Finite Group Databasebeta

← #35226500

Subgroups — #35226500

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#23782316C211yesmaximal, characteristic, complemented
2#23782312C211yesmaximal, characteristic, complemented
3#23782023C211yesmaximal, characteristic, complemented
4#23781997C211yesmaximal, characteristic, complemented
5#23781954C211yesmaximal, characteristic, complemented
6#23781916C211yesmaximal, characteristic, complemented
7#23781895C211yesmaximal, characteristic, complemented
8#23781879C211yesmaximal, characteristic, complemented
9#23781812C211yesmaximal, characteristic, complemented
10#23781800C211yesmaximal, characteristic, complemented
11#23781776C211yesmaximal, characteristic, complemented
12#23781739C211yesmaximal, characteristic, complemented
13#23781664C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#481×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×#14579699C2211yescharacteristic
18C2×D4×D163C2211yescharacteristic
19C2×#14579688C2211yescharacteristic, complemented
20C2×D1304C2211yescharacteristic
21C2×#14579684C2211yescharacteristic
22C2×#14579683C2211yescharacteristic, complemented
23C2×C8×D163C2211yescharacteristic
24#19104734C2211yescharacteristic, complemented
25#19104722C2211yescharacteristic
26#19104720—21no—
27#19104718—21no—
28#19104717—21no—
29#19104715C2211yescharacteristic
30#19104713—21no—
31#19104710C2211yescharacteristic
32#19104709C2211yescharacteristic, complemented
33#19104706—21no—
34#19104703—21no—
35#19104697C2211yescharacteristic
36#19104697C2211yescharacteristic, complemented
37C4×C22⋊D163C2211yescharacteristic
38C4⋊C4×D163C2211yescharacteristic
39C22⋊C4×D163—21no—
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104658C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49#19104655C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic, complemented
52#121×C163C2211yescharacteristic
53C2×C4⋊C4×C163C2211yescharacteristic, complemented
54#106×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62#14579699—44no—
63C2×Dic326—21no—
64C2×Dic326C2311yescharacteristic
65C2×D652—21no—
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163D411yescharacteristic
72D4×D163—22no—
73D4×D163—41no—
74C2×C4×D163—21no—
75C2×C4×D163—21no—
76C2×C4×D163—21no—
77C2×C4×D163—21no—
78C2×C4×D163C2311yescharacteristic
79#14579688—41no—
80D1304—22no—
81#14579684—41no—
82#14579683—41no—
83C2×C163⋊C8C2311yescharacteristic
84C2×C163⋊C8C2311yescharacteristic
85C8×D163—41no—
86C4⋊Dic163—21no—
87C4⋊Dic163C2311yescharacteristic
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4D411yescharacteristic
91C22⋊Dic163—21no—
92C22⋊Dic163—21no—
93C22⋊Dic163D411yescharacteristic
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4—21no—
99(C2×D163)⋊C4D411yescharacteristic
100C4×Dic163C2311yescharacteristic
101C2×D4×C163—21no—
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163—21no—
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22⋊C4×C163—21no—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163—42no—
115C22×D163—42no—
116C22×D163—81no—
117C22×D163C2×D411yescharacteristic
118Dic326—21no—
119Dic326—21no—
120Dic326—42no—
121D652—21no—
122D652—21no—
123D652—21no—
124D652—42no—
125D652#4112yescomplemented
126C22⋊D163—22no—
127C22⋊D163—42no—
128C22⋊D163—42no—
129C22⋊D163—42no—
130C2×Dic163—21no—
131C2×Dic163—21no—
132C2×Dic163C2×D411yescharacteristic
133C2×Dic163C2×D411yescharacteristic
134C2×Dic163C2×D411yescharacteristic
135C4×D163—21no—
136C4×D163—21no—
137C4×D163—42no—
138C4×D163—42no—
139C4×D163—42no—
140C4×D163—81no—
141C163⋊C8—21no—
142C163⋊C8—21no—
143C23×C163—21no—
144C23×C163C2×D411yescharacteristic
145D4×C163—21no—
146D4×C163—21no—
147D4×C163—81no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—42no—
154C2×C4×C163C2×D411yescharacteristic
155C2×C4×C163C2411yescharacteristic
156C8×C163—21no—
157C8×C163—21no—
158C22×C163—21no—
159C22×C163—21no—
160C22×C163—21no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163—41no—
164C22×C163C22×D411yescharacteristic
165C2×D163—21no—
166C2×D163—21no—
167C2×D163—22no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—41no—
171C2×D163—41no—
172C2×D163—41no—
173C2×D163—84no—
174C2×D163AGL(1,ℤ/8)12yes—
175C4×C163—21no—
176C4×C163—41no—
177C4×C163—41no—
178C4×C163—41no—
179C4×C163C2×#4111yescharacteristic
180C4×C163H(5,𝔽2)11yescharacteristic
181Dic163—21no—
182Dic163—21no—
183Dic163—21no—
184Dic163—41no—
185Dic163—41no—
186C2×C163—41no—
187C2×C163—81no—
188C2×C163#54611yescharacteristic
189C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C163#57711yescharacteristic
191D163—22no—
192D163—42no—
193D163—81no—
194C163#330611yescharacteristic, complemented, Sylow
195#3306—1631nomaximal, Sylow
196#547—1631no—
197#546—1631no—
198#489—1631no—
199#483—1631no—
200#481D16311yescharacteristic, complemented
201#476—1631no—
202#469—1631no—
203#465—1631no—
204#463—1631no—
205#449—1631no—
206#447—1631no—
207#442—1631no—
208#435—1631no—
209#417—1631no—
210C2×#102—1631no—
211C2×#41—3261no—
212C2×#41—1631no—
213C22×D4—1631no—
214C2×SD16—1631no—
215C2×D8—1631no—
216C2×D8—1631no—
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⋊C4C2×D16311yescharacteristic, complemented
236C2×C22⋊C4—3261no—
237#106—1631no—
238#106C2×D16311yescharacteristic, complemented
239#106C2×D16311yescharacteristic, complemented
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, complemented
253#97—1631no—
254C24—3261no—
255#41—3262no—
256#41—6521no—
257#41—6524no—
258C2×Q8—3261no—
259C2×Q8—1631no—
260C2×D4—21no—
261C2×D4—3261no—
262C2×D4—3262no—
263C2×D4—3261no—
264C2×D4—3261no—
265C2×D4—3261no—
266C2×D4—6521no—
267C2×D4—1631no—
268C2×D4—1631no—
269C2×D4—1631no—
270C2×D4C22×D16311yescharacteristic
271C22×C4—21no—
272C22×C4—3261no—
273C22×C4—3261no—
274C22×C4—3261no—
275C22×C4—3261no—
276C22×C4—1631no—
277C22×C4C22×D16311yescharacteristic
278SD16—6521no—
279D8—3262no—
280D8—6521no—
281M16—6521no—
282C2×C8—6521no—
283C2×C8—1631no—
284C2×C8—1631no—
285C2×C8C22×D16311yescharacteristic
286C2×C8C22×D16311yescharacteristic
287C4⋊C4—21no—
288C4⋊C4—3261no—
289C4⋊C4—3261no—
290C4⋊C4—3261no—
291C4⋊C4—1631no—
292C4⋊C4—1631no—
293C4⋊C4C22×D16311yescharacteristic
294C4⋊C4C22×D16311yescharacteristic
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—13041no—
309C23—3261no—
310C23—3261no—
311C23—6522no—
312C23—6522no—
313C23—1631no—
314C23D4×D16311yescharacteristic
315Q8—3261no—
316Q8—3261no—
317Q8—6522no—
318D4—21no—
319D4—21no—
320D4—81no—
321D4—3262no—
322D4—3261no—
323D4—3261no—
324D4—3261no—
325D4—6522no—
326D4—6522no—
327D4—6522no—
328D4—6522no—
329D4—1632no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—13041no—
337C2×C4—3261no—
338C2×C4—3261no—
339C2×C4—3261no—
340C2×C4—3261no—
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—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—13044no—
360C22—3261no—
361C22—3261no—
362C22—3262no—
363C22—6521no—
364C22—6521no—
365C22—6521no—
366C22—6521no—
367C22—6521no—
368C22—1632no—
369C22C2×D4×D16311yescharacteristic
370C4—21no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—3261no—
375C4—3261no—
376C4—3261no—
377C4—6521no—
378C4—6521no—
379C4#41×D16311yescharacteristic
380C4#1910478111yescharacteristic
381C2—41no—
382C2—81no—
383C2—13041no—
384C2—3262no—
385C2—6522no—
386C2#2378230711yescharacteristic
387C2AGL(1,ℤ/8)×D16311yescharacteristic
388C2#2378247111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#481×C163C2
socleC22×C163C22×D4
hypercenter#481D163
hypocenterC163#3306
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #23782316C2semidirect
2. #23782312C2semidirect
3. #23782023C2semidirect
4. #23781997C2semidirect
5. #23781954C2semidirect
6. #23781916C2semidirect
7. #23781895C2semidirect
8. #23781879C2semidirect
9. #23781812C2semidirect
10. #23781800C2semidirect
11. #23781776C2semidirect
12. #23781739C2semidirect
13. #23781664C2semidirect
14. #102×D163C2semidirect
15. #481×C163C2semidirect
19. C2×#14579688C22semidirect
22. C2×#14579683C22semidirect
24. #19104734C22semidirect
32. #19104709C22semidirect
36. #19104697C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104660C22semidirect
43. #19104658C22semidirect
44. #19104658C22semidirect
49. #19104655C22semidirect
51. #19104645C22semidirect
53. C2×C4⋊C4×C163C22semidirect
54. #106×C163C22semidirect
55. #106×C163C22semidirect
58. C22⋊C8×C163C22semidirect
125. D652#41semidirect
194. C163#3306semidirect
200. #481D163semidirect
235. C2×C4⋊C4C2×D163semidirect
238. #106C2×D163semidirect
239. #106C2×D163semidirect
252. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823162
#237823122
#237820232
#237819972
#237819542
#237819162
#237818952
#237818792
#237818122
#237818002
#237817762
#237817392
#237816642
#102×D1632
#481×C1632
#3306163