Finite Group Databasebeta

← #35226392

Subgroups — #35226392

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#23782367C211yesmaximal, characteristic, complemented
2#23782306C211yesmaximal, characteristic, complemented
3#23782021C211yesmaximal, characteristic, complemented
4#23781998C211yesmaximal, characteristic, complemented
5#23781991C211yesmaximal, characteristic, complemented
6#23781964C211yesmaximal, characteristic, complemented
7#23781900C211yesmaximal, characteristic, complemented
8#23781882C211yesmaximal, characteristic, complemented
9#23781809C211yesmaximal, characteristic, complemented
10#23781798C211yesmaximal, characteristic, complemented
11#23781745C211yesmaximal, characteristic, complemented
12C8×C22⋊D163C211yesmaximal, characteristic, complemented
13#23781660C211yesmaximal, characteristic, complemented
14C2×#19104657C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C2×C8×D163C2211yescharacteristic, complemented
21#19104728—21no—
22#19104726C2211yescharacteristic, complemented
23#19104724C2211yescharacteristic
24#19104720—21no—
25#19104719—21no—
26#19104710—21no—
27#19104710C2211yescharacteristic
28#19104707—21no—
29#19104706C2211yescharacteristic, complemented
30#19104705—21no—
31D4×Dic163—21no—
32D4×Dic163C2211yescharacteristic, complemented
33#19104697C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35#19104693C2211yescharacteristic, complemented
36C2×C4⋊Dic163C2211yescharacteristic
37C22⋊C4×D163—21no—
38#19104668C2211yescharacteristic, complemented
39#19104668C2211yescharacteristic, 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
51C8×Dic163C2211yescharacteristic, 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—
62C2×D652—22no—
63C2×D652—21no—
64C2×D652—21no—
65C2×D652—21no—
66C2×D652—41no—
67C2×D652C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163D411yescharacteristic
71C22×Dic163—21no—
72C22×Dic163—21no—
73C2×C4×D163—21no—
74C2×C4×D163—21no—
75C2×C4×D163C2311yescharacteristic
76#14579684—22no—
77#14579684—41no—
78C2×C163⋊C8C2311yescharacteristic
79C2×C163⋊C8C2311yescharacteristic
80C8×D163—41no—
81C4⋊Dic163—21no—
82C4⋊Dic163—21no—
83C4⋊Dic163C2311yescharacteristic
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163D411yescharacteristic
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)⋊C4D411yescharacteristic
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C2×Q8×C163C2311yescharacteristic
99C2×D4×C163—21no—
100C2×D4×C163C2311yescharacteristic
101C22×C4×C163C2311yescharacteristic
102SD16×C163—41no—
103C2×C8×C163—41no—
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117D652—21no—
118D652—21no—
119D652—21no—
120D652—42no—
121D652—42no—
122D652—81no—
123D652#4112yes—
124C22⋊D163—22no—
125C22⋊D163—81no—
126C2×Dic163—21no—
127C2×Dic163—21no—
128C2×Dic163—81no—
129C2×Dic163—81no—
130C2×Dic163C2×D411yescharacteristic
131C2×Dic163C2×D411yescharacteristic
132C2×Dic163C2×D411yescharacteristic
133C4×D163—21no—
134C4×D163—21no—
135C4×D163—42no—
136C4×D163—81no—
137C163⋊C8—21no—
138C163⋊C8—21no—
139C23×C163—21no—
140C23×C163C2×D411yescharacteristic
141Q8×C163—21no—
142Q8×C163—21no—
143D4×C163—21no—
144D4×C163—21no—
145D4×C163—42no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—21no—
150C2×C4×C163—21no—
151C2×C4×C163C2×D411yescharacteristic
152C2×C4×C163C2411yescharacteristic
153C8×C163—21no—
154C8×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163#13512yes—
161C22×C163C22×D411yescharacteristic
162C2×D163—21no—
163C2×D163—22no—
164C2×D163—21no—
165C2×D163—21no—
166C2×D163—41no—
167C2×D163—41no—
168C2×D163—41no—
169C2×D163—41no—
170C2×D163—41no—
171C2×D163—84no—
172C4×C163—21no—
173C4×C163—41no—
174C4×C163—41no—
175C4×C163—41no—
176C4×C163C2×#4111yescharacteristic
177C4×C163H(5,𝔽2)11yescharacteristic
178Dic163—21no—
179Dic163—21no—
180Dic163—21no—
181Dic163—41no—
182Dic163—41no—
183C2×C163—22no—
184C2×C163—81no—
185C2×C163#54611yescharacteristic
186C2×C163C2×#13511yescharacteristic
187C2×C163#57611yescharacteristic
188D163—41no—
189D163—42no—
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#465D16311yescharacteristic, complemented
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×SD16C2×D16311yescharacteristic, complemented
211C2×D8—1631no—
212C2×D8—1631no—
213C22×C8—1631no—
214C22×C8C2×D16311yescharacteristic
215#126—3261no—
216#125—1631no—
217#124—1631no—
218#123—3261no—
219#123—3261no—
220C22⋊Q8C2×D16311yescharacteristic
221#121—3261no—
222#121—3261no—
223#121—1631no—
224#121—1631no—
225#121C2×D16311yescharacteristic
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#106C2×D16311yescharacteristic, complemented
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, complemented
248C22⋊C8—1631no—
249C22⋊C8—1631no—
250C4×C8—1631no—
251C24—3261no—
252#41—3262no—
253#41—6521no—
254C2×Q8C22×D16311yescharacteristic
255C2×D4—21no—
256C2×D4—3261no—
257C2×D4—3262no—
258C2×D4—3261no—
259C2×D4—3261no—
260C2×D4—3261no—
261C2×D4—6521no—
262C2×D4—1631no—
263C2×D4—1631no—
264C2×D4—1631no—
265C2×D4C22×D16311yescharacteristic
266C22×C4—3261no—
267C22×C4—3261no—
268C22×C4—3261no—
269C22×C4—3261no—
270C22×C4—1631no—
271C22×C4C22×D16311yescharacteristic
272SD16—41no—
273D8—3262no—
274D8—6521no—
275C2×C8—41no—
276C2×C8—6521no—
277C2×C8—1631no—
278C2×C8—1631no—
279C2×C8C22×D16311yescharacteristic
280C2×C8C22×D16311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—3261no—
283C4⋊C4—3261no—
284C4⋊C4—3261no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4—1631no—
288C4⋊C4C22×D16311yescharacteristic
289C4⋊C4C22×D16311yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
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—13041no—
305C23—3261no—
306C23—3261no—
307C23—6522no—
308C23—6522no—
309C23—1631no—
310C23D4×D16311yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—13041no—
317D4—13041no—
318D4—3262no—
319D4—3261no—
320D4—3261no—
321D4—3261no—
322D4—6522no—
323D4—6522no—
324D4—1632no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—13041no—
331C2×C4—13041no—
332C2×C4—13041no—
333C2×C4—3261no—
334C2×C4—3261no—
335C2×C4—3261no—
336C2×C4—3261no—
337C2×C4—6522no—
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—41no—
350C22—41no—
351C22—41no—
352C22—13044no—
353C22—3261no—
354C22—3261no—
355C22—3261no—
356C22—3262no—
357C22—6521no—
358C22—6521no—
359C22—6521no—
360C22—6521no—
361C22—6521no—
362C22#1910475612yes—
363C22C2×D4×D16311yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—3261no—
369C4—3261no—
370C4—3261no—
371C4—6521no—
372C4—6521no—
373C4#41×D16311yescharacteristic
374C4#1910478011yescharacteristic
375C2—22no—
376C2—81no—
377C2—13041no—
378C2—6522no—
379C2—6521no—
380C2#2378230411yescharacteristic
381C2C2×#1910475611yescharacteristic
382C2#2378246711yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
hypercenter#465D163
hypocenterC163#3297
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. #23782367C2semidirect
2. #23782306C2semidirect
3. #23782021C2semidirect
4. #23781998C2semidirect
5. #23781991C2semidirect
6. #23781964C2semidirect
7. #23781900C2semidirect
8. #23781882C2semidirect
9. #23781809C2semidirect
10. #23781798C2semidirect
11. #23781745C2semidirect
12. C8×C22⋊D163C2semidirect
13. #23781660C2semidirect
14. C2×#19104657C2semidirect
15. #465×C163C2semidirect
20. C2×C8×D163C22semidirect
22. #19104726C22semidirect
29. #19104706C22semidirect
32. D4×Dic163C22semidirect
35. #19104693C22semidirect
38. #19104668C22semidirect
39. #19104668C22semidirect
40. Dic163⋊C8C22semidirect
41. #19104659C22semidirect
42. #19104659C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
52. C2×SD16×C163C22semidirect
56. #106×C163C22semidirect
57. #103×C163C22semidirect
58. #102×C163C22semidirect
191. C163#3297semidirect
200. #465D163semidirect
210. C2×SD16C2×D163semidirect
236. #106C2×D163semidirect
240. #103C2×D163semidirect
247. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823672
#237823062
#237820212
#237819982
#237819912
#237819642
#237819002
#237818822
#237818092
#237817982
#237817452
C8×C22⋊D1632
#237816602
C2×#191046572
#465×C1632
#3297163