Finite Group Databasebeta

← #35224559

Subgroups — #35224559

Order 20864.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782200C211yesmaximal, characteristic, complemented
2C4×#14579698C211yesmaximal, characteristic, complemented
3#23782009C211yesmaximal, characteristic, complemented
4#23781992C211yesmaximal, characteristic, complemented
5#23781969C211yesmaximal, characteristic, complemented
6#23781888C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic, complemented
8#23781828C211yesmaximal, characteristic, complemented
9#23781811C211yesmaximal, characteristic
10#23781759C211yesmaximal, characteristic, complemented
11#23781756C211yesmaximal, characteristic, complemented
12#23781752C211yesmaximal, characteristic, complemented
13#23781745C211yesmaximal, characteristic, complemented
14#23781717C211yesmaximal, characteristic, complemented
15#478×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×Dic652C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21#19104730—21no—
22#19104723C2211yescharacteristic
23#19104720—21no—
24#19104715C2211yescharacteristic
25#19104710C2211yescharacteristic
26C4×Dic326C2211yescharacteristic
27#19104700C2211yescharacteristic
28#19104700C2211yescharacteristic
29D4×Dic163C2211yescharacteristic
30#19104697—21no—
31#19104697C2211yescharacteristic
32C4×C22⋊D163—21no—
33(C22⋊D163)⋊C4—21no—
34#19104693—21no—
35#19104693C2211yescharacteristic
36#19104691—21no—
37C2×C4×Dic163—21no—
38C42×D163C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104666C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104658C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48(C2×D163)⋊C8C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52#125×C163C2211yescharacteristic
53C4×D4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—22no—
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63C2×D652—21no—
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C22×Dic163—21no—
67C22×Dic163—21no—
68C2×C4×D163—21no—
69C2×C4×D163D411yescharacteristic
70C2×C4×D163D411yescharacteristic
71C2×C4×D163C2311yescharacteristic
72Dic652—22no—
73#14579688—22no—
74#14579687—22no—
75#14579684—22no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C4⋊Dic163—21no—
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83C22⋊Dic163—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163D411yescharacteristic
95C4×Dic163D411yescharacteristic
96C4×Dic163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C22×C4×C163—21no—
99C2×C8×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C4⋊C4×C163—21no—
102C4⋊C4×C163—21no—
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C22⋊C4×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163C2×D411yescharacteristic
110Dic326—21no—
111Dic326—21no—
112Dic326—21no—
113Dic326#4112yes—
114D652—21no—
115D652—21no—
116D652—42no—
117C22⋊D163—24no—
118C2×Dic163—24no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—42no—
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C4×D163—21no—
127C4×D163—21no—
128C4×D163—22no—
129C4×D163—41no—
130C4×D163—42no—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C23×C163—21no—
134D4×C163—21no—
135D4×C163#4112yes—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—42no—
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C8×C163—21no—
144C8×C163—21no—
145C22×C163—22no—
146C22×C163—41no—
147C22×C163—41no—
148C22×C163C22×D411yescharacteristic
149C2×D163—21no—
150C2×D163—21no—
151C2×D163—21no—
152C2×D163—41no—
153C2×D163—41no—
154C2×D163—41no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—41no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163C2×#4111yescharacteristic
161C4×C163C2×#4111yescharacteristic
162Dic163—21no—
163Dic163—22no—
164Dic163—21no—
165Dic163—41no—
166Dic163—41no—
167Dic163#13512yes—
168C2×C163—42no—
169C2×C163#52511yescharacteristic
170C2×C163C2×#13511yescharacteristic
171C2×C163#57511yescharacteristic
172D163—41no—
173D163—81no—
174C163#310211yescharacteristic, complemented, Sylow
175#3102—1631nomaximal, Sylow
176#529—1631no—
177C4×#41—1631no—
178#485—1631no—
179#482—1631no—
180#479—1631no—
181#478D16311yescharacteristic, complemented
182#464—1631no—
183#463—1631no—
184#452—1631no—
185#449—1631no—
186C4×Q16—1631no—
187C4×SD16—1631no—
188C4×SD16—1631no—
189C4×D8—1631no—
190#431—1631no—
191C2×#41—1631no—
192C2×Q16—1631no—
193C2×SD16—1631no—
194C2×SD16—1631no—
195C2×D8—1631no—
196#126—3261no—
197#125C2×D16311yescharacteristic
198#124—1631no—
199#123—3261no—
200C22⋊Q8—1631no—
201#121—1631no—
202C4×Q8—1631no—
203C4×Q8—1631no—
204C4×Q8—1631no—
205C4×D4—3261no—
206C4×D4—3261no—
207C4×D4—3261no—
208C4×D4—1631no—
209C4×D4—1631no—
210C4×D4C2×D16311yescharacteristic
211#117—3261no—
212#117—3261no—
213#117—1631no—
214C2×C42—3261no—
215C2×C42—1631no—
216#107—1631no—
217#107C2×D16311yescharacteristic
218#106—1631no—
219#106C2×D16311yescharacteristic
220C4⋊C8—1631no—
221C4⋊C8C2×D16311yescharacteristic
222#103—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#102—1631no—
227#102—1631no—
228#102C2×D16311yescharacteristic
229#102C2×D16311yescharacteristic
230C22⋊C8—1631no—
231C22⋊C8—1631no—
232C4×C8—1631no—
233C4×C8—1631no—
234#41—3262no—
235#41—3262no—
236C2×Q8—1631no—
237C2×Q8—1631no—
238C2×D4—3261no—
239C2×D4—3261no—
240C2×D4—1631no—
241C2×D4C22×D16311yescharacteristic
242C22×C4—21no—
243C22×C4—3261no—
244C22×C4—3261no—
245C22×C4—3261no—
246C22×C4—1631no—
247C22×C4—1631no—
248C22×C4—1631no—
249Q16—3262no—
250SD16—3262no—
251SD16—3262no—
252D8—3262no—
253C2×C8—1631no—
254C2×C8—1631no—
255C2×C8C22×D16311yescharacteristic
256C2×C8C22×D16311yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—1631no—
264C4⋊C4C22×D16311yescharacteristic
265C4⋊C4C22×D16311yescharacteristic
266C4⋊C4C22×D16311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—3261no—
269C22⋊C4—3261no—
270C22⋊C4—3261no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C42—3261no—
276C42—3261no—
277C42—3261no—
278C42—3261no—
279C42—1631no—
280C42—1631no—
281C42—1631no—
282C42C22×D16311yescharacteristic
283C23—21no—
284C23—3261no—
285C23—1631no—
286Q8—3261no—
287Q8—3261no—
288Q8—3261no—
289Q8—1632no—
290D4—21no—
291D4—3264no—
292D4—3261no—
293D4—3261no—
294D4—6522no—
295D4#1457970012yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—3262no—
301C2×C4—3264no—
302C2×C4—3261no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—3261no—
306C2×C4—3261no—
307C2×C4—6521no—
308C2×C4—6522no—
309C2×C4—6522no—
310C2×C4—1631no—
311C2×C4—1631no—
312C2×C4—1631no—
313C2×C4D4×D16311yescharacteristic
314C2×C4D4×D16311yescharacteristic
315C2×C4C23×D16311yescharacteristic
316C8—21no—
317C8—21no—
318C8—3261no—
319C8—3261no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—3261no—
324C22—3261no—
325C22—3261no—
326C22—6521no—
327C22—6521no—
328C22—6521no—
329C22C2×D4×D16311yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—3261no—
336C4—3262no—
337C4—3261no—
338C4—6521no—
339C4—6521no—
340C4—1632no—
341C4#41×D16311yescharacteristic
342C4C2×#1457970011yescharacteristic
343C2—42no—
344C2—13041no—
345C2—6521no—
346C2#2378217811yescharacteristic
347C2#135×D16311yescharacteristic
348C2#2378246111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#478×C163C2
socleC22×C163C22×D4
prime radical#23781811C2
hypercenter#478D163
hypocenterC163#3102
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #23782200C2semidirect
2. C4×#14579698C2semidirect
3. #23782009C2semidirect
4. #23781992C2semidirect
5. #23781969C2semidirect
6. #23781888C2semidirect
7. #23781880C2semidirect
8. #23781828C2semidirect
10. #23781759C2semidirect
11. #23781756C2semidirect
12. #23781752C2semidirect
13. #23781745C2semidirect
14. #23781717C2semidirect
15. #478×C163C2semidirect
174. C163#3102semidirect
181. #478D163semidirect

Maximal subgroups

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

subgroupindex
#237822002
C4×#145796982
#237820092
#237819922
#237819692
#237818882
#237818802
#237818282
#237818112
#237817592
#237817562
#237817522
#237817452
#237817172
#478×C1632
#3102163