Finite Group Databasebeta

← #35224784

Subgroups — #35224784

Order 20864.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782275C211yesmaximal, characteristic, complemented
2#23782139C211yesmaximal, characteristic
3#23782006C211yesmaximal, characteristic, complemented
4#23781968C211yesmaximal, characteristic, complemented
5#23781964C211yesmaximal, characteristic, complemented
6#23781869C211yesmaximal, characteristic, complemented
7#23781869C211yesmaximal, characteristic, complemented
8#23781818C211yesmaximal, characteristic
9#23781815C211yesmaximal, characteristic
10#23781768C211yesmaximal, characteristic, complemented
11#23781764C211yesmaximal, characteristic, complemented
12#23781752C211yesmaximal, characteristic, complemented
13#23781750C211yesmaximal, characteristic, complemented
14#23781723C211yesmaximal, characteristic, complemented
15#485×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C2×#14579688C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21#19104733C2211yescharacteristic, complemented
22#19104728—21no—
23#19104728—21no—
24#19104726—21no—
25#19104726C2211yescharacteristic, complemented
26#19104725—21no—
27#19104714C2211yescharacteristic
28Q8×Dic163C2211yescharacteristic, complemented
29#19104700C2211yescharacteristic, complemented
30C4×D652—21no—
31C4×D652—21no—
32#19104697C2211yescharacteristic
33#19104697C2211yescharacteristic
34#19104697C2211yescharacteristic
35#19104697C2211yescharacteristic
36#19104694C2211yescharacteristic
37C22⋊C4×D163—21no—
38C22⋊C4×D163—21no—
39#19104668C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42Dic163⋊C8C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104658C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50#19104645C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22⋊Q8×C163C2211yescharacteristic
53#117×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56#103×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic
59C23×D163—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652—22no—
62C2×D652—22no—
63C2×D652—21no—
64C2×D652C2311yescharacteristic
65C2×D652C2311yescharacteristic
66C2×C4×D163—21no—
67C2×C4×D163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70#14579688—22no—
71#14579688—22no—
72#14579686—22no—
73#14579686—22no—
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163D411yescharacteristic
79C4⋊Dic163D411yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163D411yescharacteristic
88C22⋊Dic163D411yescharacteristic
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96C4×Dic163C2311yescharacteristic
97C2×Q8×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C4⋊C4×C163—21no—
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C42×C163—21no—
108C22×D163—21no—
109C22×D163—21no—
110C22×D163—24no—
111C22×D163—81no—
112C22×D163—81no—
113Dic326—21no—
114Dic326—21no—
115D652—24no—
116D652—21no—
117D652—21no—
118D652#4112yes—
119D652#4112yes—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C4×D163—81no—
127C4×D163—81no—
128C4×D163—81no—
129C4×D163—81no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163C2×D411yescharacteristic
133Q8×C163—21no—
134Q8×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C8×C163—21no—
144C8×C163—21no—
145C22×C163—21no—
146C22×C163—21no—
147C22×C163—21no—
148C22×C163C22×D411yescharacteristic
149C2×D163—22no—
150C2×D163—22no—
151C2×D163—41no—
152C2×D163—41no—
153C2×D163—41no—
154C2×D163—41no—
155C2×D163—84no—
156C4×C163—21no—
157C4×C163—41no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163C2×#4111yescharacteristic
162C4×C163C2×#4111yescharacteristic
163Dic163—21no—
164Dic163—21no—
165Dic163—21no—
166Dic163—21no—
167Dic163—41no—
168Dic163—41no—
169C2×C163—41no—
170C2×C163#52511yescharacteristic
171C2×C163#57711yescharacteristic
172C2×C163#57711yescharacteristic
173D163—42no—
174D163—42no—
175C163#312311yescharacteristic, complemented, Sylow
176#3123—1631nomaximal, Sylow
177#541—1631no—
178#518—1631no—
179#485—1631no—
180#485D16311yescharacteristic, complemented
181#478—1631no—
182#478—1631no—
183#461—1631no—
184#461—1631no—
185#450—1631no—
186#450—1631no—
187#440—1631no—
188#440—1631no—
189C4×SD16—1631no—
190C4×SD16—1631no—
191#433—1631no—
192C22×D4—1631no—
193C2×SD16—1631no—
194C2×SD16—1631no—
195C2×SD16—1631no—
196C2×SD16—1631no—
197C4⋊Q8—1631no—
198#126—3261no—
199#126—3261no—
200#125—3261no—
201#125—3261no—
202#125—1631no—
203C22⋊Q8—1631no—
204C22⋊Q8C2×D16311yescharacteristic
205C4×Q8—1631no—
206C4×Q8—1631no—
207C4×D4—3261no—
208C4×D4—3261no—
209C4×D4—1631no—
210C4×D4—1631no—
211C4×D4—1631no—
212C4×D4—1631no—
213#117—1631no—
214#117C2×D16311yescharacteristic
215C2×C22⋊C4—3261no—
216C2×C22⋊C4—3261no—
217#107—1631no—
218#107C2×D16311yescharacteristic, complemented
219#106—1631no—
220#106C2×D16311yescharacteristic, complemented
221C4⋊C8—1631no—
222C4⋊C8—1631no—
223#103—1631no—
224#103—1631no—
225#103C2×D16311yescharacteristic, complemented
226#103C2×D16311yescharacteristic, complemented
227#102—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231C22⋊C8—1631no—
232C22⋊C8C2×D16311yescharacteristic
233#97—1631no—
234C4×C8—1631no—
235C24—3261no—
236C2×Q8—1631no—
237C2×Q8C22×D16311yescharacteristic
238C2×D4—3262no—
239C2×D4—3262no—
240C2×D4—3261no—
241C2×D4—1631no—
242C2×D4—1631no—
243C22×C4—3261no—
244C22×C4—3261no—
245C22×C4—3261no—
246C22×C4—3261no—
247C22×C4C22×D16311yescharacteristic
248SD16—3262no—
249SD16—3262no—
250SD16—3262no—
251SD16—3262no—
252C2×C8—1631no—
253C2×C8—1631no—
254C2×C8C22×D16311yescharacteristic
255C2×C8C22×D16311yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4C22×D16311yescharacteristic
268C4⋊C4C22×D16311yescharacteristic
269C4⋊C4C22×D16311yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—1631no—
278C22⋊C4—1631no—
279C42—21no—
280C42—3261no—
281C42—3261no—
282C42—1631no—
283C42—1631no—
284C23—13041no—
285C23—13041no—
286C23—3264no—
287C23—3261no—
288C23—3261no—
289C23D4×D16311yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—3261no—
293Q8—3261no—
294D4—3264no—
295D4—3261no—
296D4—3261no—
297D4—1632no—
298D4—1632no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—13041no—
306C2×C4—13041no—
307C2×C4—13041no—
308C2×C4—13041no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—1631no—
312C2×C4—1631no—
313C2×C4—1631no—
314C2×C4—1631no—
315C2×C4D4×D16311yescharacteristic
316C2×C4C23×D16311yescharacteristic
317C8—21no—
318C8—21no—
319C8—3261no—
320C8—3261no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—13044no—
325C22—3262no—
326C22—3262no—
327C22—6521no—
328C22—6521no—
329C22—6521no—
330C22—6521no—
331C22C2×D4×D16311yescharacteristic
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3261no—
338C4—3261no—
339C4—3261no—
340C4—3261no—
341C4—6521no—
342C4—6521no—
343C4#41×D16311yescharacteristic
344C4#41×D16311yescharacteristic
345C2—41no—
346C2—6522no—
347C2—6522no—
348C2#2378217311yescharacteristic
349C2#2378247211yescharacteristic
350C2#2378247411yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#485×C163C2
socleC22×C163C22×D4
prime radicalC22×D652C22
hypercenter#485D163
hypocenterC163#3123
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #23782275C2semidirect
3. #23782006C2semidirect
4. #23781968C2semidirect
5. #23781964C2semidirect
6. #23781869C2semidirect
7. #23781869C2semidirect
10. #23781768C2semidirect
11. #23781764C2semidirect
12. #23781752C2semidirect
13. #23781750C2semidirect
14. #23781723C2semidirect
15. #485×C163C2semidirect
21. #19104733C22semidirect
25. #19104726C22semidirect
28. Q8×Dic163C22semidirect
29. #19104700C22semidirect
39. #19104668C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. Dic163⋊C8C22semidirect
43. #19104658C22semidirect
44. #19104658C22semidirect
50. #19104645C22semidirect
51. C8×Dic163C22semidirect
54. #107×C163C22semidirect
55. #106×C163C22semidirect
56. #103×C163C22semidirect
57. #103×C163C22semidirect
175. C163#3123semidirect
180. #485D163semidirect
218. #107C2×D163semidirect
220. #106C2×D163semidirect
225. #103C2×D163semidirect
226. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237822752
#237821392
#237820062
#237819682
#237819642
#237818692
#237818182
#237818152
#237817682
#237817642
#237817522
#237817502
#237817232
#485×C1632
#3123163