Finite Group Databasebeta

← #35226613

Subgroups — #35226613

Order 20864.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782370C211yesmaximal, characteristic, complemented
2#23782309C211yesmaximal, characteristic
3#23782023C211yesmaximal, characteristic, complemented
4#23782000C211yesmaximal, characteristic, complemented
5#23781977C211yesmaximal, characteristic, complemented
6#23781943C211yesmaximal, characteristic, complemented
7#23781910C211yesmaximal, characteristic, complemented
8#23781906C211yesmaximal, characteristic, complemented
9#23781886C211yesmaximal, characteristic, complemented
10#23781828C211yesmaximal, characteristic, complemented
11#23781816C211yesmaximal, characteristic, complemented
12#23781762C211yesmaximal, characteristic, complemented
13#23781736C211yesmaximal, characteristic, complemented
14#23781663C211yesmaximal, characteristic, complemented
15C2×#102×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×Dic652C2211yescharacteristic, complemented
18C2×#14579687C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579682C2211yescharacteristic, complemented
21#19104734C2211yescharacteristic, complemented
22#19104725—21no—
23#19104722C2211yescharacteristic
24#19104716—21no—
25#19104715C2211yescharacteristic, complemented
26#19104714—21no—
27#19104714C2211yescharacteristic
28#19104714C2211yescharacteristic
29#19104713—21no—
30#19104707—21no—
31#19104706C2211yescharacteristic
32#19104704—21no—
33#19104700C2211yescharacteristic, complemented
34D4×Dic163C2211yescharacteristic
35C4×C22⋊D163—21no—
36C4×C22⋊D163C2211yescharacteristic
37C4⋊C4×D163C2211yescharacteristic, complemented
38C2×C22⋊Dic163—21no—
39#19104669C2211yescharacteristic, complemented
40#19104668C2211yescharacteristic, complemented
41#19104668C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104658C2211yescharacteristic, complemented
46#19104658C2211yescharacteristic, complemented
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C2×C4⋊C4×C163C2211yescharacteristic
55#102×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—41no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63C2×Dic326C2311yescharacteristic
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163D411yescharacteristic
67C22×Dic163—21no—
68C22×Dic163—21no—
69C2×C4×D163—21no—
70C2×C4×D163C2311yescharacteristic
71Dic652—41no—
72#14579687—22no—
73#14579687—41no—
74#14579682—41no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4D411yescharacteristic
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92C22⋊Dic163D411yescharacteristic
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4D411yescharacteristic
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C24×C163—21no—
98C2×D4×C163—22no—
99C2×D4×C163—21no—
100C2×D4×C163—41no—
101C2×D4×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163C2311yescharacteristic
105C2×C8×C163—41no—
106C2×C8×C163C2311yescharacteristic
107C2×C8×C163C2311yescharacteristic
108C4⋊C4×C163—21no—
109C4⋊C4×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—81no—
117C22⋊D163—22no—
118C22⋊D163—42no—
119C22⋊D163—81no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—42no—
125C2×Dic163—81no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—21no—
130C4×D163—21no—
131C4×D163—81no—
132C163⋊C8—21no—
133C163⋊C8—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—42no—
144D4×C163#4112yes—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—81no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C8×C163—21no—
153C8×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—22no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—84no—
163C22×C163#13712yes—
164C22×C163C22×D411yescharacteristic
165C2×D163—21no—
166C2×D163—21no—
167C2×D163—21no—
168C4×C163—21no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163C2×#4111yescharacteristic
172C4×C163H(5,𝔽2)11yescharacteristic
173Dic163—21no—
174Dic163—21no—
175Dic163—21no—
176Dic163—41no—
177Dic163—41no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—22no—
181C2×C163—42no—
182C2×C163—81no—
183C2×C163#54611yescharacteristic
184C2×C163C2×#13711yescharacteristic
185C2×C163#57611yescharacteristic
186D163—41no—
187C163#331411yescharacteristic, complemented, Sylow
188#3314—1631nomaximal, Sylow
189#554—1631no—
190#546—1631no—
191#489—1631no—
192#484—1631no—
193#480—1631no—
194#474—1631no—
195#468—1631no—
196#467—1631no—
197#464—1631no—
198#452—1631no—
199#450—1631no—
200#440—1631no—
201#435—1631no—
202#417—1631no—
203C2×#102D16311yescharacteristic, complemented
204C2×#41—1631no—
205C22×D4C2×D16311yescharacteristic
206C2×Q16—1631no—
207C2×SD16—1631no—
208C2×SD16—1631no—
209C2×M16—1631no—
210C22×C8C2×D16311yescharacteristic
211C4⋊Q8—1631no—
212#125—3261no—
213#124—1631no—
214#123—3261no—
215C22⋊Q8—3261no—
216C22⋊Q8—3261no—
217C22⋊Q8—1631no—
218C22⋊Q8—1631no—
219C22⋊Q8—1631no—
220#121—3261no—
221#121—1631no—
222#120—3261no—
223C4×Q8—1631no—
224C4×D4—3261no—
225C4×D4—1631no—
226C4×D4—1631no—
227C2×C4⋊C4—1631no—
228C2×C4⋊C4C2×D16311yescharacteristic
229C2×C22⋊C4—3261no—
230#107—1631no—
231#107—1631no—
232#107—1631no—
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#103—1631no—
238#102—1631no—
239#102—1631no—
240#102C2×D16311yescharacteristic
241#102C2×D16311yescharacteristic
242#102C2×D16311yescharacteristic
243#102C2×D16311yescharacteristic
244C22⋊C8—1631no—
245C22⋊C8—1631no—
246#97—1631no—
247C24—21no—
248#41—3262no—
249#41—6521no—
250C2×Q8—3261no—
251C2×Q8—1631no—
252C2×Q8—1631no—
253C2×D4—22no—
254C2×D4—21no—
255C2×D4—41no—
256C2×D4—3261no—
257C2×D4—3261no—
258C2×D4—1631no—
259C2×D4C22×D16311yescharacteristic
260C2×D4C22×D16311yescharacteristic
261C22×C4—21no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—1631no—
266C22×C4C22×D16311yescharacteristic
267Q16—6521no—
268SD16—3262no—
269SD16—6521no—
270M16—6521no—
271C2×C8—41no—
272C2×C8—1631no—
273C2×C8—1631no—
274C2×C8C22×D16311yescharacteristic
275C2×C8C22×D16311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3261no—
278C4⋊C4—3261no—
279C4⋊C4—3261no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—1631no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4C22×D16311yescharacteristic
287C4⋊C4C22×D16311yescharacteristic
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—1631no—
296C22⋊C4—1631no—
297C42—3261no—
298C42—1631no—
299C23—21no—
300C23—21no—
301C23—42no—
302C23—42no—
303C23—81no—
304C23—1631no—
305C23D4×D16311yescharacteristic
306Q8—13041no—
307Q8—3261no—
308Q8—3261no—
309Q8—3261no—
310Q8—3261no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—13041no—
316D4—3262no—
317D4—6522no—
318D4#1457969912yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—13041no—
325C2×C4—13041no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—6522no—
333C2×C4—1631no—
334C2×C4—1631no—
335C2×C4—1631no—
336C2×C4D4×D16311yescharacteristic
337C2×C4C23×D16311yescharacteristic
338C8—21no—
339C8—21no—
340C8—3261no—
341C8—3261no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—84no—
351C22—3261no—
352C22—3261no—
353C22—3261no—
354C22#1910476412yes—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—3261no—
360C4—3261no—
361C4—3261no—
362C4—6521no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4C2×#1457969911yescharacteristic
367C4#1910478011yescharacteristic
368C2—22no—
369C2—42no—
370C2—81no—
371C2—6521no—
372C2#2378230511yescharacteristic
373C2C2×#1910476411yescharacteristic
374C2#2378246611yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#102×C163C2
socleC22×C163C22×D4
prime radical#23782309C2
hypercenterC2×#102D163
hypocenterC163#3314
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. #23782370C2semidirect
3. #23782023C2semidirect
4. #23782000C2semidirect
5. #23781977C2semidirect
6. #23781943C2semidirect
7. #23781910C2semidirect
8. #23781906C2semidirect
9. #23781886C2semidirect
10. #23781828C2semidirect
11. #23781816C2semidirect
12. #23781762C2semidirect
13. #23781736C2semidirect
14. #23781663C2semidirect
15. C2×#102×C163C2semidirect
17. C2×Dic652C22semidirect
20. C2×#14579682C22semidirect
21. #19104734C22semidirect
25. #19104715C22semidirect
33. #19104700C22semidirect
37. C4⋊C4×D163C22semidirect
39. #19104669C22semidirect
40. #19104668C22semidirect
41. #19104668C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104660C22semidirect
44. #19104660C22semidirect
45. #19104658C22semidirect
46. #19104658C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
187. C163#3314semidirect
203. C2×#102D163semidirect

Maximal subgroups

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

subgroupindex
#237823702
#237823092
#237820232
#237820002
#237819772
#237819432
#237819102
#237819062
#237818862
#237818282
#237818162
#237817622
#237817362
#237816632
C2×#102×C1632
#3314163