Finite Group Databasebeta

← #35226379

Subgroups — #35226379

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#23782362C211yesmaximal, characteristic, complemented
2#23782303C211yesmaximal, characteristic, complemented
3#23782021C211yesmaximal, characteristic, complemented
4#23781999C211yesmaximal, characteristic, complemented
5#23781990C211yesmaximal, characteristic, complemented
6#23781967C211yesmaximal, characteristic, complemented
7#23781899C211yesmaximal, characteristic, complemented
8#23781891C211yesmaximal, characteristic, complemented
9#23781882C211yesmaximal, characteristic, complemented
10#23781797C211yesmaximal, characteristic, complemented
11D8×Dic163C211yesmaximal, characteristic, complemented
12(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
13#23781661C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#449×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C2×#14579684C2211yescharacteristic
18C22×C163⋊C8C2211yescharacteristic, complemented
19C2×C8×D163C2211yescharacteristic
20#19104728—21no—
21#19104726C2211yescharacteristic, complemented
22#19104724C2211yescharacteristic
23#19104718—21no—
24#19104716—21no—
25#19104711C2211yescharacteristic
26#19104709C2211yescharacteristic
27#19104707—21no—
28#19104706—21no—
29#19104706C2211yescharacteristic
30#19104703—21no—
31D4×Dic163C2211yescharacteristic
32D4×Dic163C2211yescharacteristic, complemented
33C4×C22⋊D163C2211yescharacteristic
34(C22⋊D163)⋊C4—21no—
35#19104694C2211yescharacteristic, complemented
36C4⋊C4×D163C2211yescharacteristic
37C2×(C2×D163)⋊C4—21no—
38#19104669C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, complemented
40#19104667C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104659C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic, complemented
52C2×#41×C163C2211yescharacteristic
53C2×SD16×C163C2211yescharacteristic, complemented
54C2×D8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic, complemented
56#103×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×D652C2311yescharacteristic
61C2×C22⋊D163—21no—
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163D411yescharacteristic
65D4×D163—22no—
66D4×D163—41no—
67C22×Dic163—21no—
68C22×Dic163—21no—
69C2×C4×D163—21no—
70C2×C4×D163C2311yescharacteristic
71#14579684—41no—
72C2×C163⋊C8—41no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C8×D163—41no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4D411yescharacteristic
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163D411yescharacteristic
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96#41×C163—22no—
97#41×C163—41no—
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163—21no—
100C2×D4×C163—21no—
101C2×D4×C163C2311yescharacteristic
102C2×D4×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163C2311yescharacteristic
105SD16×C163—41no—
106D8×C163—22no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163—42no—
113C22×D163—42no—
114C22×D163—81no—
115C22×D163C2×D411yescharacteristic
116D652—21no—
117D652—21no—
118C22⋊D163—22no—
119C22⋊D163—42no—
120C22⋊D163—42no—
121C22⋊D163—81no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—81no—
126C2×Dic163—81no—
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C4×D163—21no—
131C4×D163—21no—
132C4×D163—81no—
133C163⋊C8—21no—
134C163⋊C8—21no—
135C23×C163—21no—
136C23×C163—21no—
137C23×C163C2×D411yescharacteristic
138Q8×C163—21no—
139Q8×C163—21no—
140D4×C163—21no—
141D4×C163—21no—
142D4×C163—21no—
143D4×C163—42no—
144D4×C163—81no—
145D4×C163#4112yes—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—42no—
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—21no—
156C22×C163—22no—
157C22×C163—21no—
158C22×C163—21no—
159C22×C163—41no—
160C22×C163—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163—41no—
164C22×C163C22×D411yescharacteristic
165C2×D163—21no—
166C2×D163—21no—
167C2×D163—41no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—84no—
171C2×D163#13512yes—
172C4×C163—21no—
173C4×C163—41no—
174C4×C163—41no—
175C4×C163C2×#4111yescharacteristic
176C4×C163H(5,𝔽2)11yescharacteristic
177Dic163—21no—
178Dic163—21no—
179Dic163—21no—
180Dic163—41no—
181Dic163—41no—
182Dic163—41no—
183C2×C163—42no—
184C2×C163—41no—
185C2×C163—81no—
186C2×C163#54611yescharacteristic
187C2×C163C2×#13511yescharacteristic
188C2×C163#57611yescharacteristic
189D163—22no—
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#478—1631no—
199#466—1631no—
200#465—1631no—
201#463—1631no—
202#449D16311yescharacteristic, complemented
203#447—1631no—
204C4×D8—1631no—
205C8×D4—1631no—
206#416—1631no—
207C2×#102—1631no—
208C2×#41C2×D16311yescharacteristic
209C22×D4—1631no—
210C2×SD16C2×D16311yescharacteristic, complemented
211C2×D8—1631no—
212C2×D8C2×D16311yescharacteristic
213C22×C8—1631no—
214C22×C8—1631no—
215#126—3261no—
216#125—1631no—
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#121C2×D16311yescharacteristic, complemented
226#120—3261no—
227C4×D4—3261no—
228C4×D4—1631no—
229C4×D4—1631no—
230C4×D4—1631no—
231#117—1631no—
232C2×C4⋊C4—1631no—
233C2×C22⋊C4—3261no—
234#107—1631no—
235#107—1631no—
236#106—1631no—
237C4⋊C8—1631no—
238#103—1631no—
239#103—1631no—
240#103C2×D16311yescharacteristic, complemented
241#102—1631no—
242#102—1631no—
243#102—1631no—
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102C2×D16311yescharacteristic
248C22⋊C8—1631no—
249C22⋊C8C2×D16311yescharacteristic, complemented
250C4×C8—1631no—
251C24—3261no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D16311yescharacteristic
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—3261no—
258C2×D4—3262no—
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—3261no—
268C22×C4—3261no—
269C22×C4—3261no—
270C22×C4—1631no—
271C22×C4C22×D16311yescharacteristic
272SD16—41no—
273D8—22no—
274D8—6521no—
275C2×C8—6521no—
276C2×C8—6521no—
277C2×C8—1631no—
278C2×C8—1631no—
279C2×C8C22×D16311yescharacteristic
280C2×C8C22×D16311yescharacteristic
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—3261no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4—1631no—
288C4⋊C4—1631no—
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—
302C42—1631no—
303C23—21no—
304C23—21no—
305C23—13041no—
306C23—3261no—
307C23—6522no—
308C23—6522no—
309C23—1631no—
310C23D4×D16311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—13041no—
319D4—3262no—
320D4—3261no—
321D4—3261no—
322D4—6522no—
323D4—6522no—
324D4#1457969812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—42no—
330C2×C4—13041no—
331C2×C4—13041no—
332C2×C4—13041no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—3261no—
338C2×C4—1631no—
339C2×C4—1631no—
340C2×C4—1631no—
341C2×C4D4×D16311yescharacteristic
342C2×C4C23×D16311yescharacteristic
343C8—21no—
344C8—21no—
345C8—3261no—
346C8—3261no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—13044no—
357C22—3261no—
358C22—3261no—
359C22—6521no—
360C22—6521no—
361C22—6521no—
362C22—1632no—
363C22C2×D4×D16311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—3261no—
368C4—3261no—
369C4—3261no—
370C4—6521no—
371C4—6521no—
372C4—6521no—
373C4C2×#1457969811yescharacteristic
374C4#1910478011yescharacteristic
375C2—41no—
376C2—42no—
377C2—81no—
378C2—13041no—
379C2—3262no—
380C2#2378230211yescharacteristic
381C2#135×D16311yescharacteristic
382C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#449×C163C2
socleC22×C163C22×D4
hypercenter#449D163
hypocenterC163#3297
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #23782362C2semidirect
2. #23782303C2semidirect
3. #23782021C2semidirect
4. #23781999C2semidirect
5. #23781990C2semidirect
6. #23781967C2semidirect
7. #23781899C2semidirect
8. #23781891C2semidirect
9. #23781882C2semidirect
10. #23781797C2semidirect
11. D8×Dic163C2semidirect
12. (C22⋊D163)⋊C8C2semidirect
13. #23781661C2semidirect
14. #102×D163C2semidirect
15. #449×C163C2semidirect
18. C22×C163⋊C8C22semidirect
21. #19104726C22semidirect
32. D4×Dic163C22semidirect
35. #19104694C22semidirect
38. #19104669C22semidirect
39. #19104668C22semidirect
40. #19104667C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104659C22semidirect
43. #19104659C22semidirect
48. #19104656C22semidirect
51. C8×Dic163C22semidirect
53. C2×SD16×C163C22semidirect
55. #121×C163C22semidirect
56. #103×C163C22semidirect
58. C22⋊C8×C163C22semidirect
191. C163#3297semidirect
202. #449D163semidirect
210. C2×SD16C2×D163semidirect
225. #121C2×D163semidirect
240. #103C2×D163semidirect
249. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823622
#237823032
#237820212
#237819992
#237819902
#237819672
#237818992
#237818912
#237818822
#237817972
D8×Dic1632
(C22⋊D163)⋊C82
#237816612
#102×D1632
#449×C1632
#3297163