Finite Group Databasebeta

← #35226380

Subgroups — #35226380

Order 20864.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782311C211yesmaximal, characteristic, complemented
2#23782022C211yesmaximal, characteristic, complemented
3#23782000C211yesmaximal, characteristic, complemented
4#23781989C211yesmaximal, characteristic, complemented
5#23781962C211yesmaximal, characteristic, complemented
6#23781899C211yesmaximal, characteristic, complemented
7#23781892C211yesmaximal, characteristic, complemented
8#23781880C211yesmaximal, characteristic, complemented
9#23781808C211yesmaximal, characteristic, complemented
10#23781798C211yesmaximal, characteristic, complemented
11C4×#14579684C211yesmaximal, characteristic, complemented
12D4×C163⋊C8C211yesmaximal, characteristic, complemented
13#23781659C211yesmaximal, characteristic, complemented
14C2×#19104655C211yesmaximal, characteristic, complemented
15#553×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×#14579687C2211yescharacteristic, complemented
19C2×#14579684C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C22×C163⋊C8C2211yescharacteristic
22C22×C163⋊C8C2211yescharacteristic, complemented
23#19104723C2211yescharacteristic
24#19104720—21no—
25#19104714C2211yescharacteristic
26#19104709C2211yescharacteristic
27#19104708—21no—
28#19104703—21no—
29C4×D652C2211yescharacteristic
30C22⋊C4×D163—21no—
31#19104668C2211yescharacteristic, complemented
32#19104668C2211yescharacteristic, complemented
33#19104666C2211yescharacteristic, complemented
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35#19104658C2211yescharacteristic, complemented
36#19104658C2211yescharacteristic, complemented
37#19104658C2211yescharacteristic, complemented
38#19104656C2211yescharacteristic, complemented
39#19104655C2211yescharacteristic
40#19104655C2211yescharacteristic
41#19104655C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44#19104655C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic, complemented
47C4×C163⋊C8C2211yescharacteristic, complemented
48#126×C163—21no—
49#125×C163C2211yescharacteristic, complemented
50#123×C163—21no—
51#121×C163—21no—
52#121×C163C2211yescharacteristic
53#121×C163C2211yescharacteristic, complemented
54C4×D4×C163—21no—
55C4×D4×C163C2211yescharacteristic
56C4×D4×C163C2211yescharacteristic, complemented
57#117×C163C2211yescharacteristic, complemented
58C2×C4⋊C4×C163C2211yescharacteristic
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62C2×Dic326C2311yescharacteristic
63C2×D652—22no—
64C2×D652—21no—
65C2×D652—41no—
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72#14579687—41no—
73#14579684—22no—
74#14579684—41no—
75C2×C163⋊C8—41no—
76C2×C163⋊C8—41no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C2×C163⋊C8C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83C22⋊Dic163—21no—
84(C2×D163)⋊C4—21no—
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—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—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116Dic326—21no—
117Dic326—21no—
118D652—21no—
119D652—21no—
120D652—21no—
121D652—42no—
122D652#4112yes—
123C22⋊D163—42no—
124C22⋊D163—81no—
125C2×Dic163—21no—
126C2×Dic163—21no—
127C4×D163—42no—
128C4×D163—81no—
129C163⋊C8—21no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C163⋊C8—21no—
133C23×C163—21no—
134C23×C163C2×D411yescharacteristic
135C23×C163C2×D411yescharacteristic
136D4×C163—22no—
137D4×C163—21no—
138D4×C163—21no—
139D4×C163—42no—
140D4×C163—81no—
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—81no—
149C2×C4×C163—81no—
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×C163#13512yes—
164C22×C163C22×D411yescharacteristic
165C2×D163—22no—
166C2×D163—41no—
167C2×D163—41no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—41no—
171C2×D163—84no—
172C4×C163—21no—
173C4×C163—21no—
174C4×C163—21no—
175C4×C163—21no—
176C4×C163—41no—
177C4×C163—41no—
178C4×C163—41no—
179C4×C163C2×#4111yescharacteristic
180C4×C163H(5,𝔽2)11yescharacteristic
181Dic163—41no—
182Dic163—41no—
183C2×C163—22no—
184C2×C163—41no—
185C2×C163—81no—
186C2×C163#54611yescharacteristic
187C2×C163C2×#13511yescharacteristic
188C2×C163#57611yescharacteristic
189D163—42no—
190D163—81no—
191C163#329711yescharacteristic, complemented, Sylow
192#3297—1631nomaximal, Sylow
193#553D16311yescharacteristic, complemented
194#546—1631no—
195#488—1631no—
196#484—1631no—
197#482—1631no—
198#478—1631no—
199#466—1631no—
200#465—1631no—
201#463—1631no—
202#449—1631no—
203#447—1631no—
204C4×D8—1631no—
205C8×D4—1631no—
206#416—1631no—
207C2×#102—1631no—
208C2×#41—1631no—
209C22×D4—1631no—
210C2×SD16—1631no—
211C2×D8—1631no—
212C2×D8—1631no—
213C22×C8—1631no—
214C22×C8—1631no—
215#126—21no—
216#125C2×D16311yescharacteristic, complemented
217#124—1631no—
218#123—21no—
219#123—3261no—
220C22⋊Q8—1631no—
221#121—21no—
222#121—3261no—
223#121—1631no—
224#121C2×D16311yescharacteristic
225#121C2×D16311yescharacteristic, complemented
226#120—3261no—
227C4×D4—21no—
228C4×D4—1631no—
229C4×D4C2×D16311yescharacteristic
230C4×D4C2×D16311yescharacteristic, complemented
231#117C2×D16311yescharacteristic, complemented
232C2×C4⋊C4C2×D16311yescharacteristic
233C2×C22⋊C4—3261no—
234#107—1631no—
235#107—1631no—
236#106—1631no—
237C4⋊C8—1631no—
238#103—1631no—
239#103—1631no—
240#103—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248C22⋊C8—1631no—
249C22⋊C8—1631no—
250C4×C8—1631no—
251C24—3261no—
252#41—3262no—
253#41—6521no—
254C2×Q8—1631no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3262no—
258C2×D4—3261no—
259C2×D4—3261no—
260C2×D4—3261no—
261C2×D4—6521no—
262C2×D4—1631no—
263C2×D4—1631no—
264C2×D4C22⋊D16311yescharacteristic
265C2×D4C22×D16311yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—3261no—
269C22×C4—3261no—
270C22×C4C22×D16311yescharacteristic
271C22×C4C22×D16311yescharacteristic
272SD16—6521no—
273D8—3262no—
274D8—6521no—
275C2×C8—6521no—
276C2×C8—6521no—
277C2×C8—1631no—
278C2×C8—1631no—
279C2×C8—1631no—
280C2×C8—1631no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—3261no—
285C4⋊C4—1631no—
286C4⋊C4C22⋊D16311yescharacteristic
287C4⋊C4C22×D16311yescharacteristic
288C4⋊C4C22×D16311yescharacteristic
289C4⋊C4C22×D16311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4—3261no—
297C22⋊C4—3261no—
298C22⋊C4—3261no—
299C22⋊C4C22⋊D16311yescharacteristic
300C22⋊C4C22⋊D16311yescharacteristic
301C42—21no—
302C42C22×D16311yescharacteristic
303C23—21no—
304C23—13041no—
305C23—3261no—
306C23—3261no—
307C23—6522no—
308C23—6522no—
309C23C2×C22⋊D16311yescharacteristic
310C23C2×C22⋊D16311yescharacteristic
311Q8—3261no—
312Q8—3261no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—13041no—
319D4—3261no—
320D4—3261no—
321D4—3261no—
322D4—6522no—
323D4—6522no—
324D4—1632no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—13041no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—6522no—
338C2×C4C2×C22⋊D16311yescharacteristic
339C2×C4C2×C22⋊D16311yescharacteristic
340C2×C4C2×C22⋊D16311yescharacteristic
341C2×C4C2×C22⋊D16311yescharacteristic
342C2×C4C23×D16311yescharacteristic
343C8—3261no—
344C8—3261no—
345C8—3261no—
346C8—3261no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—13044no—
356C22—3262no—
357C22—6521no—
358C22—6521no—
359C22—6521no—
360C22—6521no—
361C22—6521no—
362C22#1910475512yes—
363C22C22×C22⋊D16311yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—6521no—
372C4—6521no—
373C4#41×D16311yescharacteristic
374C4#1910478111yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—13041no—
379C2—6522no—
380C2#2378230311yescharacteristic
381C2C2×#1910475511yescharacteristic
382C2#2378246811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#553×C163C2
socleC22×C163C22×D4
hypercenter#553D163
hypocenterC163#3297
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. #23782311C2semidirect
2. #23782022C2semidirect
3. #23782000C2semidirect
4. #23781989C2semidirect
5. #23781962C2semidirect
6. #23781899C2semidirect
7. #23781892C2semidirect
8. #23781880C2semidirect
9. #23781808C2semidirect
10. #23781798C2semidirect
11. C4×#14579684C2semidirect
12. D4×C163⋊C8C2semidirect
13. #23781659C2semidirect
14. C2×#19104655C2semidirect
15. #553×C163C2semidirect
18. C2×#14579687C22semidirect
22. C22×C163⋊C8C22semidirect
31. #19104668C22semidirect
32. #19104668C22semidirect
33. #19104666C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. #19104658C22semidirect
36. #19104658C22semidirect
37. #19104658C22semidirect
38. #19104656C22semidirect
46. C22⋊(C163⋊C8)C22semidirect
47. C4×C163⋊C8C22semidirect
49. #125×C163C22semidirect
53. #121×C163C22semidirect
56. C4×D4×C163C22semidirect
57. #117×C163C22semidirect
191. C163#3297semidirect
193. #553D163semidirect
216. #125C2×D163semidirect
225. #121C2×D163semidirect
230. C4×D4C2×D163semidirect
231. #117C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823112
#237820222
#237820002
#237819892
#237819622
#237818992
#237818922
#237818802
#237818082
#237817982
C4×#145796842
D4×C163⋊C82
#237816592
C2×#191046552
#553×C1632
#3297163