Finite Group Databasebeta

← #35226383

Subgroups — #35226383

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#23782368C211yesmaximal, characteristic, complemented
2#23782310C211yesmaximal, characteristic, complemented
3#23782022C211yesmaximal, characteristic, complemented
4#23782002C211yesmaximal, characteristic, complemented
5#23781991C211yesmaximal, characteristic, complemented
6#23781902C211yesmaximal, characteristic, complemented
7#23781895C211yesmaximal, characteristic, complemented
8#23781881C211yesmaximal, characteristic, complemented
9#23781811C211yesmaximal, characteristic, complemented
10#23781800C211yesmaximal, characteristic, complemented
11C4×#14579684C211yesmaximal, characteristic, complemented
12#23781730C211yesmaximal, characteristic, complemented
13#23781658C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#478×C163C211yesmaximal, characteristic, complemented
16C2×#14579698C2211yescharacteristic
17C2×D4×D163C2211yescharacteristic
18C2×#14579688C2211yescharacteristic, complemented
19C2×D1304C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C2×C8×D163C2211yescharacteristic
22C2×C8×D163C2211yescharacteristic, complemented
23#19104730—21no—
24#19104723C2211yescharacteristic
25#19104720—21no—
26#19104719—21no—
27#19104715C2211yescharacteristic
28#19104710—21no—
29#19104710C2211yescharacteristic
30#19104710C2211yescharacteristic
31#19104710C2211yescharacteristic, complemented
32#19104709—21no—
33#19104705—21no—
34C4×D652C2211yescharacteristic
35C4×D652C2211yescharacteristic, complemented
36#19104697—21no—
37#19104693C2211yescharacteristic, complemented
38C4⋊C4×D163C2211yescharacteristic
39C2×(C2×D163)⋊C4—21no—
40#19104668C2211yescharacteristic, complemented
41#19104660C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic, complemented
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C4×C163⋊C8C2211yescharacteristic, complemented
52#125×C163C2211yescharacteristic, complemented
53C4×D4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56C4⋊C8×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C23×D163—21no—
60#14579698—22no—
61#14579698—41no—
62C2×Dic326C2311yescharacteristic
63C2×D652—21no—
64C2×D652—21no—
65C2×D652—21no—
66C2×D652D411yescharacteristic
67C2×D652C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71D4×D163—22no—
72D4×D163—41no—
73C22×Dic163—21no—
74C2×C4×D163—21no—
75C2×C4×D163—21no—
76C2×C4×D163C2311yescharacteristic
77C2×C4×D163C2311yescharacteristic
78#14579688—41no—
79D1304—41no—
80#14579684—22no—
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C8×D163—41no—
84C8×D163—41no—
85C4⋊Dic163—21no—
86C4⋊Dic163D411yescharacteristic
87C4⋊Dic163C2311yescharacteristic
88Dic163⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4D411yescharacteristic
98(C2×D163)⋊C4D411yescharacteristic
99C4×Dic163—21no—
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163—21no—
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C4⋊C4×C163—21no—
105C4⋊C4×C163—21no—
106C4⋊C4×C163C2311yescharacteristic
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117C22×D163C2×D411yescharacteristic
118Dic326—21no—
119Dic326—21no—
120D652—22no—
121D652—21no—
122D652—21no—
123D652—21no—
124D652—21no—
125D652—42no—
126D652—81no—
127D652—81no—
128C22⋊D163—42no—
129C22⋊D163—42no—
130C2×Dic163—21no—
131C2×Dic163—21no—
132C2×Dic163—42no—
133C2×Dic163C2×D411yescharacteristic
134C2×Dic163C2×D411yescharacteristic
135C4×D163—21no—
136C4×D163—21no—
137C4×D163—21no—
138C4×D163—21no—
139C4×D163—81no—
140C4×D163—81no—
141C163⋊C8—21no—
142C163⋊C8—21no—
143C23×C163—21no—
144D4×C163—21no—
145D4×C163#4112yes—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—81no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—22no—
156C22×C163—41no—
157C22×C163—41no—
158C22×C163C22×D411yescharacteristic
159C2×D163—21no—
160C2×D163—21no—
161C2×D163—21no—
162C2×D163—21no—
163C2×D163—21no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—41no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—84no—
171C2×D163#13512yes—
172C4×C163—21no—
173C4×C163—21no—
174C4×C163—41no—
175C4×C163—41no—
176C4×C163—41no—
177C4×C163C2×#4111yescharacteristic
178C4×C163H(5,𝔽2)11yescharacteristic
179Dic163—21no—
180Dic163—21no—
181Dic163—41no—
182Dic163—41no—
183C2×C163—42no—
184C2×C163#54611yescharacteristic
185C2×C163C2×#13511yescharacteristic
186C2×C163#57611yescharacteristic
187D163—22no—
188D163—41no—
189D163—81no—
190D163—81no—
191C163#329711yescharacteristic, complemented, Sylow
192#3297—1631nomaximal, Sylow
193#553—1631no—
194#546—1631no—
195#488—1631no—
196#484—1631no—
197#482—1631no—
198#478D16311yescharacteristic, complemented
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—3261no—
216#125C2×D16311yescharacteristic, complemented
217#124—1631no—
218#123—3261no—
219#123—3261no—
220C22⋊Q8—1631no—
221#121—3261no—
222#121—3261no—
223#121—1631no—
224#121—1631no—
225#121—1631no—
226#120—3261no—
227C4×D4—3261no—
228C4×D4—1631no—
229C4×D4—1631no—
230C4×D4C2×D16311yescharacteristic
231#117—1631no—
232C2×C4⋊C4—1631no—
233C2×C22⋊C4—3261no—
234#107—1631no—
235#107C2×D16311yescharacteristic, complemented
236#106C2×D16311yescharacteristic, complemented
237C4⋊C8C2×D16311yescharacteristic, complemented
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#102C2×D16311yescharacteristic
247#102C2×D16311yescharacteristic
248C22⋊C8—1631no—
249C22⋊C8—1631no—
250C4×C8—1631no—
251C24—3261no—
252#41—3262no—
253#41—6521no—
254C2×Q8—1631no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—3261no—
258C2×D4—3262no—
259C2×D4—3261no—
260C2×D4—3261no—
261C2×D4—6521no—
262C2×D4—1631no—
263C2×D4—1631no—
264C2×D4—1631no—
265C2×D4C22×D16311yescharacteristic
266C22×C4—21no—
267C22×C4—3261no—
268C22×C4—3261no—
269C22×C4—3261no—
270C22×C4—1631no—
271C22×C4—1631no—
272SD16—6521no—
273D8—3262no—
274D8—6521no—
275C2×C8—6521no—
276C2×C8—6521no—
277C2×C8—1631no—
278C2×C8—1631no—
279C2×C8C22×D16311yescharacteristic
280C2×C8C22×D16311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—3261no—
284C4⋊C4—3261no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4C22×D16311yescharacteristic
288C4⋊C4C22×D16311yescharacteristic
289C4⋊C4C22×D16311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—3261no—
296C22⋊C4—3261no—
297C22⋊C4—3261no—
298C22⋊C4—3261no—
299C22⋊C4—1631no—
300C22⋊C4—1631no—
301C42—3261no—
302C42C22×D16311yescharacteristic
303C23—21no—
304C23—13041no—
305C23—3261no—
306C23—3261no—
307C23—6522no—
308C23—6522no—
309C23—1631no—
310C23—1631no—
311Q8—3261no—
312Q8—3261no—
313D4—21no—
314D4—13041no—
315D4—13041no—
316D4—3262no—
317D4—3261no—
318D4—3261no—
319D4—3261no—
320D4—3261no—
321D4—6522no—
322D4—6522no—
323D4—6522no—
324D4#1457970012yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—81no—
329C2×C4—13041no—
330C2×C4—13041no—
331C2×C4—3261no—
332C2×C4—3261no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—6522no—
338C2×C4—1631no—
339C2×C4—1631no—
340C2×C4D4×D16311yescharacteristic
341C2×C4D4×D16311yescharacteristic
342C2×C4C23×D16311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3261no—
346C8—3261no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—13044no—
351C22—3261no—
352C22—3261no—
353C22—3261no—
354C22—3261no—
355C22—3261no—
356C22—6521no—
357C22—6521no—
358C22—6521no—
359C22—6521no—
360C22—6521no—
361C22—6521no—
362C22—1632no—
363C22C2×D4×D16311yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—3261no—
370C4—3261no—
371C4—6521no—
372C4—6521no—
373C4C2×#1457970011yescharacteristic
374C4#1910478111yescharacteristic
375C2—42no—
376C2—13041no—
377C2—13041no—
378C2—3262no—
379C2—6521no—
380C2#2378230611yescharacteristic
381C2#135×D16311yescharacteristic
382C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#478×C163C2
socleC22×C163C22×D4
hypercenter#478D163
hypocenterC163#3297
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #23782368C2semidirect
2. #23782310C2semidirect
3. #23782022C2semidirect
4. #23782002C2semidirect
5. #23781991C2semidirect
6. #23781902C2semidirect
7. #23781895C2semidirect
8. #23781881C2semidirect
9. #23781811C2semidirect
10. #23781800C2semidirect
11. C4×#14579684C2semidirect
12. #23781730C2semidirect
13. #23781658C2semidirect
14. #102×D163C2semidirect
15. #478×C163C2semidirect
18. C2×#14579688C22semidirect
22. C2×C8×D163C22semidirect
31. #19104710C22semidirect
35. C4×D652C22semidirect
37. #19104693C22semidirect
40. #19104668C22semidirect
41. #19104660C22semidirect
42. #19104658C22semidirect
43. #19104658C22semidirect
48. #19104655C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C4×C163⋊C8C22semidirect
52. #125×C163C22semidirect
54. #107×C163C22semidirect
55. #106×C163C22semidirect
56. C4⋊C8×C163C22semidirect
191. C163#3297semidirect
198. #478D163semidirect
216. #125C2×D163semidirect
235. #107C2×D163semidirect
236. #106C2×D163semidirect
237. C4⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823682
#237823102
#237820222
#237820022
#237819912
#237819022
#237818952
#237818812
#237818112
#237818002
C4×#145796842
#237817302
#237816582
#102×D1632
#478×C1632
#3297163