Finite Group Databasebeta

← #35226506

Subgroups — #35226506

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#23782320C211yesmaximal, characteristic, complemented
2#23782035C211yesmaximal, characteristic, complemented
3#23782006C211yesmaximal, characteristic, complemented
4#23781995C211yesmaximal, characteristic, complemented
5#23781962C211yesmaximal, characteristic, complemented
6#23781909C211yesmaximal, characteristic, complemented
7#23781892C211yesmaximal, characteristic, complemented
8#23781870C211yesmaximal, characteristic, complemented
9#23781808C211yesmaximal, characteristic, complemented
10#23781808C211yesmaximal, characteristic, complemented
11#23781775C211yesmaximal, characteristic, complemented
12#23781737C211yesmaximal, characteristic, complemented
13#23781664C211yesmaximal, characteristic, complemented
14#23781659C211yesmaximal, characteristic, complemented
15#553×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C2×#14579699C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C2×#14579683C2211yescharacteristic
22C22×C163⋊C8C2211yescharacteristic
23#19104735C2211yescharacteristic
24#19104717—21no—
25#19104717—21no—
26#19104714C2211yescharacteristic
27#19104714C2211yescharacteristic
28#19104706—21no—
29C4×D652C2211yescharacteristic
30C4⋊C4×D163—21no—
31#19104668C2211yescharacteristic
32#19104666C2211yescharacteristic
33#19104666C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35#19104658C2211yescharacteristic
36#19104658C2211yescharacteristic
37#19104658C2211yescharacteristic
38#19104658C2211yescharacteristic
39#19104658C2211yescharacteristic
40#19104658C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44#19104655C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic
47#19104646C2211yescharacteristic
48#126×C163—21no—
49#125×C163C2211yescharacteristic
50#123×C163—21no—
51#121×C163—21no—
52#121×C163C2211yescharacteristic
53#121×C163C2211yescharacteristic
54C4×D4×C163—21no—
55C4×D4×C163C2211yescharacteristic
56C4×D4×C163C2211yescharacteristic
57#117×C163C2211yescharacteristic
58C2×C4⋊C4×C163C2211yescharacteristic
59#14579699—22no—
60#14579699—22no—
61#14579699—41no—
62#14579699—41no—
63C2×Dic326C2311yescharacteristic
64C2×Dic326C2311yescharacteristic
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71#14579687—41no—
72#14579687—41no—
73#14579684—22no—
74#14579683—41no—
75C2×C163⋊C8—41no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87(C2×D163)⋊C4—21no—
88C2×D4×C163—21no—
89C2×D4×C163—21no—
90C2×D4×C163D411yescharacteristic
91C2×D4×C163C2311yescharacteristic
92C22×C4×C163—21no—
93C22×C4×C163—21no—
94C22×C4×C163C2311yescharacteristic
95C22×C4×C163C2311yescharacteristic
96C4⋊C4×C163—21no—
97C4⋊C4×C163—21no—
98C4⋊C4×C163—21no—
99C4⋊C4×C163D411yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C22⋊C4×C163D411yescharacteristic
108C22⋊C4×C163D411yescharacteristic
109C42×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116D652—21no—
117D652#4112yes—
118C22⋊D163—42no—
119C22⋊D163—42no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C4×D163—42no—
124C4×D163—42no—
125C4×D163—42no—
126C163⋊C8—21no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C23×C163—21no—
131C23×C163C2×D411yescharacteristic
132C23×C163C2×D411yescharacteristic
133D4×C163—21no—
134D4×C163—21no—
135D4×C163—41no—
136D4×C163—42no—
137D4×C163—81no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—81no—
146C2×C4×C163—81no—
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C22×C163—21no—
153C22×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163C22×D411yescharacteristic
162C2×D163—22no—
163C2×D163—41no—
164C2×D163—41no—
165C4×C163—21no—
166C4×C163—21no—
167C4×C163—21no—
168C4×C163—21no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163C2×#4111yescharacteristic
173C4×C163H(5,𝔽2)11yescharacteristic
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177C2×C163—41no—
178C2×C163—41no—
179C2×C163—81no—
180C2×C163#54611yescharacteristic
181C2×C163#57511yescharacteristic
182C2×C163#57711yescharacteristic
183D163—42no—
184C163#330711yescharacteristic, complemented, Sylow
185#3307—1631nomaximal, Sylow
186#553D16311yescharacteristic, complemented
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#449—1631no—
197#442—1631no—
198#435—1631no—
199#417—1631no—
200#416—1631no—
201C2×#41—1631no—
202C2×#41—1631no—
203C2×SD16—1631no—
204C2×SD16—1631no—
205C2×D8—1631no—
206C2×M16—1631no—
207C22×C8—1631no—
208C4⋊Q8—1631no—
209#126—21no—
210#125C2×D16311yescharacteristic
211#123—21no—
212#123—3261no—
213#123—3261no—
214C22⋊Q8—1631no—
215C22⋊Q8—1631no—
216#121—21no—
217#121—3261no—
218#121C2×D16311yescharacteristic
219#121C2×D16311yescharacteristic
220C4×D4—21no—
221C4×D4—1631no—
222C4×D4C2×D16311yescharacteristic
223C4×D4C2×D16311yescharacteristic
224#117C2×D16311yescharacteristic
225C2×C4⋊C4—3261no—
226C2×C4⋊C4C2×D16311yescharacteristic
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#103—1631no—
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#102—1631no—
241C22⋊C8—1631no—
242C22⋊C8—1631no—
243#97—1631no—
244#41—3262no—
245#41—3262no—
246#41—6521no—
247#41—6521no—
248C2×Q8—1631no—
249C2×Q8—1631no—
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—21no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4—3261no—
262C22×C4C22×D16311yescharacteristic
263C22×C4C22×D16311yescharacteristic
264SD16—6521no—
265SD16—6521no—
266D8—3262no—
267M16—6521no—
268C2×C8—6521no—
269C2×C8—1631no—
270C2×C8—1631no—
271C2×C8—1631no—
272C2×C8—1631no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—3261no—
277C4⋊C4—3261no—
278C4⋊C4—3261no—
279C4⋊C4—3261no—
280C4⋊C4—1631no—
281C4⋊C4C22⋊D16311yescharacteristic
282C4⋊C4C22×D16311yescharacteristic
283C4⋊C4C22×D16311yescharacteristic
284C4⋊C4C22×D16311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4C22⋊D16311yescharacteristic
293C22⋊C4C22⋊D16311yescharacteristic
294C42—21no—
295C42C22×D16311yescharacteristic
296C23—21no—
297C23—3261no—
298C23C2×C22⋊D16311yescharacteristic
299C23C2×C22⋊D16311yescharacteristic
300Q8—3261no—
301Q8—3261no—
302Q8—3261no—
303Q8—3261no—
304D4—21no—
305D4—21no—
306D4—41no—
307D4—42no—
308D4—81no—
309D4—3261no—
310D4—6522no—
311D4—6522no—
312D4—1632no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—6522no—
326C2×C4—6522no—
327C2×C4—6522no—
328C2×C4C2×C22⋊D16311yescharacteristic
329C2×C4C2×C22⋊D16311yescharacteristic
330C2×C4C2×C22⋊D16311yescharacteristic
331C2×C4C2×C22⋊D16311yescharacteristic
332C2×C4C23×D16311yescharacteristic
333C8—3261no—
334C8—3261no—
335C8—3261no—
336C8—3261no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—3262no—
347C22—6521no—
348C22—6521no—
349C22C22×C22⋊D16311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—6521no—
358C4—6521no—
359C4—6521no—
360C4#41×D16311yescharacteristic
361C4#1910478111yescharacteristic
362C2—41no—
363C2—41no—
364C2—81no—
365C2—6522no—
366C2#2378230311yescharacteristic
367C2#2378246311yescharacteristic
368C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#553×C163C2
socleC22×C163C22×D4
hypercenter#553D163
hypocenterC163#3307
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #23782320C2semidirect
2. #23782035C2semidirect
3. #23782006C2semidirect
4. #23781995C2semidirect
5. #23781962C2semidirect
6. #23781909C2semidirect
7. #23781892C2semidirect
8. #23781870C2semidirect
9. #23781808C2semidirect
10. #23781808C2semidirect
11. #23781775C2semidirect
12. #23781737C2semidirect
13. #23781664C2semidirect
14. #23781659C2semidirect
15. #553×C163C2semidirect
184. C163#3307semidirect
186. #553D163semidirect

Maximal subgroups

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

subgroupindex
#237823202
#237820352
#237820062
#237819952
#237819622
#237819092
#237818922
#237818702
#237818082
#237817752
#237817372
#237816642
#237816592
#553×C1632
#3307163