Finite Group Databasebeta

← #35224885

Subgroups — #35224885

Order 20864.

404 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782292C211yesmaximal, characteristic, complemented
2#23782228C211yesmaximal, characteristic
3#23782029C211yesmaximal, characteristic, complemented
4#23782028C211yesmaximal, characteristic, complemented
5#23782027C211yesmaximal, characteristic, complemented
6#23782023C211yesmaximal, characteristic, complemented
7#23782020C211yesmaximal, characteristic, complemented
8#23782018C211yesmaximal, characteristic, complemented
9#23782010C211yesmaximal, characteristic, complemented
10#23781668C211yesmaximal, characteristic, complemented
11#23781663C211yesmaximal, characteristic, complemented
12#23781658C211yesmaximal, characteristic, complemented
13#102×D163C211yesmaximal, characteristic, complemented
14#23781619C211yesmaximal, characteristic, complemented
15#486×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20#19104734C2211yescharacteristic, complemented
21#19104733—21no—
22#19104731C2211yescharacteristic
23#19104727C2211yescharacteristic, complemented
24#19104726C2211yescharacteristic, complemented
25#19104725—21no—
26#19104723C2211yescharacteristic
27#19104722C2211yescharacteristic
28#19104713—21no—
29#19104711—21no—
30#19104707—21no—
31#19104707—21no—
32#19104703—21no—
33#19104703—21no—
34#19104693C2211yescharacteristic
35#19104692C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic
37#19104660C2211yescharacteristic, complemented
38#19104660C2211yescharacteristic, complemented
39#19104658C2211yescharacteristic, complemented
40#19104658C2211yescharacteristic, complemented
41#19104657C2211yescharacteristic
42#19104657C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104656C2211yescharacteristic
45#19104656C2211yescharacteristic, complemented
46#19104656C2211yescharacteristic, complemented
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49#19104646C2211yescharacteristic, complemented
50#19104645C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C4⋊Q8×C163C2211yescharacteristic, complemented
53#127×C163C2211yescharacteristic
54#102×C163C2211yescharacteristic
55#102×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic, complemented
58C4×C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579698—22no—
61#14579698—22no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69D4×D163—22no—
70D4×D163—22no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74#14579682—22no—
75C2×C163⋊C8C2311yescharacteristic
76C2×C163⋊C8C2311yescharacteristic
77C8×D163—22no—
78C4⋊Dic163—21no—
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87Dic163⋊C4D411yescharacteristic
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94(C2×D163)⋊C4D411yescharacteristic
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C2×Q8×C163—21no—
98C2×D4×C163—21no—
99C2×D4×C163—21no—
100C2×D4×C163C2311yescharacteristic
101C2×D4×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C4⋊C4×C163—21no—
105C4⋊C4×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—42no—
110C22×D163—81no—
111C22×D163—81no—
112C22×D163C2×D411yescharacteristic
113Dic326—21no—
114Dic326—21no—
115Dic326—81no—
116D652—21no—
117D652—21no—
118C22⋊D163—42no—
119C22⋊D163—42no—
120C22⋊D163—81no—
121C22⋊D163—81no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—42no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—81no—
130C4×D163#4112yes—
131C4×D163#4112yes—
132C163⋊C8—21no—
133C163⋊C8—21no—
134C23×C163—21no—
135C23×C163—21no—
136Q8×C163—81no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—21no—
141D4×C163—81no—
142D4×C163—81no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C8×C163—21no—
149C8×C163—21no—
150C22×C163—41no—
151C22×C163—41no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163C22×D411yescharacteristic
157C2×D163—21no—
158C2×D163—21no—
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—84no—
163C2×D163#12412yes—
164C4×C163—21no—
165C4×C163—21no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163C2×#4111yescharacteristic
169C4×C163C2×#4111yescharacteristic
170Dic163—21no—
171Dic163—21no—
172Dic163—41no—
173Dic163—41no—
174Dic163—41no—
175Dic163#12412yes—
176C2×C163—81no—
177C2×C163—81no—
178C2×C163C2×#12411yescharacteristic
179C2×C163#57611yescharacteristic
180C2×C163#57711yescharacteristic
181D163—22no—
182D163—81no—
183C163#313611yescharacteristic, complemented, Sylow
184#3136—1631nomaximal, Sylow
185#544—1631no—
186#535—1631no—
187#490—1631no—
188#490—1631no—
189#489—1631no—
190#489—1631no—
191#488—1631no—
192#488—1631no—
193#486—1631no—
194#486D16311yescharacteristic, complemented
195#418—1631no—
196#417—1631no—
197#416—1631no—
198C2×#102—1631no—
199#405—1631no—
200C2×#41—1631no—
201C22×D4—1631no—
202C2×M16—1631no—
203C22×C8—1631no—
204C4⋊Q8—3261no—
205C4⋊Q8—1631no—
206C4⋊Q8C2×D16311yescharacteristic, complemented
207#127—1631no—
208#127C2×D16311yescharacteristic
209#125—3261no—
210#125—1631no—
211#125—1631no—
212#124—1631no—
213#124—1631no—
214C22⋊Q8—3261no—
215C22⋊Q8—3261no—
216#121—3261no—
217#121—3261no—
218#120—3261no—
219#120—3261no—
220#117—1631no—
221#117—1631no—
222C2×C4⋊C4—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#102—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102C2×D16311yescharacteristic
236#102C2×D16311yescharacteristic
237#102C2×D16311yescharacteristic, complemented
238#102C2×D16311yescharacteristic, complemented
239#97—1631no—
240#97—1631no—
241C4×C8—1631no—
242C4×C8C2×D16311yescharacteristic, complemented
243C24—3261no—
244#41—3262no—
245#41—3262no—
246C2×Q8—21no—
247C2×Q8—3261no—
248C2×Q8—1631no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—3261no—
252C2×D4—3262no—
253C2×D4—3262no—
254C2×D4—3261no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—1631no—
258C2×D4C22×D16311yescharacteristic
259C2×D4C22×D16311yescharacteristic
260C22×C4—3261no—
261C22×C4—3261no—
262C22×C4—1631no—
263M16—3262no—
264C2×C8—3262no—
265C2×C8—1631no—
266C2×C8—1631no—
267C2×C8C22×D16311yescharacteristic
268C2×C8C22×D16311yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—3261no—
271C4⋊C4—3261no—
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—3261no—
275C4⋊C4—3261no—
276C4⋊C4—1631no—
277C4⋊C4—1631no—
278C4⋊C4—1631no—
279C4⋊C4—1631no—
280C4⋊C4C22×D16311yescharacteristic
281C4⋊C4C22×D16311yescharacteristic
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—3261no—
287C22⋊C4—1631no—
288C22⋊C4—1631no—
289C42—3261no—
290C42—1631no—
291C42C22×D16311yescharacteristic
292C23—21no—
293C23—21no—
294C23—13041no—
295C23—13041no—
296C23—3261no—
297C23—6522no—
298C23—1631no—
299Q8—81no—
300Q8—13041no—
301Q8—3261no—
302Q8—3261no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—81no—
308D4—81no—
309D4—13041no—
310D4—13041no—
311D4—3261no—
312D4—3261no—
313D4—6522no—
314D4—6522no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—13041no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4—6522no—
322C2×C4—1631no—
323C2×C4—1631no—
324C2×C4—1631no—
325C2×C4—1632no—
326C2×C4—1632no—
327C2×C4D4×D16311yescharacteristic
328C2×C4D4×D16311yescharacteristic
329C2×C4C23×D16311yescharacteristic
330C8—21no—
331C8—21no—
332C8—3261no—
333C8—3261no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—13044no—
341C22—3261no—
342C22—3261no—
343C22—6521no—
344C22—6521no—
345C22—6521no—
346C22—1632no—
347C22C2×D4×D16311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—3261no—
353C4—3261no—
354C4—6521no—
355C4—6521no—
356C4—6521no—
357C4—1632no—
358C4#41×D16311yescharacteristic
359C4#41×D16311yescharacteristic
360C2—81no—
361C2—81no—
362C2—13041no—
363C2—3262no—
364C2#124×D16311yescharacteristic
365C2#2378246611yescharacteristic
366C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#486×C163C2
socleC22×C163C22×D4
prime radical#23782228C2
hypercenter#486D163
hypocenterC163#3136
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #23782292C2semidirect
3. #23782029C2semidirect
4. #23782028C2semidirect
5. #23782027C2semidirect
6. #23782023C2semidirect
7. #23782020C2semidirect
8. #23782018C2semidirect
9. #23782010C2semidirect
10. #23781668C2semidirect
11. #23781663C2semidirect
12. #23781658C2semidirect
13. #102×D163C2semidirect
14. #23781619C2semidirect
15. #486×C163C2semidirect
20. #19104734C22semidirect
23. #19104727C22semidirect
24. #19104726C22semidirect
37. #19104660C22semidirect
38. #19104660C22semidirect
39. #19104658C22semidirect
40. #19104658C22semidirect
45. #19104656C22semidirect
46. #19104656C22semidirect
49. #19104646C22semidirect
50. #19104645C22semidirect
51. C8×Dic163C22semidirect
52. C4⋊Q8×C163C22semidirect
56. #102×C163C22semidirect
57. #102×C163C22semidirect
58. C4×C8×C163C22semidirect
183. C163#3136semidirect
194. #486D163semidirect
206. C4⋊Q8C2×D163semidirect
237. #102C2×D163semidirect
238. #102C2×D163semidirect
242. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822922
#237822282
#237820292
#237820282
#237820272
#237820232
#237820202
#237820182
#237820102
#237816682
#237816632
#237816582
#102×D1632
#237816192
#486×C1632
#3136163