
57. C4⋊C8×C163C22semidirect
58. C22⋊C8×C163C22semidirect
59. C22⋊C8×C163C22semidirect
60. #97×C163C22semidirect
88. #14579682D4semidirect
132. C22⋊D163M16semidirect
155. C163⋊C8C2×D4semidirect
171. C8×C163C2×D4semidirect
177. C22×C163C2×M16semidirect
189. C2×D163C2×M16semidirect
200. Dic163C2×M16semidirect
209. C163D4×M16semidirect
219. #435D163semidirect
238. C2×M16C2×D163semidirect
242. C22×C8C2×D163semidirect
258. C4⋊C8C2×D163semidirect
265. C22⋊C8C2×D163semidirect
266. C22⋊C8C2×D163semidirect
268. #97C2×D163semidirect
381. C8D4×D163semidirect
398. C22M16×D163semidirect

Maximal subgroups

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

subgroupindex
C22×#145796822
C2×M16×D1632
C4×D4×D1632
#237817412
#237817402
#237817382
#237817372
#237817352
C8×C22⋊D1632
(C22⋊D163)⋊C82
#237816872
#237816272
#237816172
#435×C1632
D4×M16163