Finite Group Databasebeta

← #35226355

Subgroups — #35226355

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#23782373C211yesmaximal, characteristic, complemented
2#23782299C211yesmaximal, characteristic
3#23782010C211yesmaximal, characteristic, complemented
4#23781985C211yesmaximal, characteristic, complemented
5#23781981C211yesmaximal, characteristic, complemented
6#23781954C211yesmaximal, characteristic, complemented
7#23781897C211yesmaximal, characteristic, complemented
8#23781863C211yesmaximal, characteristic, complemented
9#23781815C211yesmaximal, characteristic, complemented
10#23781814C211yesmaximal, characteristic, complemented
11#23781750C211yesmaximal, characteristic, complemented
12C8×C22⋊D163C211yesmaximal, characteristic, complemented
13C2×#19104657C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#465×C163C211yesmaximal, characteristic, complemented
16C22×D652C2211yescharacteristic
17C2×D4×D163—21no—
18C2×D4×D163C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×#14579686C2211yescharacteristic, complemented
21C2×C8×D163C2211yescharacteristic, complemented
22#19104734C2211yescharacteristic, complemented
23#19104731C2211yescharacteristic
24#19104725—21no—
25#19104715—21no—
26#19104712—21no—
27#19104711C2211yescharacteristic, complemented
28#19104710C2211yescharacteristic
29#19104708—21no—
30#19104705—21no—
31#19104703—21no—
32Q8×Dic163C2211yescharacteristic, complemented
33#19104697C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35(C22⋊D163)⋊C4—21no—
36C2×C4⋊Dic163C2211yescharacteristic, complemented
37C4⋊C4×D163C2211yescharacteristic, complemented
38#19104666C2211yescharacteristic, complemented
39#19104666C2211yescharacteristic, complemented
40Dic163⋊C8C2211yescharacteristic, complemented
41#19104657C2211yescharacteristic
42#19104657C2211yescharacteristic
43#19104657C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic, complemented
45#19104656C2211yescharacteristic, complemented
46#19104656C2211yescharacteristic, complemented
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic, complemented
49C22⋊(C163⋊C8)C2211yescharacteristic, complemented
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51C8×Dic163C2211yescharacteristic, complemented
52C2×SD16×C163C2211yescharacteristic, complemented
53C22×C8×C163C2211yescharacteristic, complemented
54C22⋊Q8×C163C2211yescharacteristic, complemented
55#121×C163C2211yescharacteristic
56#106×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58#102×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C23×D163—21no—
61C2×Dic326—21no—
62C2×D652—22no—
63C2×D652—21no—
64C2×D652—21no—
65C2×D652—41no—
66C2×D652C2311yescharacteristic
67C2×D652C2311yescharacteristic
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C2×C22⋊D163D411yescharacteristic
72D4×D163—22no—
73D4×D163—41no—
74D4×D163—44no—
75C22×Dic163—21no—
76C2×C4×D163—21no—
77C2×C4×D163—21no—
78C2×C4×D163C2311yescharacteristic
79#14579686—22no—
80#14579686—41no—
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C8×D163—41no—
84C4⋊Dic163—21no—
85C4⋊Dic163—21no—
86C4⋊Dic163C2311yescharacteristic
87C4⋊Dic163C2311yescharacteristic
88Dic163⋊C4—21no—
89Dic163⋊C4—21no—
90Dic163⋊C4—21no—
91Dic163⋊C4D411yescharacteristic
92C22⋊Dic163D411yescharacteristic
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4D411yescharacteristic
97C4×Dic163—21no—
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—42no—
117C22×D163—42no—
118C22×D163—81no—
119C22×D163—81no—
120C22×D163C2×D411yescharacteristic
121Dic326—81no—
122D652—21no—
123D652—21no—
124D652—21no—
125D652—42no—
126D652—42no—
127D652#4112yes—
128C22⋊D163—42no—
129C22⋊D163—42no—
130C22⋊D163—81no—
131C22⋊D163SD1614yescomplemented
132C2×Dic163—21no—
133C2×Dic163—21no—
134C2×Dic163—81no—
135C2×Dic163C2×D411yescharacteristic
136C2×Dic163C2×D411yescharacteristic
137C2×Dic163C2×D411yescharacteristic
138C4×D163—21no—
139C4×D163—21no—
140C4×D163—42no—
141C4×D163—81no—
142C163⋊C8—21no—
143C163⋊C8—21no—
144C23×C163—21no—
145C23×C163C2×D411yescharacteristic
146Q8×C163—21no—
147Q8×C163—21no—
148D4×C163—21no—
149D4×C163—21no—
150D4×C163—42no—
151C2×C4×C163—21no—
152C2×C4×C163—21no—
153C2×C4×C163—21no—
154C2×C4×C163—21no—
155C2×C4×C163—21no—
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#481—1631no—
204#476—1631no—
205#465—1631no—
206#465D16311yescharacteristic, complemented
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×SD16C2×D16311yescharacteristic, complemented
220C22×C8—1631no—
221C22×C8C2×D16311yescharacteristic, complemented
222C4⋊Q8—1631no—
223#127—1631no—
224#125—3261no—
225C22⋊Q8—3261no—
226C22⋊Q8—3261no—
227C22⋊Q8—1631no—
228C22⋊Q8C2×D16311yescharacteristic, complemented
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⋊C4—1631no—
240#106—1631no—
241#106—1631no—
242#106C2×D16311yescharacteristic, complemented
243C4⋊C8—1631no—
244#103C2×D16311yescharacteristic, complemented
245#102—1631no—
246#102—1631no—
247#102—1631no—
248#102—1631no—
249#102—1631no—
250#102—1631no—
251#102—1631no—
252#102—1631no—
253#102C2×D16311yescharacteristic, complemented
254C22⋊C8—1631no—
255C22⋊C8—1631no—
256C4×C8—1631no—
257C24—3261no—
258C24—3261no—
259C2×Q8—3261no—
260C2×Q8C22×D16311yescharacteristic
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—3261no—
277C22×C4—3261no—
278C22×C4—3261no—
279C22×C4—1631no—
280C22×C4C22×D16311yescharacteristic
281SD16—41no—
282SD16—3262no—
283SD16—6521no—
284C2×C8—41no—
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⋊C4—1631no—
299C4⋊C4C22×D16311yescharacteristic
300C4⋊C4C22×D16311yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
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—21no—
322Q8—21no—
323Q8—13041no—
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—13041no—
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#1910478011yescharacteristic
387C2—22no—
388C2—81no—
389C2—13041no—
390C2—3262no—
391C2—6522no—
392C2#2378230411yescharacteristic
393C2C2×SD16×D16311yescharacteristic
394C2#2378246711yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#465×C163C2
socleC22×C163C22×D4
prime radical#23782299C2
hypercenter#465D163
hypocenterC163#3295
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. #23782373C2semidirect
3. #23782010C2semidirect
4. #23781985C2semidirect
5. #23781981C2semidirect
6. #23781954C2semidirect
7. #23781897C2semidirect
8. #23781863C2semidirect
9. #23781815C2semidirect
10. #23781814C2semidirect
11. #23781750C2semidirect
12. C8×C22⋊D163C2semidirect
13. C2×#19104657C2semidirect
14. #102×D163C2semidirect
15. #465×C163C2semidirect
20. C2×#14579686C22semidirect
21. C2×C8×D163C22semidirect
22. #19104734C22semidirect
27. #19104711C22semidirect
32. Q8×Dic163C22semidirect
36. C2×C4⋊Dic163C22semidirect
37. C4⋊C4×D163C22semidirect
38. #19104666C22semidirect
39. #19104666C22semidirect
40. Dic163⋊C8C22semidirect
43. #19104657C22semidirect
44. #19104657C22semidirect
45. #19104656C22semidirect
46. #19104656C22semidirect
48. #19104655C22semidirect
49. C22⋊(C163⋊C8)C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. C8×Dic163C22semidirect
52. C2×SD16×C163C22semidirect
53. C22×C8×C163C22semidirect
54. C22⋊Q8×C163C22semidirect
56. #106×C163C22semidirect
57. #103×C163C22semidirect
58. #102×C163C22semidirect
131. C22⋊D163SD16semidirect
197. C163#3295semidirect
206. #465D163semidirect
219. C2×SD16C2×D163semidirect
221. C22×C8C2×D163semidirect
228. C22⋊Q8C2×D163semidirect
242. #106C2×D163semidirect
244. #103C2×D163semidirect
253. #102C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823732
#237822992
#237820102
#237819852
#237819812
#237819542
#237818972
#237818632
#237818152
#237818142
#237817502
C8×C22⋊D1632
C2×#191046572
#102×D1632
#465×C1632
#3295163