Finite Group Databasebeta

← #35226501

Subgroups — #35226501

Order 20864.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782315C211yesmaximal, characteristic, complemented
2#23782309C211yesmaximal, characteristic, complemented
3#23782023C211yesmaximal, characteristic, complemented
4#23781996C211yesmaximal, characteristic, complemented
5#23781982C211yesmaximal, characteristic, complemented
6#23781953C211yesmaximal, characteristic, complemented
7#23781915C211yesmaximal, characteristic, complemented
8#23781892C211yesmaximal, characteristic, complemented
9#23781878C211yesmaximal, characteristic, complemented
10#23781807C211yesmaximal, characteristic, complemented
11#23781777C211yesmaximal, characteristic, complemented
12#23781733C211yesmaximal, characteristic, complemented
13#23781663C211yesmaximal, characteristic, complemented
14C2×#19104656C211yesmaximal, characteristic, complemented
15#447×C163C211yesmaximal, characteristic, complemented
16C2×#14579698—21no—
17C2×#14579698C2211yescharacteristic
18C2×#14579687C2211yescharacteristic, complemented
19C2×#14579682C2211yescharacteristic, complemented
20C22×C163⋊C8C2211yescharacteristic
21#19104734C2211yescharacteristic, complemented
22#19104722C2211yescharacteristic
23#19104717—21no—
24#19104716—21no—
25#19104716—21no—
26#19104714C2211yescharacteristic
27#19104712—21no—
28#19104707—21no—
29#19104706C2211yescharacteristic
30#19104706C2211yescharacteristic, complemented
31#19104704—21no—
32D4×Dic163C2211yescharacteristic
33D4×Dic163C2211yescharacteristic, complemented
34C4×C22⋊D163C2211yescharacteristic
35C2×C4⋊Dic163C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic, complemented
37C2×C22⋊Dic163—21no—
38#19104667C2211yescharacteristic, complemented
39#19104667C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104658C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C22×D4×C163C2211yescharacteristic
53C2×D8×C163C2211yescharacteristic
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59#14579698—22no—
60#14579698—41no—
61#14579698—44no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64C2×C22⋊D163—21no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163—21no—
67C2×C22⋊D163D411yescharacteristic
68C22×Dic163—21no—
69C22×Dic163—21no—
70C22×Dic163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73C2×C4×D163C2311yescharacteristic
74#14579687—41no—
75#14579682—41no—
76C2×C163⋊C8—41no—
77C2×C163⋊C8C2311yescharacteristic
78C2×C163⋊C8C2311yescharacteristic
79C4⋊Dic163—21no—
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92C22⋊Dic163—21no—
93C22⋊Dic163—21no—
94C22⋊Dic163D411yescharacteristic
95(C2×D163)⋊C4D411yescharacteristic
96C4×Dic163C2311yescharacteristic
97C24×C163—21no—
98C2×D4×C163—22no—
99C2×D4×C163—21no—
100C2×D4×C163—21no—
101C2×D4×C163—41no—
102C2×D4×C163C2311yescharacteristic
103C2×D4×C163C2311yescharacteristic
104C2×D4×C163C2311yescharacteristic
105C22×C4×C163C2311yescharacteristic
106D8×C163—22no—
107D8×C163—41no—
108C2×C8×C163C2311yescharacteristic
109C2×C8×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22⋊C4×C163—21no—
112C22×D163C2×D411yescharacteristic
113Dic326—21no—
114Dic326—21no—
115Dic326—42no—
116C22⋊D163—22no—
117C22⋊D163—42no—
118C22⋊D163—42no—
119C22⋊D163—81no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—42no—
125C2×Dic163—42no—
126C2×Dic163—42no—
127C2×Dic163—81no—
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C2×Dic163C2×D411yescharacteristic
131C4×D163—21no—
132C4×D163—21no—
133C4×D163—42no—
134C163⋊C8—21no—
135C163⋊C8—21no—
136C23×C163—21no—
137C23×C163—21no—
138C23×C163—21no—
139C23×C163—42no—
140C23×C163—42no—
141C23×C163—81no—
142C23×C163C2×D411yescharacteristic
143D4×C163—21no—
144D4×C163—21no—
145D4×C163—21no—
146D4×C163—21no—
147D4×C163—21no—
148D4×C163—42no—
149D4×C163—42no—
150D4×C163#4112yescomplemented
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163C2×D411yescharacteristic
155C2×C4×C163C2411yescharacteristic
156C8×C163—21no—
157C8×C163—21no—
158C22×C163—21no—
159C22×C163—21no—
160C22×C163—22no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163—41no—
166C22×C163—41no—
167C22×C163—41no—
168C22×C163—41no—
169C22×C163—84no—
170C22×C163AGL(1,ℤ/8)12yes—
171C22×C163C22×D411yescharacteristic
172C2×D163—21no—
173C2×D163—21no—
174C2×D163—21no—
175C4×C163—21no—
176C4×C163—41no—
177C4×C163C2×#4111yescharacteristic
178C4×C163H(5,𝔽2)11yescharacteristic
179Dic163—21no—
180Dic163—21no—
181Dic163—21no—
182Dic163—41no—
183Dic163—41no—
184Dic163—41no—
185Dic163—41no—
186C2×C163—22no—
187C2×C163—42no—
188C2×C163—81no—
189C2×C163—81no—
190C2×C163#54611yescharacteristic
191C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C163#57711yescharacteristic
193D163—41no—
194C163#330611yescharacteristic, complemented, Sylow
195#3306—1631nomaximal, Sylow
196#547—1631no—
197#546—1631no—
198#489—1631no—
199#483—1631no—
200#481—1631no—
201#476—1631no—
202#469—1631no—
203#465—1631no—
204#463—1631no—
205#449—1631no—
206#447D16311yescharacteristic, complemented
207#442—1631no—
208#435—1631no—
209#417—1631no—
210C2×#102—1631no—
211C2×#41—3261no—
212C2×#41—1631no—
213C22×D4C2×D16311yescharacteristic
214C2×SD16—1631no—
215C2×D8C2×D16311yescharacteristic
216C2×D8C2×D16311yescharacteristic
217C2×M16—1631no—
218C22×C8—1631no—
219C4⋊Q8—1631no—
220#124—1631no—
221#123—3261no—
222#123—3261no—
223#123—3261no—
224C22⋊Q8—3261no—
225C22⋊Q8—1631no—
226#121—3261no—
227#121—1631no—
228#121—1631no—
229#121C2×D16311yescharacteristic
230#120—3261no—
231C4×D4—1631no—
232C4×D4—1631no—
233C4×D4—1631no—
234C2×C4⋊C4—1631no—
235C2×C4⋊C4—1631no—
236C2×C22⋊C4—3261no—
237#106—1631no—
238#106—1631no—
239#106—1631no—
240C4⋊C8—1631no—
241#103—1631no—
242#103—1631no—
243#103—1631no—
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102C2×D16311yescharacteristic
250#102C2×D16311yescharacteristic
251C22⋊C8—1631no—
252C22⋊C8C2×D16311yescharacteristic
253#97—1631no—
254C24—21no—
255#41—3262no—
256#41—6521no—
257#41—6524no—
258C2×Q8—3261no—
259C2×Q8—1631no—
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—3261no—
265C2×D4—3261no—
266C2×D4—3261no—
267C2×D4—1631no—
268C2×D4C22×D16311yescharacteristic
269C2×D4C22×D16311yescharacteristic
270C2×D4C22×D16311yescharacteristic
271C22×C4—3261no—
272C22×C4—3261no—
273C22×C4—3261no—
274C22×C4—3261no—
275C22×C4—3261no—
276C22×C4—1631no—
277C22×C4C22×D16311yescharacteristic
278SD16—6521no—
279D8—22no—
280D8—41no—
281M16—6521no—
282C2×C8—6521no—
283C2×C8—1631no—
284C2×C8—1631no—
285C2×C8C22×D16311yescharacteristic
286C2×C8C22×D16311yescharacteristic
287C4⋊C4—3261no—
288C4⋊C4—3261no—
289C4⋊C4—3261no—
290C4⋊C4—3261no—
291C4⋊C4—1631no—
292C4⋊C4—1631no—
293C4⋊C4—1631no—
294C4⋊C4—1631no—
295C4⋊C4C22×D16311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—3261no—
298C22⋊C4—3261no—
299C22⋊C4—3261no—
300C22⋊C4—3261no—
301C22⋊C4—3261no—
302C22⋊C4—3261no—
303C22⋊C4—3261no—
304C22⋊C4—1631no—
305C22⋊C4—1631no—
306C42—1631no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—42no—
312C23—81no—
313C23—1631no—
314C23D4×D16311yescharacteristic
315Q8—3261no—
316Q8—3261no—
317Q8—6522no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—42no—
324D4—42no—
325D4—13041no—
326D4—3262no—
327D4—6522no—
328D4—6522no—
329D4#1457969812yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—13041no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—3261no—
338C2×C4—3261no—
339C2×C4—3261no—
340C2×C4—6522no—
341C2×C4—6522no—
342C2×C4—6522no—
343C2×C4—6522no—
344C2×C4—1631no—
345C2×C4—1631no—
346C2×C4—1631no—
347C2×C4D4×D16311yescharacteristic
348C2×C4C23×D16311yescharacteristic
349C8—21no—
350C8—21no—
351C8—3261no—
352C8—3261no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—3261no—
366C22—3261no—
367C22—3261no—
368C22#1910475912yes—
369C22C2×D4×D16311yescharacteristic
370C4—21no—
371C4—41no—
372C4—3261no—
373C4—3261no—
374C4—3261no—
375C4—6521no—
376C4—6521no—
377C4—6521no—
378C4—6521no—
379C4C2×#1457969811yescharacteristic
380C4#1910478011yescharacteristic
381C2—22no—
382C2—42no—
383C2—81no—
384C2—81no—
385C2—6521no—
386C2#2378230211yescharacteristic
387C2C2×#1910475911yescharacteristic
388C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#447×C163C2
socleC22×C163C22×D4
hypercenter#447D163
hypocenterC163#3306
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23782315C2semidirect
2. #23782309C2semidirect
3. #23782023C2semidirect
4. #23781996C2semidirect
5. #23781982C2semidirect
6. #23781953C2semidirect
7. #23781915C2semidirect
8. #23781892C2semidirect
9. #23781878C2semidirect
10. #23781807C2semidirect
11. #23781777C2semidirect
12. #23781733C2semidirect
13. #23781663C2semidirect
14. C2×#19104656C2semidirect
15. #447×C163C2semidirect
18. C2×#14579687C22semidirect
19. C2×#14579682C22semidirect
21. #19104734C22semidirect
30. #19104706C22semidirect
33. D4×Dic163C22semidirect
36. C4⋊C4×D163C22semidirect
38. #19104667C22semidirect
39. #19104667C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104660C22semidirect
43. #19104660C22semidirect
44. #19104658C22semidirect
49. #19104656C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
150. D4×C163#41semidirect
194. C163#3306semidirect
206. #447D163semidirect
329. D4#14579698semidirect

Maximal subgroups

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

subgroupindex
#237823152
#237823092
#237820232
#237819962
#237819822
#237819532
#237819152
#237818922
#237818782
#237818072
#237817772
#237817332
#237816632
C2×#191046562
#447×C1632
#3306163