Finite Group Databasebeta

← #35226358

Subgroups — #35226358

Order 20864.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782370C211yesmaximal, characteristic, complemented
2#23782299C211yesmaximal, characteristic
3#23782010C211yesmaximal, characteristic, complemented
4#23781985C211yesmaximal, characteristic, complemented
5#23781954C211yesmaximal, characteristic, complemented
6#23781895C211yesmaximal, characteristic, complemented
7#23781892C211yesmaximal, characteristic, complemented
8#23781864C211yesmaximal, characteristic, complemented
9#23781818C211yesmaximal, characteristic, complemented
10#23781817C211yesmaximal, characteristic, complemented
11#23781752C211yesmaximal, characteristic, complemented
12(C22⋊D163)⋊C8C211yesmaximal, characteristic, complemented
13C2×#19104655C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#481×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C2×D4×D163—21no—
18C2×D4×D163C2211yescharacteristic
19C2×#14579688C2211yescharacteristic
20C2×#14579688C2211yescharacteristic, complemented
21C2×#14579687C2211yescharacteristic, complemented
22C22×C163⋊C8C2211yescharacteristic, complemented
23C2×C8×D163C2211yescharacteristic, complemented
24#19104734C2211yescharacteristic, complemented
25#19104731C2211yescharacteristic
26#19104725—21no—
27#19104715C2211yescharacteristic, complemented
28#19104714—21no—
29#19104714C2211yescharacteristic, complemented
30#19104713—21no—
31#19104710C2211yescharacteristic
32#19104708—21no—
33#19104705—21no—
34#19104703—21no—
35#19104700C2211yescharacteristic, complemented
36#19104697C2211yescharacteristic
37C4×C22⋊D163—21no—
38C4×C22⋊D163C2211yescharacteristic
39C4⋊C4×D163C2211yescharacteristic, complemented
40#19104666C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104658C2211yescharacteristic, complemented
43#19104657C2211yescharacteristic
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic, complemented
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic, complemented
49#19104655C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52#121×C163C2211yescharacteristic
53C2×C4⋊C4×C163C2211yescharacteristic, complemented
54#106×C163C2211yescharacteristic, complemented
55#106×C163C2211yescharacteristic, complemented
56#102×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C23×D163—21no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63C2×D652—22no—
64C2×D652—21no—
65C2×D652—21no—
66C2×D652—41no—
67C2×D652C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163—21no—
72C2×C22⋊D163D411yescharacteristic
73D4×D163—22no—
74D4×D163—41no—
75D4×D163—44no—
76C2×C4×D163—21no—
77C2×C4×D163—21no—
78C2×C4×D163C2311yescharacteristic
79#14579688—22no—
80#14579688—41no—
81#14579687—41no—
82C2×C163⋊C8—41no—
83C2×C163⋊C8C2311yescharacteristic
84C2×C163⋊C8C2311yescharacteristic
85C8×D163—41no—
86C4⋊Dic163—21no—
87C4⋊Dic163C2311yescharacteristic
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91Dic163⋊C4—21no—
92Dic163⋊C4D411yescharacteristic
93C22⋊Dic163—21no—
94C22⋊Dic163D411yescharacteristic
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4—21no—
98(C2×D163)⋊C4D411yescharacteristic
99C4×Dic163—21no—
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—42no—
117C22×D163—42no—
118C22×D163—81no—
119C22×D163—81no—
120C22×D163C2×D411yescharacteristic
121Dic326—21no—
122Dic326—21no—
123Dic326—81no—
124D652—21no—
125D652—21no—
126D652—21no—
127D652—42no—
128D652—42no—
129D652#4112yes—
130C22⋊D163—42no—
131C22⋊D163—42no—
132C22⋊D163—81no—
133C22⋊D163SD1614yescomplemented
134C2×Dic163—21no—
135C2×Dic163—21no—
136C2×Dic163C2×D411yescharacteristic
137C2×Dic163C2×D411yescharacteristic
138C2×Dic163C2×D411yescharacteristic
139C4×D163—21no—
140C4×D163—21no—
141C4×D163—42no—
142C4×D163—81no—
143C163⋊C8—21no—
144C163⋊C8—21no—
145C23×C163—21no—
146C23×C163C2×D411yescharacteristic
147D4×C163—21no—
148D4×C163—21no—
149D4×C163—42no—
150C2×C4×C163—21no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163—81no—
156C2×C4×C163C2×D411yescharacteristic
157C2×C4×C163C2411yescharacteristic
158C8×C163—21no—
159C8×C163—21no—
160C22×C163—21no—
161C22×C163—21no—
162C22×C163—41no—
163C22×C163—41no—
164C22×C163—41no—
165C22×C163C2×SD1612yes—
166C22×C163C22×D411yescharacteristic
167C2×D163—21no—
168C2×D163—21no—
169C2×D163—22no—
170C2×D163—41no—
171C2×D163—41no—
172C2×D163—41no—
173C2×D163—41no—
174C2×D163—41no—
175C2×D163—84no—
176C2×D163—84no—
177C2×D163C2×SD1612yes—
178C4×C163—21no—
179C4×C163—41no—
180C4×C163—41no—
181C4×C163—41no—
182C4×C163C2×#4111yescharacteristic
183C4×C163H(5,𝔽2)11yescharacteristic
184Dic163—21no—
185Dic163—21no—
186Dic163—41no—
187Dic163—41no—
188Dic163C2×SD1612yes—
189C2×C163—22no—
190C2×C163—81no—
191C2×C163#54611yescharacteristic
192C2×C163C22×SD1611yescharacteristic
193C2×C163#57611yescharacteristic
194D163—22no—
195D163—42no—
196D163—81no—
197C163#329511yescharacteristic, complemented, Sylow
198#3295—1631nomaximal, Sylow
199#554—1631no—
200D42—1631no—
201#486—1631no—
202#481—1631no—
203#481D16311yescharacteristic, complemented
204#476—1631no—
205#465—1631no—
206#465—1631no—
207#460—1631no—
208#450—1631no—
209#450—1631no—
210C4×SD16—1631no—
211C8×D4—1631no—
212C2×#102—1631no—
213C2×#102—1631no—
214C22×D4—3261no—
215C22×D4—1631no—
216C22×D4—1631no—
217C2×SD16—1631no—
218C2×SD16—1631no—
219C2×SD16—1631no—
220C22×C8—1631no—
221C22×C8—1631no—
222C4⋊Q8—1631no—
223#127—1631no—
224#125—3261no—
225C22⋊Q8—3261no—
226C22⋊Q8—3261no—
227C22⋊Q8—1631no—
228C22⋊Q8—1631no—
229#121—3261no—
230#121—1631no—
231#121C2×D16311yescharacteristic
232#120—3261no—
233#120—3261no—
234C4×Q8—1631no—
235C4×D4—3261no—
236C4×D4—1631no—
237C4×D4—1631no—
238C2×C4⋊C4—1631no—
239C2×C4⋊C4C2×D16311yescharacteristic, complemented
240#106—1631no—
241#106C2×D16311yescharacteristic, complemented
242#106C2×D16311yescharacteristic, complemented
243C4⋊C8—1631no—
244#103—1631no—
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102—1631no—
250#102—1631no—
251#102—1631no—
252#102C2×D16311yescharacteristic, complemented
253#102C2×D16311yescharacteristic, complemented
254C22⋊C8—1631no—
255C22⋊C8C2×D16311yescharacteristic, complemented
256C4×C8—1631no—
257C24—3261no—
258C24—3261no—
259C2×Q8—3261no—
260C2×Q8—1631no—
261C2×D4—21no—
262C2×D4—3262no—
263C2×D4—3261no—
264C2×D4—3262no—
265C2×D4—3261no—
266C2×D4—3261no—
267C2×D4—3261no—
268C2×D4—3261no—
269C2×D4—6521no—
270C2×D4—6521no—
271C2×D4—6524no—
272C2×D4—1631no—
273C2×D4—1631no—
274C2×D4—1631no—
275C2×D4C22×D16311yescharacteristic
276C22×C4—21no—
277C22×C4—3261no—
278C22×C4—3261no—
279C22×C4—1631no—
280C22×C4C22×D16311yescharacteristic
281SD16—3262no—
282SD16—6521no—
283SD16—6521no—
284C2×C8—6521no—
285C2×C8—6521no—
286C2×C8—1631no—
287C2×C8—1631no—
288C2×C8C22×D16311yescharacteristic
289C2×C8C22×D16311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3261no—
292C4⋊C4—3261no—
293C4⋊C4—3261no—
294C4⋊C4—3261no—
295C4⋊C4—3261no—
296C4⋊C4—1631no—
297C4⋊C4—1631no—
298C4⋊C4C22×D16311yescharacteristic
299C4⋊C4C22×D16311yescharacteristic
300C4⋊C4C22×D16311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—3261no—
303C22⋊C4—3261no—
304C22⋊C4—3261no—
305C22⋊C4—3261no—
306C22⋊C4—1631no—
307C22⋊C4—1631no—
308C42—3261no—
309C42—1631no—
310C23—21no—
311C23—13041no—
312C23—13041no—
313C23—3261no—
314C23—3261no—
315C23—6522no—
316C23—6522no—
317C23—6522no—
318C23—6522no—
319C23—1631no—
320C23D4×D16311yescharacteristic
321Q8—13041no—
322Q8—3261no—
323Q8—3261no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—13041no—
328D4—3261no—
329D4—3261no—
330D4—3261no—
331D4—6522no—
332D4—6522no—
333D4—6522no—
334D4—6522no—
335D4—1634no—
336D4—1632no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—81no—
343C2×C4—13041no—
344C2×C4—3261no—
345C2×C4—3261no—
346C2×C4—3261no—
347C2×C4—3261no—
348C2×C4—6522no—
349C2×C4—1631no—
350C2×C4—1631no—
351C2×C4—1631no—
352C2×C4D4×D16311yescharacteristic
353C2×C4C23×D16311yescharacteristic
354C8—21no—
355C8—21no—
356C8—3261no—
357C8—3261no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—13044no—
364C22—13044no—
365C22—3261no—
366C22—3261no—
367C22—3262no—
368C22—6521no—
369C22—6521no—
370C22—6521no—
371C22—6521no—
372C22—6521no—
373C22—1632no—
374C22SD16×D16312yes—
375C22C2×D4×D16311yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—3261no—
381C4—3261no—
382C4—6521no—
383C4—6521no—
384C4—1632no—
385C4#41×D16311yescharacteristic
386C4#1910478111yescharacteristic
387C2—22no—
388C2—81no—
389C2—13041no—
390C2—3262no—
391C2—6522no—
392C2#2378230711yescharacteristic
393C2C2×SD16×D16311yescharacteristic
394C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#481×C163C2
socleC22×C163C22×D4
prime radical#23782299C2
hypercenter#481D163
hypocenterC163#3295
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. #23782370C2semidirect
3. #23782010C2semidirect
4. #23781985C2semidirect
5. #23781954C2semidirect
6. #23781895C2semidirect
7. #23781892C2semidirect
8. #23781864C2semidirect
9. #23781818C2semidirect
10. #23781817C2semidirect
11. #23781752C2semidirect
12. (C22⋊D163)⋊C8C2semidirect
13. C2×#19104655C2semidirect
14. #102×D163C2semidirect
15. #481×C163C2semidirect
20. C2×#14579688C22semidirect
21. C2×#14579687C22semidirect
22. C22×C163⋊C8C22semidirect
23. C2×C8×D163C22semidirect
24. #19104734C22semidirect
27. #19104715C22semidirect
29. #19104714C22semidirect
35. #19104700C22semidirect
39. C4⋊C4×D163C22semidirect
40. #19104666C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104658C22semidirect
44. #19104657C22semidirect
45. #19104656C22semidirect
48. #19104655C22semidirect
49. #19104655C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
53. C2×C4⋊C4×C163C22semidirect
54. #106×C163C22semidirect
55. #106×C163C22semidirect
56. #102×C163C22semidirect
57. #102×C163C22semidirect
58. C22⋊C8×C163C22semidirect
133. C22⋊D163SD16semidirect
197. C163#3295semidirect
203. #481D163semidirect
239. C2×C4⋊C4C2×D163semidirect
241. #106C2×D163semidirect
242. #106C2×D163semidirect
252. #102C2×D163semidirect
253. #102C2×D163semidirect
255. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823702
#237822992
#237820102
#237819852
#237819542
#237818952
#237818922
#237818642
#237818182
#237818172
#237817522
(C22⋊D163)⋊C82
C2×#191046552
#102×D1632
#481×C1632
#3295163