Finite Group Databasebeta

← #35224025

Subgroups — #35224025

Order 20864.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19104764C211yesmaximal, characteristic, complemented
2C2×#19104759C211yesmaximal, characteristic
3#124×D163C211yesmaximal, characteristic, complemented
4C4×#14579698C211yesmaximal, characteristic, complemented
5#23781890C211yesmaximal, characteristic, complemented
6#23781888C211yesmaximal, characteristic, complemented
7#23781886C211yesmaximal, characteristic, complemented
8#23781884C211yesmaximal, characteristic, complemented
9#23781882C211yesmaximal, characteristic, complemented
10#23781880C211yesmaximal, characteristic, complemented
11#23781878C211yesmaximal, characteristic, complemented
12#23781687C211yesmaximal, characteristic, complemented
13#23781672C211yesmaximal, characteristic, complemented
14#23781668C211yesmaximal, characteristic, complemented
15#463×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19#19104764—24no—
20#19104759—24no—
21C2×Dic652C2211yescharacteristic, complemented
22C2×#14579689C2211yescharacteristic, complemented
23C2×#14579688C2211yescharacteristic
24C2×#14579687C2211yescharacteristic
25C2×#14579687C2211yescharacteristic, complemented
26C2×#14579684C2211yescharacteristic
27C2×#14579682C2211yescharacteristic
28C2×#14579682C2211yescharacteristic
29#19104724C2211yescharacteristic
30#19104723C2211yescharacteristic
31#19104722C2211yescharacteristic
32#19104721—21no—
33#19104721—21no—
34C4×Dic326C2211yescharacteristic, complemented
35#19104700C2211yescharacteristic, complemented
36D4×Dic163C2211yescharacteristic, complemented
37C4×C22⋊D163—21no—
38(C22⋊D163)⋊C4—21no—
39#19104693C2211yescharacteristic
40#19104691—21no—
41C22⋊C4×D163—21no—
42C22⋊C4×D163—21no—
43C2×C4×Dic163—21no—
44C42×D163C2211yescharacteristic
45C4⋊(C163⋊C8)C2211yescharacteristic, complemented
46Dic163⋊C8C2211yescharacteristic, complemented
47Dic163⋊C8C2211yescharacteristic, complemented
48#19104660C2211yescharacteristic, complemented
49#19104659C2211yescharacteristic, complemented
50#19104658C2211yescharacteristic, complemented
51#19104657C2211yescharacteristic
52#19104656C2211yescharacteristic
53#19104655C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic, complemented
55C2×D8×C163C2211yescharacteristic
56#124×C163C2211yescharacteristic
57C4×D4×C163C2211yescharacteristic, complemented
58C4⋊C8×C163C2211yescharacteristic, complemented
59#103×C163C2211yescharacteristic, complemented
60#102×C163C2211yescharacteristic
61C23×D163—21no—
62#14579698—22no—
63#14579698D414yes—
64C2×Dic326—21no—
65C2×Dic326C2311yescharacteristic
66C2×Dic326C2311yescharacteristic
67Q8×D163—22no—
68Q8×D163—22no—
69C2×D652C2311yescharacteristic
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72D4×D163—22no—
73D4×D163—22no—
74C22×Dic163—21no—
75C22×Dic163—21no—
76C2×C4×D163—21no—
77C2×C4×D163D411yescharacteristic
78C2×C4×D163D411yescharacteristic
79C2×C4×D163C2311yescharacteristic
80Dic652—22no—
81#14579689—22no—
82#14579688—22no—
83#14579687—22no—
84#14579687—22no—
85#14579684—22no—
86#14579682—22no—
87#14579682—22no—
88C2×C163⋊C8C2311yescharacteristic
89C2×C163⋊C8C2311yescharacteristic
90C4⋊Dic163C2311yescharacteristic
91Dic163⋊C4—21no—
92Dic163⋊C4—21no—
93C22⋊Dic163—21no—
94C22⋊Dic163—21no—
95C22⋊Dic163—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4—21no—
99(C2×D163)⋊C4—21no—
100(C2×D163)⋊C4—21no—
101C4×Dic163—21no—
102C4×Dic163—21no—
103C4×Dic163—21no—
104C4×Dic163D411yescharacteristic
105C4×Dic163D411yescharacteristic
106C4×Dic163C2311yescharacteristic
107C2×Q8×C163C2311yescharacteristic
108C2×D4×C163C2311yescharacteristic
109C2×D4×C163C2311yescharacteristic
110C22×C4×C163—21no—
111SD16×C163—22no—
112D8×C163—22no—
113C2×C8×C163C2311yescharacteristic
114C2×C8×C163C2311yescharacteristic
115C4⋊C4×C163C2311yescharacteristic
116C22⋊C4×C163—21no—
117C22⋊C4×C163—21no—
118C22⋊C4×C163—21no—
119C42×C163C2311yescharacteristic
120C22×D163—21no—
121C22×D163—42no—
122C22×D163—42no—
123C22×D163—42no—
124C22×D163C2×D411yescharacteristic
125Dic326—21no—
126Dic326—21no—
127Dic326—21no—
128Dic326—42no—
129Dic326C2×D412yes—
130D652—21no—
131D652—21no—
132C22⋊D163—24no—
133C22⋊D163—42no—
134C2×Dic163—21no—
135C2×Dic163—24no—
136C2×Dic163—21no—
137C2×Dic163—21no—
138C2×Dic163—42no—
139C2×Dic163C2×D411yescharacteristic
140C2×Dic163C2×D411yescharacteristic
141C2×Dic163C2×D411yescharacteristic
142C4×D163—41no—
143C4×D163—41no—
144C4×D163—42no—
145C4×D163C2×D412yes—
146C4×D163#4112yes—
147C163⋊C8—21no—
148C163⋊C8—21no—
149C23×C163—21no—
150C23×C163—21no—
151Q8×C163—21no—
152Q8×C163—21no—
153D4×C163—21no—
154D4×C163—21no—
155D4×C163—21no—
156D4×C163C2×D412yes—
157C2×C4×C163—21no—
158C2×C4×C163—21no—
159C2×C4×C163—42no—
160C2×C4×C163C2×D411yescharacteristic
161C2×C4×C163C2×D411yescharacteristic
162C2×C4×C163C2411yescharacteristic
163C8×C163—21no—
164C8×C163—21no—
165C22×C163—22no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163—41no—
170C22×C163—41no—
171C22×C163C22×D411yescharacteristic
172C2×D163—21no—
173C2×D163—21no—
174C2×D163—41no—
175C2×D163—41no—
176C2×D163—41no—
177C2×D163—84no—
178C2×D163#12112yes—
179C4×C163—21no—
180C4×C163—21no—
181C4×C163—41no—
182C4×C163—41no—
183C4×C163C22×D411yescharacteristic
184C4×C163C2×#4111yescharacteristic
185Dic163—22no—
186Dic163—21no—
187Dic163—21no—
188Dic163—41no—
189Dic163—41no—
190Dic163#12112yes—
191C2×C163—42no—
192C2×C163—81no—
193C2×C163C2×#12111yescharacteristic
194C2×C163#57511yescharacteristic
195C2×C163#57511yescharacteristic
196D163—22no—
197D163—81no—
198C163#304011yescharacteristic, complemented, Sylow
199#3040—1631nomaximal, Sylow
200C2×#137—1631no—
201C2×AGL(1,ℤ/8)—1631no—
202C2×#124—1631no—
203C4×#41—1631no—
204#464—1631no—
205#464—1631no—
206#464—1631no—
207#464—1631no—
208#463—1631no—
209#463—1631no—
210#463—1631no—
211#463D16311yescharacteristic, complemented
212#423—1631no—
213#419—1631no—
214#418—1631no—
215C2×#41—1631no—
216C22×Q8—1631no—
217C22×D4—1631no—
218#137—3264no—
219AGL(1,ℤ/8)—3264no—
220C2×Q16—1631no—
221C2×Q16—1631no—
222C2×SD16—1631no—
223C2×SD16—1631no—
224C2×SD16—1631no—
225C2×SD16C2×D16311yescharacteristic, complemented
226C2×D8—1631no—
227C2×D8C2×D16311yescharacteristic
228C2×M16—1631no—
229C2×M16—1631no—
230#124—3261no—
231#124—3261no—
232#124—1631no—
233#124—1631no—
234#124—1631no—
235#124C2×D16311yescharacteristic
236C4×Q8—1631no—
237C4×Q8—1631no—
238C4×D4—3261no—
239C4×D4—3261no—
240C4×D4—1631no—
241C4×D4C2×D16311yescharacteristic, complemented
242#117—3261no—
243#117—1631no—
244C2×C22⋊C4—3261no—
245C2×C22⋊C4—3261no—
246C2×C42—3261no—
247C2×C42—1631no—
248C4⋊C8—1631no—
249C4⋊C8—1631no—
250C4⋊C8—1631no—
251C4⋊C8C2×D16311yescharacteristic, complemented
252#103—1631no—
253#103—1631no—
254#103—1631no—
255#103C2×D16311yescharacteristic, complemented
256#102—1631no—
257#102—1631no—
258#102—1631no—
259#102C2×D16311yescharacteristic
260C24—3261no—
261#41—3262no—
262#41—1634no—
263C2×Q8—3261no—
264C2×Q8—3262no—
265C2×Q8—3262no—
266C2×Q8—1631no—
267C2×Q8—1631no—
268C2×Q8C22×D16311yescharacteristic
269C2×D4—3261no—
270C2×D4—3262no—
271C2×D4—3262no—
272C2×D4—3261no—
273C2×D4—1631no—
274C2×D4C22×D16311yescharacteristic
275C2×D4C22×D16311yescharacteristic
276C22×C4—21no—
277C22×C4—3261no—
278C22×C4—3261no—
279C22×C4—3261no—
280C22×C4—1631no—
281C22×C4—1631no—
282C22×C4—1631no—
283Q16—3262no—
284Q16—3262no—
285SD16—22no—
286SD16—3262no—
287SD16—3262no—
288SD16—3262no—
289D8—22no—
290D8—3262no—
291M16—3262no—
292M16—3262no—
293C2×C8—1631no—
294C2×C8—1631no—
295C2×C8C22×D16311yescharacteristic
296C2×C8C22×D16311yescharacteristic
297C4⋊C4—3261no—
298C4⋊C4—3261no—
299C4⋊C4—1631no—
300C4⋊C4C22×D16311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C22⋊C4—3261no—
305C22⋊C4—3261no—
306C22⋊C4—3261no—
307C22⋊C4—3261no—
308C22⋊C4—3261no—
309C22⋊C4—3261no—
310C22⋊C4—3261no—
311C22⋊C4—3261no—
312C42—3261no—
313C42—3261no—
314C42—3261no—
315C42—1631no—
316C42—1631no—
317C42—1631no—
318C42C22×D16311yescharacteristic
319C23—21no—
320C23—21no—
321C23—3261no—
322C23—6522no—
323C23—6522no—
324C23—6522no—
325C23—1631no—
326Q8—21no—
327Q8—21no—
328Q8—3261no—
329Q8—3261no—
330Q8—3261no—
331Q8—6522no—
332Q8—1632no—
333D4—21no—
334D4—21no—
335D4—21no—
336D4—3264no—
337D4—3261no—
338D4—3261no—
339D4—6522no—
340D4D4×D16312yes—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—3261no—
345C2×C4—3264no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—6521no—
349C2×C4—6521no—
350C2×C4—6522no—
351C2×C4—6522no—
352C2×C4—1631no—
353C2×C4—1631no—
354C2×C4—1631no—
355C2×C4—1632no—
356C2×C4—1632no—
357C2×C4D4×D16311yescharacteristic
358C2×C4D4×D16311yescharacteristic
359C2×C4C23×D16311yescharacteristic
360C8—21no—
361C8—21no—
362C8—3261no—
363C8—3261no—
364C22—22no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—13044no—
371C22—3261no—
372C22—3261no—
373C22—6521no—
374C22—6521no—
375C22—6521no—
376C22—1632no—
377C22C2×D4×D16311yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—3262no—
383C4—3261no—
384C4—3261no—
385C4—6521no—
386C4—6521no—
387C4—1632no—
388C4C2×D4×D16311yescharacteristic
389C4#41×D16311yescharacteristic
390C2—42no—
391C2—81no—
392C2—13041no—
393C2—3262no—
394C2#121×D16311yescharacteristic
395C2#2378246111yescharacteristic
396C2#2378246211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#463×C163C2
socleC22×C163C22×D4
prime radicalC2×#19104759C2
hypercenter#463D163
hypocenterC163#3040
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19104764C2semidirect
3. #124×D163C2semidirect
4. C4×#14579698C2semidirect
5. #23781890C2semidirect
6. #23781888C2semidirect
7. #23781886C2semidirect
8. #23781884C2semidirect
9. #23781882C2semidirect
10. #23781880C2semidirect
11. #23781878C2semidirect
12. #23781687C2semidirect
13. #23781672C2semidirect
14. #23781668C2semidirect
15. #463×C163C2semidirect
21. C2×Dic652C22semidirect
22. C2×#14579689C22semidirect
25. C2×#14579687C22semidirect
34. C4×Dic326C22semidirect
35. #19104700C22semidirect
36. D4×Dic163C22semidirect
45. C4⋊(C163⋊C8)C22semidirect
46. Dic163⋊C8C22semidirect
47. Dic163⋊C8C22semidirect
48. #19104660C22semidirect
49. #19104659C22semidirect
50. #19104658C22semidirect
54. C2×SD16×C163C22semidirect
57. C4×D4×C163C22semidirect
58. C4⋊C8×C163C22semidirect
59. #103×C163C22semidirect
198. C163#3040semidirect
211. #463D163semidirect
225. C2×SD16C2×D163semidirect
241. C4×D4C2×D163semidirect
251. C4⋊C8C2×D163semidirect
255. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
C2×#191047642
C2×#191047592
#124×D1632
C4×#145796982
#237818902
#237818882
#237818862
#237818842
#237818822
#237818802
#237818782
#237816872
#237816722
#237816682
#463×C1632
#3040163