Finite Group Databasebeta

← #35226527

Subgroups — #35226527

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782319C211yesmaximal, characteristic, complemented
2#23782024C211yesmaximal, characteristic, complemented
3#23781998C211yesmaximal, characteristic, complemented
4#23781976C211yesmaximal, characteristic, complemented
5#23781960C211yesmaximal, characteristic, complemented
6#23781913C211yesmaximal, characteristic, complemented
7#23781899C211yesmaximal, characteristic, complemented
8#23781877C211yesmaximal, characteristic, complemented
9#23781810C211yesmaximal, characteristic, complemented
10#23781804C211yesmaximal, characteristic, complemented
11#23781763C211yesmaximal, characteristic, complemented
12#23781737C211yesmaximal, characteristic, complemented
13#23781666C211yesmaximal, characteristic, complemented
14C2×#19104659C211yesmaximal, characteristic, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C2×#14579686C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C2×#14579683C2211yescharacteristic
21C22×C163⋊C8C2211yescharacteristic
22#19104735C2211yescharacteristic
23#19104719—21no—
24#19104719—21no—
25#19104715—21no—
26#19104709C2211yescharacteristic
27#19104709C2211yescharacteristic
28C4×D652C2211yescharacteristic
29C2×C4⋊Dic163C2211yescharacteristic
30C2×C4⋊Dic163C2211yescharacteristic
31#19104668C2211yescharacteristic
32#19104668C2211yescharacteristic
33#19104668C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35#19104659C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104659C2211yescharacteristic
39#19104659C2211yescharacteristic
40#19104659C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104656C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104655C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic
47#19104646C2211yescharacteristic
48C2×#41×C163C2211yescharacteristic
49C22×Q8×C163C2211yescharacteristic
50#124×C163—21no—
51#124×C163C2211yescharacteristic
52C22⋊Q8×C163—21no—
53C22⋊Q8×C163C2211yescharacteristic
54#121×C163—21no—
55#121×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic
57C4×D4×C163—21no—
58C4×D4×C163C2211yescharacteristic
59#14579698—44no—
60C2×Dic326—21no—
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C22×Dic163—21no—
65C22×Dic163—21no—
66C2×C4×D163—21no—
67#14579686—22no—
68#14579686—41no—
69#14579684—41no—
70#14579683—41no—
71C2×C163⋊C8—41no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C2×C163⋊C8C2311yescharacteristic
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163—21no—
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84(C2×D163)⋊C4—21no—
85#41×C163—22no—
86#41×C163—41no—
87C2×Q8×C163—21no—
88C2×Q8×C163—22no—
89C2×Q8×C163—41no—
90C2×Q8×C163C2311yescharacteristic
91C2×Q8×C163C2311yescharacteristic
92C2×D4×C163—21no—
93C2×D4×C163—21no—
94C2×D4×C163D411yescharacteristic
95C2×D4×C163C2311yescharacteristic
96C22×C4×C163—21no—
97C22×C4×C163—21no—
98C22×C4×C163C2311yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C4⋊C4×C163—21no—
101C4⋊C4×C163—21no—
102C4⋊C4×C163D411yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C22⋊C4×C163—21no—
108C22⋊C4×C163D411yescharacteristic
109C22⋊C4×C163D411yescharacteristic
110C42×C163—21no—
111C42×C163C2311yescharacteristic
112C22×D163—21no—
113Dic326—42no—
114D652—21no—
115D652—21no—
116C22⋊D163—42no—
117C22⋊D163—42no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—42no—
122C2×Dic163—42no—
123C4×D163—42no—
124C163⋊C8—21no—
125C163⋊C8—21no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C23×C163—21no—
129C23×C163C2×D411yescharacteristic
130C23×C163C2×D411yescharacteristic
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163—21no—
134Q8×C163—42no—
135Q8×C163#4112yescomplemented
136D4×C163—22no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—42no—
140D4×C163—42no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—42no—
149C2×C4×C163—42no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2×D411yescharacteristic
153C2×C4×C163C2×D411yescharacteristic
154C2×C4×C163C2411yescharacteristic
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—21no—
159C22×C163—21no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163AGL(1,ℤ/8)12yes—
164C22×C163C22×D411yescharacteristic
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—41no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—22no—
171C4×C163—21no—
172C4×C163—21no—
173C4×C163—41no—
174C4×C163—41no—
175C4×C163C2×#4111yescharacteristic
176C4×C163H(5,𝔽2)11yescharacteristic
177Dic163—41no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—22no—
181C2×C163—41no—
182C2×C163—81no—
183C2×C163#54611yescharacteristic
184C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C163#57811yescharacteristic
186D163—81no—
187C163#330811yescharacteristic, complemented, Sylow
188#3308—1631nomaximal, Sylow
189#548D16311yescharacteristic, complemented
190#547—1631no—
191#489—1631no—
192#484—1631no—
193#480—1631no—
194#477—1631no—
195#468—1631no—
196#466—1631no—
197#463—1631no—
198#449—1631no—
199#448—1631no—
200#440—1631no—
201#435—1631no—
202#417—1631no—
203C2×#103—1631no—
204C2×#41—3261no—
205C2×#41C2×D16311yescharacteristic
206C22×Q8C2×D16311yescharacteristic
207C2×SD16—1631no—
208C2×SD16—1631no—
209C2×D8—1631no—
210C2×M16—1631no—
211C22×C8—1631no—
212C4⋊Q8—1631no—
213#124—21no—
214#124C2×D16311yescharacteristic
215#123—3261no—
216#123—3261no—
217C22⋊Q8—21no—
218C22⋊Q8—3261no—
219C22⋊Q8C2×D16311yescharacteristic
220#121—21no—
221#121—1631no—
222#121—1631no—
223#121C2×D16311yescharacteristic
224C4×Q8C2×D16311yescharacteristic
225C4×D4—21no—
226C4×D4—1631no—
227C4×D4C2×D16311yescharacteristic
228C2×C4⋊C4—1631no—
229C2×C4⋊C4—1631no—
230#107—1631no—
231#107—1631no—
232#107—1631no—
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#103—1631no—
238#103—1631no—
239#103—1631no—
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244C22⋊C8—1631no—
245C22⋊C8—1631no—
246#97—1631no—
247#41—22no—
248#41—41no—
249#41—6524no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—3261no—
254C2×Q8C22×D16311yescharacteristic
255C2×Q8C22×D16311yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3261no—
259C2×D4—3261no—
260C2×D4—1631no—
261C2×D4C22⋊D16311yescharacteristic
262C2×D4C22×D16311yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3261no—
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4C22×D16311yescharacteristic
269C22×C4C22×D16311yescharacteristic
270SD16—3262no—
271SD16—6521no—
272D8—6521no—
273M16—6521no—
274C2×C8—6521no—
275C2×C8—1631no—
276C2×C8—1631no—
277C2×C8—1631no—
278C2×C8—1631no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4C22⋊D16311yescharacteristic
288C4⋊C4C22×D16311yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4C22⋊D16311yescharacteristic
297C22⋊C4C22⋊D16311yescharacteristic
298C42—21no—
299C42C22×D16311yescharacteristic
300C23—21no—
301C23—3261no—
302C23C2×C22⋊D16311yescharacteristic
303C23C2×C22⋊D16311yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—6522no—
309Q8#1457969812yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—3261no—
316D4—3261no—
317D4—6522no—
318D4—6522no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—6522no—
332C2×C4—6522no—
333C2×C4—6522no—
334C2×C4C2×C22⋊D16311yescharacteristic
335C2×C4C2×C22⋊D16311yescharacteristic
336C2×C4C2×C22⋊D16311yescharacteristic
337C2×C4C2×C22⋊D16311yescharacteristic
338C2×C4C23×D16311yescharacteristic
339C8—3261no—
340C8—3261no—
341C8—3261no—
342C8—3261no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22#1910476012yes—
355C22C22×C22⋊D16311yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4C2×#1457969811yescharacteristic
367C4#1910478011yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—13041no—
372C2#2378230911yescharacteristic
373C2C2×#1910476011yescharacteristic
374C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#548×C163C2
socleC22×C163C22×D4
hypercenter#548D163
hypocenterC163#3308
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23782319C2semidirect
2. #23782024C2semidirect
3. #23781998C2semidirect
4. #23781976C2semidirect
5. #23781960C2semidirect
6. #23781913C2semidirect
7. #23781899C2semidirect
8. #23781877C2semidirect
9. #23781810C2semidirect
10. #23781804C2semidirect
11. #23781763C2semidirect
12. #23781737C2semidirect
13. #23781666C2semidirect
14. C2×#19104659C2semidirect
15. #548×C163C2semidirect
135. Q8×C163#41semidirect
187. C163#3308semidirect
189. #548D163semidirect
309. Q8#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823192
#237820242
#237819982
#237819762
#237819602
#237819132
#237818992
#237818772
#237818102
#237818042
#237817632
#237817372
#237816662
C2×#191046592
#548×C1632
#3308163