Finite Group Databasebeta

← #35226612

Subgroups — #35226612

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#23782373C211yesmaximal, characteristic, complemented
2#23782312C211yesmaximal, characteristic
3#23782023C211yesmaximal, characteristic, complemented
4#23782002C211yesmaximal, characteristic, complemented
5#23781976C211yesmaximal, characteristic, complemented
6#23781944C211yesmaximal, characteristic, complemented
7#23781905C211yesmaximal, characteristic, complemented
8#23781885C211yesmaximal, characteristic, complemented
9#23781825C211yesmaximal, characteristic, complemented
10#23781814C211yesmaximal, characteristic, complemented
11#23781768C211yesmaximal, characteristic, complemented
12#23781734C211yesmaximal, characteristic, complemented
13#23781667C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#468×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×#14579689C2211yescharacteristic, complemented
19C2×#14579686C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic
21#19104734C2211yescharacteristic, complemented
22#19104725—21no—
23#19104722C2211yescharacteristic
24#19104717—21no—
25#19104715—21no—
26#19104715C2211yescharacteristic
27#19104712—21no—
28#19104711C2211yescharacteristic
29#19104706—21no—
30#19104703—21no—
31Q8×Dic163C2211yescharacteristic, complemented
32#19104697C2211yescharacteristic
33C4×C22⋊D163C2211yescharacteristic
34(C22⋊D163)⋊C4—21no—
35C2×C4⋊Dic163C2211yescharacteristic, complemented
36C4⋊C4×D163C2211yescharacteristic
37C22⋊C4×D163—21no—
38#19104668C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40Dic163⋊C8C2211yescharacteristic, complemented
41#19104660C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic
53C2×M16×C163C2211yescharacteristic, complemented
54C22⋊Q8×C163C2211yescharacteristic, complemented
55#121×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163D411yescharacteristic
68D4×D163—22no—
69D4×D163—41no—
70C22×Dic163—21no—
71C2×C4×D163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2311yescharacteristic
75#14579689—41no—
76#14579686—22no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C8×D163—41no—
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163—21no—
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4D411yescharacteristic
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163D411yescharacteristic
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163—21no—
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102SD16×C163—41no—
103M16×C163—41no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163—42no—
113C22×D163—42no—
114C22×D163—81no—
115C22×D163C2×D411yescharacteristic
116Dic326—21no—
117Dic326—21no—
118Dic326—81no—
119D652—21no—
120D652#4112yes—
121C22⋊D163—22no—
122C22⋊D163—42no—
123C22⋊D163—42no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C2×Dic163—81no—
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C2×Dic163C2×D411yescharacteristic
131C4×D163—21no—
132C4×D163—21no—
133C4×D163—42no—
134C4×D163—81no—
135C4×D163—81no—
136C163⋊C8—21no—
137C163⋊C8—21no—
138C23×C163—21no—
139C23×C163C2×D411yescharacteristic
140Q8×C163—21no—
141Q8×C163—21no—
142D4×C163—21no—
143D4×C163—21no—
144D4×C163—81no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C8×C163—21no—
153C8×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163C22×D411yescharacteristic
161C2×D163—21no—
162C2×D163—21no—
163C2×D163—22no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—84no—
167C2×D163#13712yes—
168C4×C163—21no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163C2×#4111yescharacteristic
173C4×C163H(5,𝔽2)11yescharacteristic
174Dic163—21no—
175Dic163—21no—
176Dic163—21no—
177Dic163—41no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—41no—
181C2×C163—81no—
182C2×C163#54611yescharacteristic
183C2×C163C2×#13711yescharacteristic
184C2×C163#57611yescharacteristic
185D163—22no—
186D163—42no—
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#468D16311yescharacteristic, complemented
196#467—1631no—
197#464—1631no—
198#452—1631no—
199#450—1631no—
200#440—1631no—
201#435—1631no—
202#417—1631no—
203C2×#102—1631no—
204C2×#41—1631no—
205C22×D4—1631no—
206C2×Q16—1631no—
207C2×SD16—1631no—
208C2×SD16C2×D16311yescharacteristic
209C2×M16C2×D16311yescharacteristic, complemented
210C22×C8—1631no—
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⋊Q8C2×D16311yescharacteristic, complemented
220#121—3261no—
221#121C2×D16311yescharacteristic
222#120—3261no—
223C4×Q8—1631no—
224C4×D4—3261no—
225C4×D4—1631no—
226C4×D4—1631no—
227C2×C4⋊C4—1631no—
228C2×C4⋊C4—1631no—
229C2×C22⋊C4—3261no—
230#107—1631no—
231#107—1631no—
232#107C2×D16311yescharacteristic, complemented
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#103C2×D16311yescharacteristic, complemented
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102C2×D16311yescharacteristic
244C22⋊C8—1631no—
245C22⋊C8—1631no—
246#97—1631no—
247C24—3261no—
248#41—3262no—
249#41—6521no—
250C2×Q8—3261no—
251C2×Q8—1631no—
252C2×Q8C22×D16311yescharacteristic
253C2×D4—21no—
254C2×D4—3262no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—6521no—
258C2×D4—1631no—
259C2×D4—1631no—
260C2×D4C22×D16311yescharacteristic
261C22×C4—3261no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—1631no—
266C22×C4C22×D16311yescharacteristic
267Q16—6521no—
268SD16—41no—
269SD16—3262no—
270M16—41no—
271C2×C8—6521no—
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—21no—
289C22⋊C4—21no—
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—13041no—
301C23—3261no—
302C23—6522no—
303C23—6522no—
304C23—1631no—
305C23D4×D16311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—13041no—
309Q8—3261no—
310Q8—3261no—
311D4—21no—
312D4—21no—
313D4—81no—
314D4—3262no—
315D4—3261no—
316D4—6522no—
317D4—6522no—
318D4—1632no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—13041no—
325C2×C4—13041no—
326C2×C4—13041no—
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—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—13044no—
349C22—3261no—
350C22—3261no—
351C22—3262no—
352C22—6521no—
353C22—6521no—
354C22—1632no—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3261no—
361C4—3261no—
362C4—3261no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4#41×D16311yescharacteristic
367C4#1910478011yescharacteristic
368C2—41no—
369C2—81no—
370C2—3262no—
371C2—6522no—
372C2#2378230411yescharacteristic
373C2#137×D16311yescharacteristic
374C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#468×C163C2
socleC22×C163C22×D4
prime radical#23782312C2
hypercenter#468D163
hypocenterC163#3314
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #23782373C2semidirect
3. #23782023C2semidirect
4. #23782002C2semidirect
5. #23781976C2semidirect
6. #23781944C2semidirect
7. #23781905C2semidirect
8. #23781885C2semidirect
9. #23781825C2semidirect
10. #23781814C2semidirect
11. #23781768C2semidirect
12. #23781734C2semidirect
13. #23781667C2semidirect
14. #102×D163C2semidirect
15. #468×C163C2semidirect
18. C2×#14579689C22semidirect
21. #19104734C22semidirect
31. Q8×Dic163C22semidirect
35. C2×C4⋊Dic163C22semidirect
38. #19104668C22semidirect
39. #19104668C22semidirect
40. Dic163⋊C8C22semidirect
41. #19104660C22semidirect
42. #19104660C22semidirect
43. #19104658C22semidirect
49. C22⋊(C163⋊C8)C22semidirect
51. #19104645C22semidirect
53. C2×M16×C163C22semidirect
54. C22⋊Q8×C163C22semidirect
56. #107×C163C22semidirect
57. #103×C163C22semidirect
187. C163#3314semidirect
195. #468D163semidirect
209. C2×M16C2×D163semidirect
219. C22⋊Q8C2×D163semidirect
232. #107C2×D163semidirect
237. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823732
#237823122
#237820232
#237820022
#237819762
#237819442
#237819052
#237818852
#237818252
#237818142
#237817682
#237817342
#237816672
#102×D1632
#468×C1632
#3314163