Finite Group Databasebeta

← #35226503

Subgroups — #35226503

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#23782306C211yesmaximal, characteristic, complemented
3#23782026C211yesmaximal, characteristic, complemented
4#23781993C211yesmaximal, characteristic, complemented
5#23781985C211yesmaximal, characteristic, complemented
6#23781954C211yesmaximal, characteristic, complemented
7#23781917C211yesmaximal, characteristic, complemented
8#23781882C211yesmaximal, characteristic, complemented
9#23781809C211yesmaximal, characteristic, complemented
10#23781798C211yesmaximal, characteristic, complemented
11#23781774C211yesmaximal, characteristic, complemented
12#23781736C211yesmaximal, characteristic, complemented
13#23781665C211yesmaximal, characteristic, complemented
14C2×#19104657C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×#14579698—21no—
18C22×D652C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21C2×#14579682C2211yescharacteristic, complemented
22#19104734C2211yescharacteristic, complemented
23#19104724C2211yescharacteristic
24#19104720—21no—
25#19104717—21no—
26#19104716—21no—
27#19104712—21no—
28#19104710—21no—
29#19104710C2211yescharacteristic
30#19104706C2211yescharacteristic, complemented
31#19104705—21no—
32D4×Dic163C2211yescharacteristic, complemented
33#19104697C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35C2×C4⋊Dic163C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic, complemented
37C22⋊C4×D163—21no—
38#19104666C2211yescharacteristic, complemented
39#19104666C2211yescharacteristic, complemented
40Dic163⋊C8C2211yescharacteristic, complemented
41#19104659C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic, complemented
53C22×C8×C163C2211yescharacteristic
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579700—22no—
61#14579700—41no—
62#14579698—44no—
63C2×Dic326—21no—
64C2×D652—22no—
65C2×D652—21no—
66C2×D652—21no—
67C2×D652—21no—
68C2×D652—41no—
69C2×D652C2311yescharacteristic
70C2×D652C2311yescharacteristic
71C2×C22⋊D163—21no—
72C2×C22⋊D163D411yescharacteristic
73C22×Dic163—21no—
74C22×Dic163—21no—
75C2×C4×D163—21no—
76C2×C4×D163—21no—
77C2×C4×D163—21no—
78C2×C4×D163C2311yescharacteristic
79#14579684—22no—
80#14579684—41no—
81#14579682—41no—
82C2×C163⋊C8C2311yescharacteristic
83C2×C163⋊C8C2311yescharacteristic
84C4⋊Dic163—21no—
85C4⋊Dic163C2311yescharacteristic
86C4⋊Dic163C2311yescharacteristic
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4D411yescharacteristic
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92C22⋊Dic163D411yescharacteristic
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4D411yescharacteristic
98C4×Dic163C2311yescharacteristic
99C2×Q8×C163C2311yescharacteristic
100C2×D4×C163—21no—
101C2×D4×C163C2311yescharacteristic
102C22×C4×C163C2311yescharacteristic
103SD16×C163—41no—
104C2×C8×C163—41no—
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163—21no—
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22⋊C4×C163—21no—
112C22×D163—21no—
113C22×D163—21no—
114C22×D163—42no—
115C22×D163—42no—
116C22×D163—81no—
117C22×D163C2×D411yescharacteristic
118Dic326—42no—
119D652—21no—
120D652—21no—
121D652—21no—
122D652—42no—
123D652—42no—
124D652—81no—
125D652#4112yescomplemented
126C22⋊D163—22no—
127C22⋊D163—42no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—42no—
131C2×Dic163—42no—
132C2×Dic163C2×D411yescharacteristic
133C2×Dic163C2×D411yescharacteristic
134C2×Dic163C2×D411yescharacteristic
135C4×D163—21no—
136C4×D163—21no—
137C4×D163—42no—
138C4×D163—42no—
139C4×D163—81no—
140C163⋊C8—21no—
141C163⋊C8—21no—
142C23×C163—21no—
143C23×C163C2×D411yescharacteristic
144Q8×C163—21no—
145Q8×C163—21no—
146D4×C163—21no—
147D4×C163—21no—
148D4×C163—42no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
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—41no—
161C22×C163—41no—
162C22×C163—41no—
163C22×C163AGL(1,ℤ/8)12yes—
164C22×C163C22×D411yescharacteristic
165C2×D163—21no—
166C2×D163—22no—
167C2×D163—21no—
168C2×D163—21no—
169C2×D163—41no—
170C2×D163—41no—
171C2×D163—41no—
172C2×D163—41no—
173C2×D163—41no—
174C2×D163—84no—
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—22no—
187C2×C163—81no—
188C2×C163#54611yescharacteristic
189C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
190C2×C163#57711yescharacteristic
191D163—41no—
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#481—1631no—
201#476—1631no—
202#469—1631no—
203#465D16311yescharacteristic, complemented
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×SD16C2×D16311yescharacteristic, complemented
215C2×D8—1631no—
216C2×D8—1631no—
217C2×M16—1631no—
218C22×C8C2×D16311yescharacteristic
219C4⋊Q8—1631no—
220#124—1631no—
221#123—3261no—
222#123—3261no—
223#123—3261no—
224C22⋊Q8—3261no—
225C22⋊Q8C2×D16311yescharacteristic
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#106C2×D16311yescharacteristic, complemented
240C4⋊C8—1631no—
241#103—1631no—
242#103—1631no—
243#103C2×D16311yescharacteristic, complemented
244#102—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102—1631no—
250#102C2×D16311yescharacteristic, complemented
251C22⋊C8—1631no—
252C22⋊C8—1631no—
253#97—1631no—
254C24—3261no—
255#41—3262no—
256#41—6521no—
257#41—6524no—
258C2×Q8—3261no—
259C2×Q8C22×D16311yescharacteristic
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—3261no—
272C22×C4—3261no—
273C22×C4—3261no—
274C22×C4—3261no—
275C22×C4—3261no—
276C22×C4—1631no—
277C22×C4C22×D16311yescharacteristic
278SD16—41no—
279D8—3262no—
280D8—6521no—
281M16—6521no—
282C2×C8—41no—
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⋊C4—1631no—
294C4⋊C4C22×D16311yescharacteristic
295C4⋊C4C22×D16311yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
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—21no—
316Q8—21no—
317Q8—6522no—
318D4—21no—
319D4—21no—
320D4—42no—
321D4—13041no—
322D4—3262no—
323D4—3261no—
324D4—3261no—
325D4—3261no—
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—13041no—
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—41no—
356C22—41no—
357C22—41no—
358C22—13044no—
359C22—3261no—
360C22—3261no—
361C22—3261no—
362C22—3262no—
363C22—6521no—
364C22—6521no—
365C22—6521no—
366C22—6521no—
367C22—6521no—
368C22#1910476212yes—
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#1910478011yescharacteristic
381C2—22no—
382C2—81no—
383C2—13041no—
384C2—6522no—
385C2—6521no—
386C2#2378230411yescharacteristic
387C2C2×#1910476211yescharacteristic
388C2#2378247211yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
hypercenter#465D163
hypocenterC163#3306
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #23782315C2semidirect
2. #23782306C2semidirect
3. #23782026C2semidirect
4. #23781993C2semidirect
5. #23781985C2semidirect
6. #23781954C2semidirect
7. #23781917C2semidirect
8. #23781882C2semidirect
9. #23781809C2semidirect
10. #23781798C2semidirect
11. #23781774C2semidirect
12. #23781736C2semidirect
13. #23781665C2semidirect
14. C2×#19104657C2semidirect
15. #465×C163C2semidirect
21. C2×#14579682C22semidirect
22. #19104734C22semidirect
30. #19104706C22semidirect
32. D4×Dic163C22semidirect
36. C4⋊C4×D163C22semidirect
38. #19104666C22semidirect
39. #19104666C22semidirect
40. Dic163⋊C8C22semidirect
41. #19104659C22semidirect
42. #19104659C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
52. C2×SD16×C163C22semidirect
56. #106×C163C22semidirect
57. #103×C163C22semidirect
58. #102×C163C22semidirect
125. D652#41semidirect
194. C163#3306semidirect
203. #465D163semidirect
214. C2×SD16C2×D163semidirect
239. #106C2×D163semidirect
243. #103C2×D163semidirect
250. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823152
#237823062
#237820262
#237819932
#237819852
#237819542
#237819172
#237818822
#237818092
#237817982
#237817742
#237817362
#237816652
C2×#191046572
#465×C1632
#3306163