Finite Group Databasebeta

← #35226505

Subgroups — #35226505

Order 20864.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782367C211yesmaximal, characteristic, complemented
2#23782314C211yesmaximal, characteristic, complemented
3#23782032C211yesmaximal, characteristic, complemented
4#23782004C211yesmaximal, characteristic, complemented
5#23781996C211yesmaximal, characteristic, complemented
6#23781967C211yesmaximal, characteristic, complemented
7#23781909C211yesmaximal, characteristic, complemented
8#23781891C211yesmaximal, characteristic, complemented
9#23781868C211yesmaximal, characteristic, complemented
10#23781807C211yesmaximal, characteristic, complemented
11#23781777C211yesmaximal, characteristic, complemented
12#23781739C211yesmaximal, characteristic, complemented
13#23781666C211yesmaximal, characteristic, complemented
14#23781658C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18C2×#14579683C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20#19104733C2211yescharacteristic
21#19104728—21no—
22#19104726C2211yescharacteristic
23#19104719—21no—
24#19104718—21no—
25#19104716—21no—
26#19104714C2211yescharacteristic
27#19104711C2211yescharacteristic
28#19104707—21no—
29#19104707—21no—
30#19104706C2211yescharacteristic
31D4×Dic163—21no—
32D4×Dic163C2211yescharacteristic
33D4×Dic163C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35#19104693C2211yescharacteristic
36C2×C4⋊Dic163C2211yescharacteristic
37C2×Dic163⋊C4—21no—
38#19104668C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104667C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic
52C2×#41×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—41no—
61C2×Dic326C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163D411yescharacteristic
65C22×Dic163—21no—
66C22×Dic163—21no—
67C22×Dic163—21no—
68C22×Dic163—21no—
69C2×C4×D163C2311yescharacteristic
70#14579687—41no—
71#14579683—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C8×D163—41no—
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4D411yescharacteristic
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163D411yescharacteristic
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96#41×C163—22no—
97#41×C163—41no—
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163—21no—
100C2×D4×C163—21no—
101C2×D4×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163C2311yescharacteristic
105SD16×C163—41no—
106D8×C163—22no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114C22⋊D163—41no—
115C22⋊D163—42no—
116C22⋊D163—81no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—42no—
122C2×Dic163—42no—
123C2×Dic163—81no—
124C2×Dic163—81no—
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C4×D163—21no—
129C4×D163—21no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163—21no—
133C23×C163—21no—
134C23×C163C2×D411yescharacteristic
135Q8×C163—21no—
136Q8×C163—21no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—42no—
141D4×C163—42no—
142D4×C163#4112yes—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—42no—
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2411yescharacteristic
150C8×C163—21no—
151C8×C163—21no—
152C22×C163—21no—
153C22×C163—22no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163C22×D411yescharacteristic
162C2×D163—21no—
163C2×D163—21no—
164C2×D163—21no—
165C4×C163—21no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163C2×#4111yescharacteristic
169C4×C163H(5,𝔽2)11yescharacteristic
170Dic163—21no—
171Dic163—21no—
172Dic163—21no—
173Dic163—41no—
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177C2×C163—41no—
178C2×C163—42no—
179C2×C163—81no—
180C2×C163#54611yescharacteristic
181C2×C163#57511yescharacteristic
182C2×C163#57711yescharacteristic
183D163—41no—
184C163#330711yescharacteristic, complemented, Sylow
185#3307—1631nomaximal, Sylow
186#553—1631no—
187#547—1631no—
188#491—1631no—
189#485—1631no—
190#483—1631no—
191#478—1631no—
192#468—1631no—
193#465—1631no—
194#461—1631no—
195#449—1631no—
196#449D16311yescharacteristic, complemented
197#442—1631no—
198#435—1631no—
199#417—1631no—
200#416—1631no—
201C2×#41—1631no—
202C2×#41C2×D16311yescharacteristic
203C2×SD16—1631no—
204C2×SD16C2×D16311yescharacteristic
205C2×D8C2×D16311yescharacteristic
206C2×M16—1631no—
207C22×C8—1631no—
208C4⋊Q8—1631no—
209#126—3261no—
210#125—1631no—
211#123—3261no—
212#123—3261no—
213#123—3261no—
214C22⋊Q8—1631no—
215C22⋊Q8—1631no—
216#121—3261no—
217#121—3261no—
218#121—1631no—
219#121C2×D16311yescharacteristic
220C4×D4—3261no—
221C4×D4—1631no—
222C4×D4—1631no—
223C4×D4—1631no—
224#117—1631no—
225C2×C4⋊C4—3261no—
226C2×C4⋊C4—1631no—
227#107—1631no—
228#106—1631no—
229#106—1631no—
230C4⋊C8—1631no—
231#103—1631no—
232#103—1631no—
233#103—1631no—
234#103—1631no—
235#103—1631no—
236#103C2×D16311yescharacteristic
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#102C2×D16311yescharacteristic
241C22⋊C8—1631no—
242C22⋊C8C2×D16311yescharacteristic
243#97—1631no—
244#41—22no—
245#41—41no—
246#41—3262no—
247#41—6521no—
248C2×Q8—1631no—
249C2×Q8C22×D16311yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—3261no—
253C2×D4—3261no—
254C2×D4—1631no—
255C2×D4C22×D16311yescharacteristic
256C2×D4C22×D16311yescharacteristic
257C22×C4—21no—
258C22×C4—3261no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4—3261no—
262C22×C4—1631no—
263C22×C4C22×D16311yescharacteristic
264SD16—41no—
265SD16—6521no—
266D8—22no—
267M16—6521no—
268C2×C8—6521no—
269C2×C8—1631no—
270C2×C8—1631no—
271C2×C8C22×D16311yescharacteristic
272C2×C8C22×D16311yescharacteristic
273C4⋊C4—3261no—
274C4⋊C4—3261no—
275C4⋊C4—3261no—
276C4⋊C4—3261no—
277C4⋊C4—3261no—
278C4⋊C4—3261no—
279C4⋊C4—3261no—
280C4⋊C4—1631no—
281C4⋊C4—1631no—
282C4⋊C4—1631no—
283C4⋊C4—1631no—
284C4⋊C4C22×D16311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—3261no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—1631no—
293C22⋊C4—1631no—
294C42—3261no—
295C42—1631no—
296C23—21no—
297C23—21no—
298C23—1631no—
299C23D4×D16311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—3261no—
303Q8—3261no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—42no—
309D4—13041no—
310D4—6521no—
311D4—6522no—
312D4#1457969812yes—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—13041no—
319C2×C4—13041no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—6522no—
327C2×C4—6522no—
328C2×C4—1631no—
329C2×C4—1631no—
330C2×C4—1631no—
331C2×C4D4×D16311yescharacteristic
332C2×C4C23×D16311yescharacteristic
333C8—21no—
334C8—21no—
335C8—3261no—
336C8—3261no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3261no—
347C22—3261no—
348C22—3261no—
349C22C2×D4×D16311yescharacteristic
350C4—21no—
351C4—41no—
352C4—41no—
353C4—3261no—
354C4—3261no—
355C4—3261no—
356C4—6521no—
357C4—6521no—
358C4—6521no—
359C4—6521no—
360C4C2×#1457969811yescharacteristic
361C4#1910478011yescharacteristic
362C2—41no—
363C2—42no—
364C2—81no—
365C2—6521no—
366C2#2378230211yescharacteristic
367C2#2378246111yescharacteristic
368C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3307
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #23782367C2semidirect
2. #23782314C2semidirect
3. #23782032C2semidirect
4. #23782004C2semidirect
5. #23781996C2semidirect
6. #23781967C2semidirect
7. #23781909C2semidirect
8. #23781891C2semidirect
9. #23781868C2semidirect
10. #23781807C2semidirect
11. #23781777C2semidirect
12. #23781739C2semidirect
13. #23781666C2semidirect
14. #23781658C2semidirect
15. #449×C163C2semidirect
184. C163#3307semidirect
196. #449D163semidirect

Maximal subgroups

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

subgroupindex
#237823672
#237823142
#237820322
#237820042
#237819962
#237819672
#237819092
#237818912
#237818682
#237818072
#237817772
#237817392
#237816662
#237816582
#449×C1632
#3307163