Finite Group Databasebeta

← #35226390

Subgroups — #35226390

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782367C211yesmaximal, characteristic, complemented
2#23782309C211yesmaximal, characteristic, complemented
3#23782018C211yesmaximal, characteristic, complemented
4#23782001C211yesmaximal, characteristic, complemented
5#23781987C211yesmaximal, characteristic, complemented
6#23781963C211yesmaximal, characteristic, complemented
7#23781898C211yesmaximal, characteristic, complemented
8#23781892C211yesmaximal, characteristic, complemented
9#23781878C211yesmaximal, characteristic, complemented
10#23781807C211yesmaximal, characteristic, complemented
11D8×Dic163C211yesmaximal, characteristic, complemented
12(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
13#23781658C211yesmaximal, characteristic, complemented
14C2×#19104656C211yesmaximal, characteristic, complemented
15#447×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579687C2211yescharacteristic, complemented
18C22×C163⋊C8C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic, complemented
20#19104728—21no—
21#19104726C2211yescharacteristic, complemented
22#19104722C2211yescharacteristic
23#19104719—21no—
24#19104716—21no—
25#19104714C2211yescharacteristic
26#19104707—21no—
27#19104707—21no—
28#19104706C2211yescharacteristic
29#19104706C2211yescharacteristic, complemented
30#19104704—21no—
31D4×Dic163—21no—
32D4×Dic163C2211yescharacteristic
33D4×Dic163C2211yescharacteristic, complemented
34C4×C22⋊D163C2211yescharacteristic
35#19104693C2211yescharacteristic, complemented
36C2×C4⋊Dic163C2211yescharacteristic
37C2×C22⋊Dic163—21no—
38#19104669C2211yescharacteristic, complemented
39#19104669C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104658C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C2×D8×C163C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—41no—
61C2×Dic326C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163D411yescharacteristic
66C22×Dic163—21no—
67C22×Dic163—21no—
68C22×Dic163—21no—
69C22×Dic163—21no—
70C2×C4×D163C2311yescharacteristic
71#14579687—41no—
72C2×C163⋊C8—41no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C8×D163—41no—
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4D411yescharacteristic
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163D411yescharacteristic
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96C24×C163—21no—
97C2×D4×C163—22no—
98C2×D4×C163—21no—
99C2×D4×C163—21no—
100C2×D4×C163—41no—
101C2×D4×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C2×D4×C163C2311yescharacteristic
104C22×C4×C163C2311yescharacteristic
105D8×C163—22no—
106D8×C163—41no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114C22⋊D163—22no—
115C22⋊D163—42no—
116C22⋊D163—81no—
117C22⋊D163—81no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—42no—
123C2×Dic163—81no—
124C2×Dic163—81no—
125C2×Dic163—81no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—21no—
130C4×D163—21no—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C23×C163—21no—
134C23×C163—21no—
135C23×C163—21no—
136C23×C163—42no—
137C23×C163—42no—
138C23×C163—81no—
139C23×C163C2×D411yescharacteristic
140D4×C163—21no—
141D4×C163—21no—
142D4×C163—21no—
143D4×C163—21no—
144D4×C163—21no—
145D4×C163—42no—
146D4×C163—42no—
147D4×C163#4112yes—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—22no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—84no—
167C22×C163#13512yes—
168C22×C163C22×D411yescharacteristic
169C2×D163—21no—
170C2×D163—21no—
171C2×D163—21no—
172C4×C163—21no—
173C4×C163—41no—
174C4×C163C2×#4111yescharacteristic
175C4×C163H(5,𝔽2)11yescharacteristic
176Dic163—21no—
177Dic163—21no—
178Dic163—21no—
179Dic163—41no—
180Dic163—41no—
181Dic163—41no—
182Dic163—41no—
183C2×C163—22no—
184C2×C163—42no—
185C2×C163—81no—
186C2×C163—81no—
187C2×C163#54611yescharacteristic
188C2×C163C2×#13511yescharacteristic
189C2×C163#57611yescharacteristic
190D163—41no—
191C163#329711yescharacteristic, complemented, Sylow
192#3297—1631nomaximal, Sylow
193#553—1631no—
194#546—1631no—
195#488—1631no—
196#484—1631no—
197#482—1631no—
198#478—1631no—
199#466—1631no—
200#465—1631no—
201#463—1631no—
202#449—1631no—
203#447D16311yescharacteristic, complemented
204C4×D8—1631no—
205C8×D4—1631no—
206#416—1631no—
207C2×#102—1631no—
208C2×#41—1631no—
209C22×D4C2×D16311yescharacteristic
210C2×SD16—1631no—
211C2×D8C2×D16311yescharacteristic
212C2×D8C2×D16311yescharacteristic
213C22×C8—1631no—
214C22×C8—1631no—
215#126—3261no—
216#125—1631no—
217#124—1631no—
218#123—3261no—
219#123—3261no—
220C22⋊Q8—1631no—
221#121—3261no—
222#121—3261no—
223#121—1631no—
224#121—1631no—
225#121C2×D16311yescharacteristic
226#120—3261no—
227C4×D4—3261no—
228C4×D4—1631no—
229C4×D4—1631no—
230C4×D4—1631no—
231#117—1631no—
232C2×C4⋊C4—1631no—
233C2×C22⋊C4—3261no—
234#107—1631no—
235#107—1631no—
236#106—1631no—
237C4⋊C8—1631no—
238#103—1631no—
239#103—1631no—
240#103—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244#102—1631no—
245#102—1631no—
246#102C2×D16311yescharacteristic
247#102C2×D16311yescharacteristic
248C22⋊C8—1631no—
249C22⋊C8C2×D16311yescharacteristic
250C4×C8—1631no—
251C24—21no—
252#41—3262no—
253#41—6521no—
254C2×Q8—1631no—
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—3261no—
260C2×D4—3261no—
261C2×D4—3261no—
262C2×D4—1631no—
263C2×D4C22×D16311yescharacteristic
264C2×D4C22×D16311yescharacteristic
265C2×D4C22×D16311yescharacteristic
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4—3261no—
269C22×C4—3261no—
270C22×C4—1631no—
271C22×C4C22×D16311yescharacteristic
272SD16—6521no—
273D8—22no—
274D8—41no—
275C2×C8—6521no—
276C2×C8—6521no—
277C2×C8—1631no—
278C2×C8—1631no—
279C2×C8C22×D16311yescharacteristic
280C2×C8C22×D16311yescharacteristic
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—3261no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4—1631no—
288C4⋊C4—1631no—
289C4⋊C4C22×D16311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4—3261no—
297C22⋊C4—3261no—
298C22⋊C4—3261no—
299C22⋊C4—1631no—
300C22⋊C4—1631no—
301C42—3261no—
302C42—1631no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—42no—
307C23—42no—
308C23—81no—
309C23—1631no—
310C23D4×D16311yescharacteristic
311Q8—3261no—
312Q8—3261no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—42no—
319D4—42no—
320D4—13041no—
321D4—13041no—
322D4—3262no—
323D4—6522no—
324D4#1457969812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—13041no—
329C2×C4—13041no—
330C2×C4—13041no—
331C2×C4—3261no—
332C2×C4—3261no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—6522no—
338C2×C4—1631no—
339C2×C4—1631no—
340C2×C4—1631no—
341C2×C4D4×D16311yescharacteristic
342C2×C4C23×D16311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3261no—
346C8—3261no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—3261no—
360C22—3261no—
361C22—3261no—
362C22#1910475412yes—
363C22C2×D4×D16311yescharacteristic
364C4—21no—
365C4—41no—
366C4—3261no—
367C4—3261no—
368C4—3261no—
369C4—6521no—
370C4—6521no—
371C4—6521no—
372C4—6521no—
373C4C2×#1457969811yescharacteristic
374C4#1910478011yescharacteristic
375C2—22no—
376C2—42no—
377C2—81no—
378C2—81no—
379C2—6521no—
380C2#2378230211yescharacteristic
381C2C2×#1910475411yescharacteristic
382C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#447×C163C2
socleC22×C163C22×D4
hypercenter#447D163
hypocenterC163#3297
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23782367C2semidirect
2. #23782309C2semidirect
3. #23782018C2semidirect
4. #23782001C2semidirect
5. #23781987C2semidirect
6. #23781963C2semidirect
7. #23781898C2semidirect
8. #23781892C2semidirect
9. #23781878C2semidirect
10. #23781807C2semidirect
11. D8×Dic163C2semidirect
12. (C22⋊D163)⋊C8C2semidirect
13. #23781658C2semidirect
14. C2×#19104656C2semidirect
15. #447×C163C2semidirect
17. C2×#14579687C22semidirect
19. C2×C8×D163C22semidirect
21. #19104726C22semidirect
29. #19104706C22semidirect
33. D4×Dic163C22semidirect
35. #19104693C22semidirect
38. #19104669C22semidirect
39. #19104669C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104660C22semidirect
43. #19104660C22semidirect
44. #19104658C22semidirect
49. #19104656C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
191. C163#3297semidirect
203. #447D163semidirect

Maximal subgroups

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

subgroupindex
#237823672
#237823092
#237820182
#237820012
#237819872
#237819632
#237818982
#237818922
#237818782
#237818072
D8×Dic1632
(C22⋊D163)⋊C82
#237816582
C2×#191046562
#447×C1632
#3297163