Finite Group Databasebeta

← #35224551

Subgroups — #35224551

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#23782194C211yesmaximal, characteristic, complemented
2C4×#14579698C211yesmaximal, characteristic, complemented
3#23782005C211yesmaximal, characteristic, complemented
4#23781988C211yesmaximal, characteristic, complemented
5#23781973C211yesmaximal, characteristic, complemented
6#23781965C211yesmaximal, characteristic, complemented
7#23781884C211yesmaximal, characteristic, complemented
8#23781824C211yesmaximal, characteristic, complemented
9#23781807C211yesmaximal, characteristic
10#23781757C211yesmaximal, characteristic, complemented
11#23781756C211yesmaximal, characteristic, complemented
12SD16×Dic163C211yesmaximal, characteristic, complemented
13D8×Dic163C211yesmaximal, characteristic, complemented
14#23781717C211yesmaximal, characteristic, complemented
15#463×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19#19104728—21no—
20#19104727C2211yescharacteristic
21#19104717—21no—
22#19104711C2211yescharacteristic
23#19104706C2211yescharacteristic
24Q8×Dic163C2211yescharacteristic
25C4×Dic326C2211yescharacteristic
26#19104700C2211yescharacteristic
27D4×Dic163C2211yescharacteristic
28D4×Dic163C2211yescharacteristic
29C4×C22⋊D163—21no—
30(C22⋊D163)⋊C4—21no—
31(C22⋊D163)⋊C4—21no—
32#19104693C2211yescharacteristic
33#19104691—21no—
34#19104691—21no—
35C2×C4×Dic163—21no—
36C42×D163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48(C2×D163)⋊C8C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C2×D8×C163C2211yescharacteristic
54#124×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—22no—
61C2×Dic326C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C22×Dic163—21no—
65C22×Dic163—21no—
66C22×Dic163—21no—
67C2×C4×D163D411yescharacteristic
68C2×C4×D163D411yescharacteristic
69C2×C4×D163C2311yescharacteristic
70#14579689—22no—
71#14579687—22no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C4⋊Dic163—21no—
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163D411yescharacteristic
93C4×Dic163D411yescharacteristic
94C4×Dic163C2311yescharacteristic
95C2×Q8×C163C2311yescharacteristic
96C2×D4×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C22×C4×C163—21no—
99SD16×C163—22no—
100D8×C163—22no—
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C42×C163C2311yescharacteristic
108C22×D163C2×D411yescharacteristic
109Dic326—21no—
110Dic326#4112yes—
111C22⋊D163—24no—
112C22⋊D163—42no—
113C2×Dic163—24no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—42no—
119C2×Dic163—42no—
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C4×D163—21no—
124C4×D163—21no—
125C4×D163—22no—
126C4×D163—41no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C23×C163—21no—
130C23×C163—21no—
131Q8×C163—21no—
132Q8×C163—21no—
133D4×C163—21no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163#4112yes—
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×C163—41no—
149C22×C163—41no—
150C22×C163—41no—
151C22×C163C22×D411yescharacteristic
152C2×D163—21no—
153C2×D163—21no—
154C2×D163—21no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—41no—
158C4×C163—41no—
159C4×C163C2×#4111yescharacteristic
160C4×C163C2×#4111yescharacteristic
161Dic163—21no—
162Dic163—22no—
163Dic163—21no—
164Dic163—41no—
165Dic163—41no—
166Dic163—41no—
167Dic163#13512yes—
168C2×C163—42no—
169C2×C163—81no—
170C2×C163#52511yescharacteristic
171C2×C163C2×#13511yescharacteristic
172C2×C163#57511yescharacteristic
173D163—41no—
174C163#310211yescharacteristic, complemented, Sylow
175#3102—1631nomaximal, Sylow
176#529—1631no—
177C4×#41—1631no—
178#485—1631no—
179#482—1631no—
180#479—1631no—
181#478—1631no—
182#464—1631no—
183#463D16311yescharacteristic, complemented
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×SD16C2×D16311yescharacteristic
195C2×D8C2×D16311yescharacteristic
196#126—3261no—
197#125—1631no—
198#124C2×D16311yescharacteristic
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#107—1631no—
218#106—1631no—
219#106—1631no—
220C4⋊C8—1631no—
221C4⋊C8C2×D16311yescharacteristic
222#103—1631no—
223#103—1631no—
224#103—1631no—
225#103C2×D16311yescharacteristic
226#102—1631no—
227#102—1631no—
228#102—1631no—
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×Q8C22×D16311yescharacteristic
238C2×D4—3261no—
239C2×D4—3261no—
240C2×D4C22×D16311yescharacteristic
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—22no—
251SD16—3262no—
252D8—22no—
253C2×C8—1631no—
254C2×C8—1631no—
255C2×C8C22×D16311yescharacteristic
256C2×C8C22×D16311yescharacteristic
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—1631no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4C22×D16311yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
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—21no—
285C23—1631no—
286Q8—21no—
287Q8—21no—
288Q8—3261no—
289Q8—1632no—
290D4—21no—
291D4—21no—
292D4—21no—
293D4—3264no—
294D4—6522no—
295D4#1457969812yes—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—42no—
299C2×C4—3262no—
300C2×C4—3264no—
301C2×C4—3261no—
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—41no—
324C22—41no—
325C22—41no—
326C22—3261no—
327C22—3261no—
328C22—3261no—
329C22C2×D4×D16311yescharacteristic
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—3261no—
335C4—3262no—
336C4—3261no—
337C4—6521no—
338C4—6521no—
339C4—6521no—
340C4—1632no—
341C4#41×D16311yescharacteristic
342C4C2×#1457969811yescharacteristic
343C2—42no—
344C2—81no—
345C2—6521no—
346C2#2378217511yescharacteristic
347C2#135×D16311yescharacteristic
348C2#2378246111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#463×C163C2
socleC22×C163C22×D4
prime radical#23781807C2
hypercenter#463D163
hypocenterC163#3102
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #23782194C2semidirect
2. C4×#14579698C2semidirect
3. #23782005C2semidirect
4. #23781988C2semidirect
5. #23781973C2semidirect
6. #23781965C2semidirect
7. #23781884C2semidirect
8. #23781824C2semidirect
10. #23781757C2semidirect
11. #23781756C2semidirect
12. SD16×Dic163C2semidirect
13. D8×Dic163C2semidirect
14. #23781717C2semidirect
15. #463×C163C2semidirect
174. C163#3102semidirect
183. #463D163semidirect

Maximal subgroups

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

subgroupindex
#237821942
C4×#145796982
#237820052
#237819882
#237819732
#237819652
#237818842
#237818242
#237818072
#237817572
#237817562
SD16×Dic1632
D8×Dic1632
#237817172
#463×C1632
#3102163