Finite Group Databasebeta

← #35223236

Subgroups — #35223236

Order 20864.

492 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
2C2×M16×D163C211yesmaximal, characteristic, complemented
3#23782174C211yesmaximal, characteristic
4#23781744C211yesmaximal, characteristic, complemented
5#23781741C211yesmaximal, characteristic, complemented
6#23781741C211yesmaximal, characteristic, complemented
7#23781739C211yesmaximal, characteristic, complemented
8#23781737C211yesmaximal, characteristic, complemented
9#23781735C211yesmaximal, characteristic, complemented
10#23781733C211yesmaximal, characteristic, complemented
11#23781690C211yesmaximal, characteristic, complemented
12#23781630C211yesmaximal, characteristic, complemented
13#23781627C211yesmaximal, characteristic, complemented
14#23781619C211yesmaximal, characteristic, complemented
15#436×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×C4×D163C2211yescharacteristic
18#19104743—24no—
19C2×#14579683C2211yescharacteristic
20C2×#14579682C2211yescharacteristic
21C2×#14579682C2211yescharacteristic
22C2×#14579682C2211yescharacteristic
23C2×#14579682C2211yescharacteristic
24C2×#14579682C2211yescharacteristic, complemented
25M16×D163—24no—
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#19104646C2211yescharacteristic, complemented
52#19104645C2211yescharacteristic, complemented
53C8×Dic163C2211yescharacteristic, complemented
54C2×M16×C163C2211yescharacteristic
55C2×M16×C163C2211yescharacteristic, complemented
56C4×D4×C163C2211yescharacteristic
57C4⋊C8×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic
59C22⋊C8×C163C2211yescharacteristic, complemented
60C4×C8×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#14579683—22no—
80#14579682—22no—
81#14579682—22no—
82#14579682—22no—
83#14579682—22no—
84#14579682—22no—
85C2×C163⋊C8—22no—
86C2×C163⋊C8C2311yescharacteristic
87C2×C163⋊C8C2311yescharacteristic
88C2×C163⋊C8C2311yescharacteristic
89C2×C163⋊C8C2311yescharacteristic
90C8×D163—22no—
91C8×D163—22no—
92C8×D163D414yescomplemented
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—
111M16×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—41no—
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—41no—
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—41no—
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—21no—
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—41no—
196C2×C163—41no—
197C2×C163C2×C4×D411yescharacteristic
198C2×C163#56811yescharacteristic
199C2×C163#56811yescharacteristic
200D163—22no—
201D163—41no—
202D163—41no—
203C163#293011yescharacteristic, complemented, Sylow
204#2930—1631nomaximal, Sylow
205C2×#131—1631no—
206C22×M16—1631no—
207#525—1631no—
208#436—1631no—
209#436—1631no—
210#436—1631no—
211#436D16311yescharacteristic, complemented
212#435—1631no—
213#435—1631no—
214#435—1631no—
215#435—1631no—
216#424—1631no—
217#408—1631no—
218#407—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×M16—1631no—
230C2×M16C2×D16311yescharacteristic
231C2×M16C2×D16311yescharacteristic, complemented
232C22×C8—1631no—
233C22×C8—1631no—
234C22×C8—1631no—
235C22×C8—1631no—
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#97—1631no—
263C4×C8—1631no—
264C4×C8C2×D16311yescharacteristic, complemented
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—22no—
288M16—3262no—
289M16—3262no—
290M16—3262no—
291M16—3262no—
292M16—3262no—
293M16—3262no—
294C2×C8—3262no—
295C2×C8—3262no—
296C2×C8—3262no—
297C2×C8—1631no—
298C2×C8—1631no—
299C2×C8—1631no—
300C2×C8—1631no—
301C2×C8—1634no—
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—6521no—
326C23—6521no—
327C23—1631no—
328C23—1631no—
329C23—1631no—
330C23C2×C4×D16311yescharacteristic
331C23C2×C4×D16311yescharacteristic
332Q8—3262no—
333D4—41no—
334D4—3262no—
335D4—3262no—
336D4—3262no—
337D4—6521no—
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—21no—
377C22—3261no—
378C22—3261no—
379C22—3261no—
380C22—3261no—
381C22—3261no—
382C22—3261no—
383C22—3261no—
384C22—3261no—
385C22—6524no—
386C22—1632no—
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—41no—
400C2—41no—
401C2—3262no—
402C2—6521no—
403C2—6521no—
404C2C4×D4×D16311yescharacteristic
405C2#2378243011yescharacteristic
406C2#2378243111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D163
commutator, socleC22×C163C23×C4
Fitting#436×C163C2
prime radical#23782174C2
cosocleC2×C4×C163C24
hypercenter#436D163
hypocenterC163#2930
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#19104743C2semidirect
2. C2×M16×D163C2semidirect
4. #23781744C2semidirect
5. #23781741C2semidirect
6. #23781741C2semidirect
7. #23781739C2semidirect
8. #23781737C2semidirect
9. #23781735C2semidirect
10. #23781733C2semidirect
11. #23781690C2semidirect
12. #23781630C2semidirect
13. #23781627C2semidirect
14. #23781619C2semidirect
15. #436×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. #19104646C22semidirect
52. #19104645C22semidirect
53. C8×Dic163C22semidirect
55. C2×M16×C163C22semidirect
57. C4⋊C8×C163C22semidirect
59. C22⋊C8×C163C22semidirect
60. C4×C8×C163C22semidirect
92. C8×D163D4semidirect
203. C163#2930semidirect
211. #436D163semidirect
231. C2×M16C2×D163semidirect
252. C4⋊C8C2×D163semidirect
260. C22⋊C8C2×D163semidirect
264. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047432
C2×M16×D1632
#237821742
#237817442
#237817412
#237817392
#237817372
#237817352
#237817332
#237816902
#237816302
#237816272
#237816192
#436×C1632
#2930163