Finite Group Databasebeta

← #35224556

Subgroups — #35224556

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#23782199C211yesmaximal, characteristic, complemented
2C4×#14579699C211yesmaximal, characteristic, complemented
3#23782007C211yesmaximal, characteristic, complemented
4#23781990C211yesmaximal, characteristic, complemented
5#23781973C211yesmaximal, characteristic, complemented
6#23781966C211yesmaximal, characteristic, complemented
7#23781888C211yesmaximal, characteristic, complemented
8#23781881C211yesmaximal, characteristic, complemented
9#23781829C211yesmaximal, characteristic, complemented
10#23781812C211yesmaximal, characteristic
11C4×Dic652C211yesmaximal, characteristic, complemented
12C4×#14579688C211yesmaximal, characteristic, complemented
13C4×#14579688C211yesmaximal, characteristic, complemented
14C4×D1304C211yesmaximal, characteristic, complemented
15#431×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×Dic652C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×#14579688C2211yescharacteristic
20C2×D1304C2211yescharacteristic
21#19104729—21no—
22#19104727C2211yescharacteristic
23#19104723C2211yescharacteristic
24#19104718—21no—
25#19104714C2211yescharacteristic
26#19104709C2211yescharacteristic
27C4×Dic326C2211yescharacteristic
28C4×Dic326C2211yescharacteristic
29C4×Dic326C2211yescharacteristic
30C4×D652C2211yescharacteristic
31C4×D652C2211yescharacteristic
32C4×D652C2211yescharacteristic
33C4×C22⋊D163—21no—
34C4×C22⋊D163—21no—
35C4×C22⋊D163—21no—
36#19104694C2211yescharacteristic
37#19104692—21no—
38#19104692—21no—
39C42×D163—21no—
40#19104669C2211yescharacteristic
41#19104669C2211yescharacteristic
42#19104667C2211yescharacteristic
43#19104667C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104660C2211yescharacteristic
46#19104660C2211yescharacteristic
47#19104660C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104657C2211yescharacteristic
51#19104657C2211yescharacteristic
52C2×C42×C163C2211yescharacteristic
53C4⋊C8×C163C2211yescharacteristic
54C4⋊C8×C163C2211yescharacteristic
55C22⋊C8×C163C2211yescharacteristic
56C22⋊C8×C163C2211yescharacteristic
57C4×C8×C163C2211yescharacteristic
58C4×C8×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—22no—
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63C2×D652C2311yescharacteristic
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C4×D163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71Dic652—22no—
72#14579688—22no—
73#14579688—22no—
74D1304—22no—
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
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×Dic163—21no—
96C4×Dic163—21no—
97C22×C4×C163D411yescharacteristic
98C22×C4×C163D411yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C42×C163D411yescharacteristic
105C42×C163D411yescharacteristic
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—21no—
110Dic326—21no—
111Dic326—21no—
112Dic326—21no—
113Dic326#4112yes—
114D652—21no—
115D652—21no—
116D652—21no—
117D652#4112yes—
118C22⋊D163—24no—
119C22⋊D163—42no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C4×D163—24no—
127C4×D163—42no—
128C4×D163—42no—
129C4×D163—42no—
130C23×C163C2×D411yescharacteristic
131C2×C4×C163—22no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—41no—
135C2×C4×C163C2×D411yescharacteristic
136C2×C4×C163C2×D411yescharacteristic
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2411yescharacteristic
141C8×C163—21no—
142C8×C163—21no—
143C8×C163—21no—
144C8×C163—21no—
145C22×C163—21no—
146C22×C163—21no—
147C22×C163—21no—
148C22×C163C22×D411yescharacteristic
149C2×D163—22no—
150C2×D163—41no—
151C2×D163—41no—
152C2×D163—41no—
153C2×D163—41no—
154C2×D163—41no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163—21no—
159C4×C163#13512yes—
160C4×C163C2×#4111yescharacteristic
161C4×C163C2×#4111yescharacteristic
162Dic163—22no—
163Dic163—41no—
164Dic163—41no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168C2×C163—41no—
169C2×C163#52511yescharacteristic
170C2×C163C2×#13511yescharacteristic
171C2×C163#57511yescharacteristic
172D163—42no—
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#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#431D16311yescharacteristic, complemented
191C2×#41—1631no—
192C2×Q16—1631no—
193C2×SD16—1631no—
194C2×SD16—1631no—
195C2×D8—1631no—
196#126—3261no—
197#125—1631no—
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×D4—1631no—
211#117—3261no—
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⋊C8C2×D16311yescharacteristic
221C4⋊C8C2×D16311yescharacteristic
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⋊C8C2×D16311yescharacteristic
231C22⋊C8C2×D16311yescharacteristic
232C4×C8C2×D16311yescharacteristic
233C4×C8C2×D16311yescharacteristic
234#41—3262no—
235#41—3262no—
236C2×Q8—1631no—
237C2×Q8—1631no—
238C2×D4—3261no—
239C2×D4—3261no—
240C2×D4—1631no—
241C2×D4—1631no—
242C22×C4—3261no—
243C22×C4—3261no—
244C22×C4—3261no—
245C22×C4—3261no—
246C22×C4D65211yescharacteristic
247C22×C4D65211yescharacteristic
248C22×C4C22×D16311yescharacteristic
249Q16—3262no—
250SD16—3262no—
251SD16—3262no—
252D8—3262no—
253C2×C8C22×D16311yescharacteristic
254C2×C8C22×D16311yescharacteristic
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⋊C4—1631no—
267C22⋊C4—3261no—
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—
279C42D65211yescharacteristic
280C42D65211yescharacteristic
281C42C22×D16311yescharacteristic
282C42C22×D16311yescharacteristic
283C23—3261no—
284C23—3261no—
285C23C2×D65211yescharacteristic
286Q8—3261no—
287Q8—3261no—
288Q8—3261no—
289Q8—1632no—
290D4—3264no—
291D4—3261no—
292D4—3261no—
293D4—3261no—
294D4—6522no—
295D4—1632no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—3264no—
301C2×C4—3261no—
302C2×C4—3261no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—3261no—
306C2×C4—3261no—
307C2×C4—6522no—
308C2×C4—6522no—
309C2×C4—6522no—
310C2×C4C2×D65211yescharacteristic
311C2×C4C2×D65211yescharacteristic
312C2×C4C2×D65211yescharacteristic
313C2×C4C2×D65211yescharacteristic
314C2×C4C2×D65211yescharacteristic
315C2×C4C23×D16311yescharacteristic
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—3262no—
324C22—6521no—
325C22—6521no—
326C22—6521no—
327C22—6521no—
328C22—6521no—
329C22C22×D65211yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—3262no—
335C4—6521no—
336C4—6521no—
337C4—6521no—
338C4—6521no—
339C4—6521no—
340C4#1910475712yes—
341C4#41×D16311yescharacteristic
342C4#41×D16311yescharacteristic
343C2—41no—
344C2—13041no—
345C2—6522no—
346C2#2378217711yescharacteristic
347C2C2×#1910475711yescharacteristic
348C2#2378246511yescharacteristic
subgroupG/H
centerC2×C4C2×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#431×C163C2
socleC22×C163C22×D4
prime radical#23781812C2
hypercenter#431D163
hypocenterC163#3102
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #431

Complements

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

NH ≅ G/Ntype
1. #23782199C2semidirect
2. C4×#14579699C2semidirect
3. #23782007C2semidirect
4. #23781990C2semidirect
5. #23781973C2semidirect
6. #23781966C2semidirect
7. #23781888C2semidirect
8. #23781881C2semidirect
9. #23781829C2semidirect
11. C4×Dic652C2semidirect
12. C4×#14579688C2semidirect
13. C4×#14579688C2semidirect
14. C4×D1304C2semidirect
15. #431×C163C2semidirect
174. C163#3102semidirect
190. #431D163semidirect

Maximal subgroups

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

subgroupindex
#237821992
C4×#145796992
#237820072
#237819902
#237819732
#237819662
#237818882
#237818812
#237818292
#237818122
C4×Dic6522
C4×#145796882
C4×D13042
#431×C1632
#3102163