Finite Group Databasebeta

← #35224548

Subgroups — #35224548

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
1C4×#14579699C211yesmaximal, characteristic, complemented
2#23782003C211yesmaximal, characteristic, complemented
3#23781986C211yesmaximal, characteristic, complemented
4#23781969C211yesmaximal, characteristic, complemented
5#23781962C211yesmaximal, characteristic, complemented
6#23781884C211yesmaximal, characteristic, complemented
7#23781877C211yesmaximal, characteristic, complemented
8#23781825C211yesmaximal, characteristic, complemented
9#23781808C211yesmaximal, characteristic
10C4×#14579689C211yesmaximal, characteristic, complemented
11C4×#14579686C211yesmaximal, characteristic, complemented
12C4×#14579687C211yesmaximal, characteristic, complemented
13C4×#14579684C211yesmaximal, characteristic, complemented
14#23781718C211yesmaximal, characteristic, complemented
15#529×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C4×Dic326C2211yescharacteristic
22C4×Dic326C2211yescharacteristic
23C4×D652C2211yescharacteristic
24C4×D652C2211yescharacteristic
25C4×C22⋊D163—21no—
26C4×C22⋊D163—21no—
27#19104694C2211yescharacteristic
28#19104692—21no—
29C42×D163—21no—
30#19104668C2211yescharacteristic
31#19104668C2211yescharacteristic
32#19104666C2211yescharacteristic
33#19104666C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35C4⋊(C163⋊C8)C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104658C2211yescharacteristic
39#19104658C2211yescharacteristic
40#19104656C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44C22⋊(C163⋊C8)C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C4×C163⋊C8C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48#126×C163—21no—
49#125×C163C2211yescharacteristic
50#124×C163C2211yescharacteristic
51#123×C163—21no—
52C22⋊Q8×C163C2211yescharacteristic
53#121×C163C2211yescharacteristic
54C4×Q8×C163C2211yescharacteristic
55C4×D4×C163—21no—
56C4×D4×C163C2211yescharacteristic
57#117×C163—21no—
58C2×C42×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—22no—
61C2×Dic326C2311yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C4×D163—21no—
65C2×C4×D163—21no—
66C2×C4×D163—21no—
67#14579689—22no—
68#14579687—22no—
69#14579686—22no—
70#14579684—22no—
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79C22⋊Dic163—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82C4×Dic163—21no—
83C4×Dic163—21no—
84C4×Dic163—21no—
85C2×Q8×C163C2311yescharacteristic
86C2×D4×C163—21no—
87C2×D4×C163C2311yescharacteristic
88C22×C4×C163—21no—
89C22×C4×C163D411yescharacteristic
90C22×C4×C163D411yescharacteristic
91C22×C4×C163C2311yescharacteristic
92C4⋊C4×C163—21no—
93C4⋊C4×C163—21no—
94C4⋊C4×C163—21no—
95C4⋊C4×C163—21no—
96C4⋊C4×C163C2311yescharacteristic
97C4⋊C4×C163C2311yescharacteristic
98C22⋊C4×C163—21no—
99C22⋊C4×C163—21no—
100C22⋊C4×C163—21no—
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C42×C163—21no—
104C42×C163D411yescharacteristic
105C42×C163D411yescharacteristic
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109Dic326—21no—
110Dic326#4112yes—
111D652—21no—
112D652#4112yes—
113C22⋊D163—24no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C4×D163—24no—
118C4×D163—42no—
119C4×D163—42no—
120C163⋊C8—21no—
121C163⋊C8—21no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124C23×C163—21no—
125C23×C163C2×D411yescharacteristic
126Q8×C163—21no—
127Q8×C163—21no—
128D4×C163—21no—
129D4×C163—21no—
130D4×C163—42no—
131C2×C4×C163—22no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—41no—
138C2×C4×C163—42no—
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2411yescharacteristic
145C22×C163—21no—
146C22×C163—21no—
147C22×C163—21no—
148C22×C163—41no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163C22×D411yescharacteristic
152C2×D163—22no—
153C2×D163—41no—
154C2×D163—41no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163—21no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163#13512yes—
163C4×C163C2×#4111yescharacteristic
164C4×C163C2×#4111yescharacteristic
165Dic163—22no—
166Dic163—41no—
167Dic163—41no—
168C2×C163—41no—
169C2×C163—81no—
170C2×C163#52511yescharacteristic
171C2×C163C2×#13511yescharacteristic
172C2×C163#57511yescharacteristic
173D163—42no—
174C163#310211yescharacteristic, complemented, Sylow
175#3102—1631nomaximal, Sylow
176#529D16311yescharacteristic, complemented
177C4×#41—1631no—
178#485—1631no—
179#482—1631no—
180#479—1631no—
181#478—1631no—
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—21no—
197#125C2×D16311yescharacteristic
198#124C2×D16311yescharacteristic
199#123—21no—
200C22⋊Q8C2×D16311yescharacteristic
201#121C2×D16311yescharacteristic
202C4×Q8—1631no—
203C4×Q8—1631no—
204C4×Q8C2×D16311yescharacteristic
205C4×D4—21no—
206C4×D4—3261no—
207C4×D4—3261no—
208C4×D4—1631no—
209C4×D4—1631no—
210C4×D4C2×D16311yescharacteristic
211#117—21no—
212#117—3261no—
213#117—1631no—
214C2×C42—3261no—
215C2×C42C2×D16311yescharacteristic
216#107—1631no—
217#107—1631no—
218#106—1631no—
219#106—1631no—
220C4⋊C8—1631no—
221C4⋊C8—1631no—
222#103—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#102—1631no—
227#102—1631no—
228#102—1631no—
229#102—1631no—
230C22⋊C8—1631no—
231C22⋊C8—1631no—
232C4×C8—1631no—
233C4×C8—1631no—
234#41—3262no—
235#41—3262no—
236C2×Q8—1631no—
237C2×Q8C22×D16311yescharacteristic
238C2×D4—21no—
239C2×D4—3261no—
240C2×D4—1631no—
241C2×D4C22×D16311yescharacteristic
242C22×C4—21no—
243C22×C4—3261no—
244C22×C4—3261no—
245C22×C4—3261no—
246C22×C4C22⋊D16311yescharacteristic
247C22×C4C22⋊D16311yescharacteristic
248C22×C4C22×D16311yescharacteristic
249Q16—3262no—
250SD16—3262no—
251SD16—3262no—
252D8—3262no—
253C2×C8—1631no—
254C2×C8—1631no—
255C2×C8—1631no—
256C2×C8—1631no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—1631no—
264C4⋊C4—1631no—
265C4⋊C4C22×D16311yescharacteristic
266C4⋊C4C22×D16311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C42—21no—
276C42—3261no—
277C42—3261no—
278C42—3261no—
279C42C22⋊D16311yescharacteristic
280C42C22⋊D16311yescharacteristic
281C42C22×D16311yescharacteristic
282C42C22×D16311yescharacteristic
283C23—21no—
284C23—3261no—
285C23C2×C22⋊D16311yescharacteristic
286Q8—21no—
287Q8—21no—
288Q8—3261no—
289Q8—1632no—
290D4—21no—
291D4—21no—
292D4—42no—
293D4—3264no—
294D4—3261no—
295D4—1632no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—3264no—
305C2×C4—3261no—
306C2×C4—3261no—
307C2×C4—3261no—
308C2×C4—6522no—
309C2×C4—6522no—
310C2×C4C2×C22⋊D16311yescharacteristic
311C2×C4C2×C22⋊D16311yescharacteristic
312C2×C4C2×C22⋊D16311yescharacteristic
313C2×C4C2×C22⋊D16311yescharacteristic
314C2×C4C2×C22⋊D16311yescharacteristic
315C2×C4C23×D16311yescharacteristic
316C8—3261no—
317C8—3261no—
318C8—3261no—
319C8—3261no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—3262no—
327C22—6521no—
328C22—6521no—
329C22C22×C22⋊D16311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3262no—
338C4—6521no—
339C4—6521no—
340C4#1910475512yes—
341C4#41×D16311yescharacteristic
342C4#41×D16311yescharacteristic
343C2—41no—
344C2—81no—
345C2—6522no—
346C2#2378217211yescharacteristic
347C2C2×#1910475511yescharacteristic
348C2#2378246311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#529×C163C2
socleC22×C163C22×D4
prime radical#23781808C2
hypercenter#529D163
hypocenterC163#3102
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C4×#14579699C2semidirect
2. #23782003C2semidirect
3. #23781986C2semidirect
4. #23781969C2semidirect
5. #23781962C2semidirect
6. #23781884C2semidirect
7. #23781877C2semidirect
8. #23781825C2semidirect
10. C4×#14579689C2semidirect
11. C4×#14579686C2semidirect
12. C4×#14579687C2semidirect
13. C4×#14579684C2semidirect
14. #23781718C2semidirect
15. #529×C163C2semidirect
174. C163#3102semidirect
176. #529D163semidirect

Maximal subgroups

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

subgroupindex
C4×#145796992
#237820032
#237819862
#237819692
#237819622
#237818842
#237818772
#237818252
#237818082
C4×#145796892
C4×#145796862
C4×#145796872
C4×#145796842
#237817182
#529×C1632
#3102163