Finite Group Databasebeta

← #35226353

Subgroups — #35226353

Order 20864.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782373C211yesmaximal, characteristic, complemented
2D4×C22⋊D163C211yesmaximal, characteristic
3#23782010C211yesmaximal, characteristic, complemented
4#23781984C211yesmaximal, characteristic, complemented
5#23781982C211yesmaximal, characteristic, complemented
6#23781953C211yesmaximal, characteristic, complemented
7#23781896C211yesmaximal, characteristic, complemented
8#23781891C211yesmaximal, characteristic, complemented
9#23781863C211yesmaximal, characteristic, complemented
10#23781814C211yesmaximal, characteristic, complemented
11SD16×Dic163C211yesmaximal, characteristic, complemented
12(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
13C2×#19104656C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#450×C163C211yesmaximal, characteristic, complemented
16C22×C22⋊D163—21no—
17C2×D4×D163C2211yescharacteristic
18C2×#14579686C2211yescharacteristic, complemented
19C22×C163⋊C8C2211yescharacteristic, complemented
20C2×C8×D163C2211yescharacteristic, complemented
21#19104734C2211yescharacteristic, complemented
22#19104731C2211yescharacteristic
23#19104725—21no—
24#19104715—21no—
25#19104712—21no—
26#19104711C2211yescharacteristic, complemented
27#19104709C2211yescharacteristic
28#19104707—21no—
29#19104706C2211yescharacteristic
30#19104704—21no—
31#19104703—21no—
32Q8×Dic163C2211yescharacteristic, complemented
33D4×Dic163C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35(C22⋊D163)⋊C4—21no—
36C2×C4⋊Dic163C2211yescharacteristic, complemented
37C4⋊C4×D163C2211yescharacteristic, complemented
38#19104667C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic, complemented
48#19104656C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic
54C2×SD16×C163C2211yescharacteristic, complemented
55C22⋊Q8×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×Dic326—21no—
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163—44no—
68C2×C22⋊D163D411yescharacteristic
69D4×D163—22no—
70D4×D163—41no—
71C22×Dic163—21no—
72C22×Dic163—21no—
73C2×C4×D163—21no—
74C2×C4×D163C2311yescharacteristic
75#14579686—41no—
76C2×C163⋊C8—41no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C8×D163—41no—
80C4⋊Dic163—21no—
81C4⋊Dic163—21no—
82C4⋊Dic163C2311yescharacteristic
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4D411yescharacteristic
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163D411yescharacteristic
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4D411yescharacteristic
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C24×C163—21no—
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163—22no—
100C2×D4×C163—21no—
101C2×D4×C163—41no—
102C2×D4×C163C2311yescharacteristic
103C2×D4×C163C2311yescharacteristic
104C22×C4×C163C2311yescharacteristic
105SD16×C163—22no—
106SD16×C163—41no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163—21no—
110C4⋊C4×C163C2311yescharacteristic
111C22⋊C4×C163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117Dic326—81no—
118D652—21no—
119D652—21no—
120C22⋊D163—42no—
121C22⋊D163—42no—
122C22⋊D163—42no—
123C22⋊D163—42no—
124C22⋊D163—81no—
125C22⋊D163SD1614yescomplemented
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—42no—
130C2×Dic163—81no—
131C2×Dic163C2×D411yescharacteristic
132C2×Dic163C2×D411yescharacteristic
133C2×Dic163C2×D411yescharacteristic
134C4×D163—21no—
135C4×D163—21no—
136C4×D163—81no—
137C163⋊C8—21no—
138C163⋊C8—21no—
139C23×C163—21no—
140C23×C163—21no—
141C23×C163—42no—
142C23×C163—42no—
143C23×C163—81no—
144C23×C163C2×D411yescharacteristic
145Q8×C163—21no—
146Q8×C163—21no—
147D4×C163—21no—
148D4×C163—21no—
149D4×C163—21no—
150D4×C163—42no—
151D4×C163#4112yes—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
156C2×C4×C163C2×D411yescharacteristic
157C2×C4×C163C2411yescharacteristic
158C8×C163—21no—
159C8×C163—21no—
160C22×C163—21no—
161C22×C163—21no—
162C22×C163—22no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—84no—
169C22×C163C2×SD1612yes—
170C22×C163C22×D411yescharacteristic
171C2×D163—21no—
172C2×D163—21no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—41no—
176C2×D163—84no—
177C2×D163C2×SD1612yes—
178C4×C163—21no—
179C4×C163—41no—
180C4×C163—41no—
181C4×C163C2×#4111yescharacteristic
182C4×C163H(5,𝔽2)11yescharacteristic
183Dic163—21no—
184Dic163—21no—
185Dic163—41no—
186Dic163—41no—
187Dic163—41no—
188Dic163C2×SD1612yes—
189C2×C163—22no—
190C2×C163—42no—
191C2×C163—81no—
192C2×C163#54611yescharacteristic
193C2×C163C22×SD1611yescharacteristic
194C2×C163#57611yescharacteristic
195D163—22no—
196D163—81no—
197C163#329511yescharacteristic, complemented, Sylow
198#3295—1631nomaximal, Sylow
199#554—1631no—
200D42—1631no—
201#486—1631no—
202#481—1631no—
203#481—1631no—
204#476—1631no—
205#465—1631no—
206#465—1631no—
207#460—1631no—
208#450—1631no—
209#450D16311yescharacteristic, complemented
210C4×SD16—1631no—
211C8×D4—1631no—
212C2×#102—1631no—
213C2×#102—1631no—
214C22×D4—3261no—
215C22×D4—1631no—
216C22×D4C2×D16311yescharacteristic
217C2×SD16—1631no—
218C2×SD16C2×D16311yescharacteristic
219C2×SD16C2×D16311yescharacteristic, complemented
220C22×C8—1631no—
221C22×C8—1631no—
222C4⋊Q8—1631no—
223#127—1631no—
224#125—3261no—
225C22⋊Q8—3261no—
226C22⋊Q8—3261no—
227C22⋊Q8—1631no—
228C22⋊Q8C2×D16311yescharacteristic, complemented
229#121—3261no—
230#121—1631no—
231#121—1631no—
232#120—3261no—
233#120—3261no—
234C4×Q8—1631no—
235C4×D4—3261no—
236C4×D4—1631no—
237C4×D4—1631no—
238C2×C4⋊C4—1631no—
239C2×C4⋊C4—1631no—
240#106—1631no—
241#106—1631no—
242#106—1631no—
243C4⋊C8—1631no—
244#103—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102—1631no—
250#102—1631no—
251#102—1631no—
252#102C2×D16311yescharacteristic
253#102C2×D16311yescharacteristic, complemented
254C22⋊C8—1631no—
255C22⋊C8C2×D16311yescharacteristic, complemented
256C4×C8—1631no—
257C24—21no—
258C24—3261no—
259C2×Q8—3261no—
260C2×Q8C22×D16311yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—3262no—
265C2×D4—3261no—
266C2×D4—3261no—
267C2×D4—3261no—
268C2×D4—3261no—
269C2×D4—3261no—
270C2×D4—6521no—
271C2×D4—6524no—
272C2×D4—1631no—
273C2×D4—1631no—
274C2×D4C22×D16311yescharacteristic
275C2×D4C22×D16311yescharacteristic
276C22×C4—3261no—
277C22×C4—3261no—
278C22×C4—3261no—
279C22×C4—1631no—
280C22×C4C22×D16311yescharacteristic
281SD16—22no—
282SD16—41no—
283SD16—6521no—
284C2×C8—6521no—
285C2×C8—6521no—
286C2×C8—1631no—
287C2×C8—1631no—
288C2×C8C22×D16311yescharacteristic
289C2×C8C22×D16311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3261no—
292C4⋊C4—3261no—
293C4⋊C4—3261no—
294C4⋊C4—3261no—
295C4⋊C4—3261no—
296C4⋊C4—1631no—
297C4⋊C4—1631no—
298C4⋊C4—1631no—
299C4⋊C4—1631no—
300C4⋊C4C22×D16311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—3261no—
303C22⋊C4—3261no—
304C22⋊C4—3261no—
305C22⋊C4—3261no—
306C22⋊C4—1631no—
307C22⋊C4—1631no—
308C42—3261no—
309C42—1631no—
310C23—21no—
311C23—21no—
312C23—42no—
313C23—42no—
314C23—81no—
315C23—13041no—
316C23—3261no—
317C23—6522no—
318C23—6522no—
319C23—1631no—
320C23D4×D16311yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—13041no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—42no—
328D4—13041no—
329D4—3261no—
330D4—3261no—
331D4—6522no—
332D4—6522no—
333D4—6522no—
334D4—6522no—
335D4—1634no—
336D4#1457969812yes—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—13041no—
342C2×C4—13041no—
343C2×C4—3261no—
344C2×C4—3261no—
345C2×C4—3261no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—6522no—
349C2×C4—1631no—
350C2×C4—1631no—
351C2×C4—1631no—
352C2×C4D4×D16311yescharacteristic
353C2×C4C23×D16311yescharacteristic
354C8—21no—
355C8—21no—
356C8—3261no—
357C8—3261no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—84no—
367C22—13044no—
368C22—3261no—
369C22—3261no—
370C22—6521no—
371C22—6521no—
372C22—6521no—
373C22—1632no—
374C22SD16×D16312yes—
375C22C2×D4×D16311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—3261no—
380C4—3261no—
381C4—6521no—
382C4—6521no—
383C4—6521no—
384C4—1632no—
385C4C2×#1457969811yescharacteristic
386C4#1910478011yescharacteristic
387C2—22no—
388C2—42no—
389C2—81no—
390C2—13041no—
391C2—3262no—
392C2#2378230211yescharacteristic
393C2C2×SD16×D16311yescharacteristic
394C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#450×C163C2
socleC22×C163C22×D4
prime radicalD4×C22⋊D163C2
hypercenter#450D163
hypocenterC163#3295
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23782373C2semidirect
3. #23782010C2semidirect
4. #23781984C2semidirect
5. #23781982C2semidirect
6. #23781953C2semidirect
7. #23781896C2semidirect
8. #23781891C2semidirect
9. #23781863C2semidirect
10. #23781814C2semidirect
11. SD16×Dic163C2semidirect
12. (C22⋊D163)⋊C8C2semidirect
13. C2×#19104656C2semidirect
14. #102×D163C2semidirect
15. #450×C163C2semidirect
18. C2×#14579686C22semidirect
19. C22×C163⋊C8C22semidirect
20. C2×C8×D163C22semidirect
21. #19104734C22semidirect
26. #19104711C22semidirect
32. Q8×Dic163C22semidirect
36. C2×C4⋊Dic163C22semidirect
37. C4⋊C4×D163C22semidirect
38. #19104667C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104659C22semidirect
43. #19104657C22semidirect
44. #19104657C22semidirect
47. #19104656C22semidirect
48. #19104656C22semidirect
49. #19104655C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
54. C2×SD16×C163C22semidirect
55. C22⋊Q8×C163C22semidirect
57. #102×C163C22semidirect
58. C22⋊C8×C163C22semidirect
125. C22⋊D163SD16semidirect
197. C163#3295semidirect
209. #450D163semidirect
219. C2×SD16C2×D163semidirect
228. C22⋊Q8C2×D163semidirect
253. #102C2×D163semidirect
255. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823732
D4×C22⋊D1632
#237820102
#237819842
#237819822
#237819532
#237818962
#237818912
#237818632
#237818142
SD16×Dic1632
(C22⋊D163)⋊C82
C2×#191046562
#102×D1632
#450×C1632
#3295163