Finite Group Databasebeta

← #35224720

Subgroups — #35224720

Order 20864.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782286C211yesmaximal, characteristic, complemented
2#23782006C211yesmaximal, characteristic, complemented
3#23781989C211yesmaximal, characteristic, complemented
4#23781955C211yesmaximal, characteristic, complemented
5#23781945C211yesmaximal, characteristic, complemented
6#23781880C211yesmaximal, characteristic, complemented
7#23781880C211yesmaximal, characteristic, complemented
8#23781816C211yesmaximal, characteristic, complemented
9#23781799C211yesmaximal, characteristic
10#23781775C211yesmaximal, characteristic, complemented
11#23781765C211yesmaximal, characteristic, complemented
12C4×#14579687C211yesmaximal, characteristic, complemented
13C4×#14579684C211yesmaximal, characteristic, complemented
14#23781726C211yesmaximal, characteristic, complemented
15#518×C163C211yesmaximal, characteristic, complemented
16C2×#14579687C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20#19104735C2211yescharacteristic, complemented
21#19104730—21no—
22#19104723C2211yescharacteristic, complemented
23#19104721—21no—
24#19104717—21no—
25#19104714C2211yescharacteristic
26#19104713—21no—
27#19104709C2211yescharacteristic
28C4×Dic326C2211yescharacteristic, complemented
29C4×D652C2211yescharacteristic, complemented
30#19104668C2211yescharacteristic, complemented
31#19104668C2211yescharacteristic, complemented
32#19104666C2211yescharacteristic, complemented
33#19104666C2211yescharacteristic, complemented
34C4⋊(C163⋊C8)C2211yescharacteristic, complemented
35C4⋊(C163⋊C8)C2211yescharacteristic, complemented
36#19104658C2211yescharacteristic, complemented
37#19104658C2211yescharacteristic, complemented
38#19104656C2211yescharacteristic
39#19104656C2211yescharacteristic
40#19104656C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104655C2211yescharacteristic, complemented
43#19104655C2211yescharacteristic, complemented
44C22⋊(C163⋊C8)C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46#19104646C2211yescharacteristic, complemented
47C4×C163⋊C8C2211yescharacteristic, complemented
48C22×D4×C163C2211yescharacteristic
49C4×D4×C163—21no—
50C4×D4×C163—21no—
51C4×D4×C163C2211yescharacteristic
52C4×D4×C163C2211yescharacteristic
53C4×D4×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55#117×C163C2211yescharacteristic
56#117×C163C2211yescharacteristic
57C2×C22⋊C4×C163—21no—
58C2×C22⋊C4×C163—21no—
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C4×D163—21no—
64#14579687—22no—
65#14579687—22no—
66#14579684—22no—
67#14579684—22no—
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C4⋊Dic163—21no—
73C4⋊Dic163C2311yescharacteristic
74C4⋊Dic163C2311yescharacteristic
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77C22⋊Dic163—21no—
78(C2×D163)⋊C4—21no—
79(C2×D163)⋊C4—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82C4×Dic163—21no—
83C24×C163—21no—
84C2×D4×C163—22no—
85C2×D4×C163—22no—
86C2×D4×C163—21no—
87C2×D4×C163C2311yescharacteristic
88C2×D4×C163C2311yescharacteristic
89C22×C4×C163—21no—
90C22×C4×C163—21no—
91C22×C4×C163—21no—
92C22×C4×C163—21no—
93C22×C4×C163C2311yescharacteristic
94C4⋊C4×C163D411yescharacteristic
95C4⋊C4×C163D411yescharacteristic
96C4⋊C4×C163C2311yescharacteristic
97C4⋊C4×C163C2311yescharacteristic
98C22⋊C4×C163—21no—
99C22⋊C4×C163—21no—
100C22⋊C4×C163—21no—
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163D411yescharacteristic
104C22⋊C4×C163D411yescharacteristic
105C42×C163—21no—
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109Dic326—21no—
110Dic326—21no—
111Dic326—81no—
112D652—21no—
113D652—21no—
114C22⋊D163—81no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C4×D163—81no—
119C163⋊C8—21no—
120C163⋊C8—21no—
121C163⋊C8—21no—
122C163⋊C8—21no—
123C23×C163—24no—
124C23×C163—21no—
125C23×C163—21no—
126C23×C163—81no—
127C23×C163—81no—
128C23×C163C2×D411yescharacteristic
129D4×C163—24no—
130D4×C163—21no—
131D4×C163—21no—
132D4×C163#4112yes—
133D4×C163#4112yes—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—81no—
139C2×C4×C163—81no—
140C2×C4×C163—81no—
141C2×C4×C163—81no—
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C22×C163—21no—
149C22×C163—22no—
150C22×C163—22no—
151C22×C163—21no—
152C22×C163—21no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163—84no—
158C22×C163C22×D411yescharacteristic
159C2×D163—41no—
160C2×D163—41no—
161C2×D163—41no—
162C4×C163—21no—
163C4×C163—21no—
164C4×C163—21no—
165C4×C163—21no—
166C4×C163—21no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163C2×#4111yescharacteristic
170C4×C163C2×#4111yescharacteristic
171Dic163—41no—
172Dic163—41no—
173Dic163—41no—
174C2×C163—41no—
175C2×C163—42no—
176C2×C163—42no—
177C2×C163#52511yescharacteristic
178C2×C163#57611yescharacteristic
179C2×C163#57711yescharacteristic
180D163—81no—
181C163#311811yescharacteristic, complemented, Sylow
182#3118—1631nomaximal, Sylow
183#542—1631no—
184#518D16311yescharacteristic, complemented
185#485—1631no—
186#482—1631no—
187#476—1631no—
188#474—1631no—
189#463—1631no—
190#463—1631no—
191#450—1631no—
192#447—1631no—
193#442—1631no—
194#440—1631no—
195C4×SD16—1631no—
196C4×D8—1631no—
197#433—1631no—
198C22×D4C2×D16311yescharacteristic
199C2×SD16—1631no—
200C2×SD16—1631no—
201C2×D8—1631no—
202C2×D8—1631no—
203C4⋊Q8—1631no—
204#126—3261no—
205#124—3261no—
206#124—1631no—
207#123—3261no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210#121—1631no—
211C4×Q8—1631no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1631no—
215C4×D4C2×D16311yescharacteristic
216C4×D4C2×D16311yescharacteristic
217C4×D4C2×D16311yescharacteristic
218C4×D4C2×D16311yescharacteristic
219#117C2×D16311yescharacteristic
220#117C2×D16311yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—1631no—
224#107—1631no—
225#106—1631no—
226#106—1631no—
227C4⋊C8—1631no—
228C4⋊C8—1631no—
229#103—1631no—
230#103—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102—1631no—
236#102—1631no—
237C22⋊C8—1631no—
238C22⋊C8—1631no—
239#97—1631no—
240C4×C8—1631no—
241C24—21no—
242C2×Q8—3261no—
243C2×Q8—1631no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4C22×D16311yescharacteristic
250C2×D4C22×D16311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3261no—
256C22×C4C22×D16311yescharacteristic
257SD16—3262no—
258SD16—3262no—
259D8—3262no—
260D8—3262no—
261C2×C8—1631no—
262C2×C8—1631no—
263C2×C8—1631no—
264C2×C8—1631no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—1631no—
269C4⋊C4—1631no—
270C4⋊C4C22⋊D16311yescharacteristic
271C4⋊C4C22⋊D16311yescharacteristic
272C4⋊C4C22×D16311yescharacteristic
273C4⋊C4C22×D16311yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4C22⋊D16311yescharacteristic
285C22⋊C4C22⋊D16311yescharacteristic
286C42—21no—
287C42—3261no—
288C42C22×D16311yescharacteristic
289C42C22×D16311yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—3261no—
296C23C2×C22⋊D16311yescharacteristic
297Q8—13041no—
298Q8—3261no—
299Q8—3261no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—13041no—
304D4—3261no—
305D4—3261no—
306D4#1457969912yes—
307D4#1457969912yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—13041no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4C2×C22⋊D16311yescharacteristic
321C2×C4C2×C22⋊D16311yescharacteristic
322C2×C4C2×C22⋊D16311yescharacteristic
323C2×C4C2×C22⋊D16311yescharacteristic
324C2×C4C2×C22⋊D16311yescharacteristic
325C2×C4C23×D16311yescharacteristic
326C8—3261no—
327C8—3261no—
328C8—3261no—
329C8—3261no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—6521no—
341C22—6521no—
342C22—6521no—
343C22C22×C22⋊D16311yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—6521no—
352C4—6521no—
353C4—6521no—
354C4C2×#1457969911yescharacteristic
355C4C2×#1457969911yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—13041no—
360C2#2378217611yescharacteristic
361C2#2378246811yescharacteristic
362C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#518×C163C2
socleC22×C163C22×D4
prime radical#23781799C2
hypercenter#518D163
hypocenterC163#3118
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #23782286C2semidirect
2. #23782006C2semidirect
3. #23781989C2semidirect
4. #23781955C2semidirect
5. #23781945C2semidirect
6. #23781880C2semidirect
7. #23781880C2semidirect
8. #23781816C2semidirect
10. #23781775C2semidirect
11. #23781765C2semidirect
12. C4×#14579687C2semidirect
13. C4×#14579684C2semidirect
14. #23781726C2semidirect
15. #518×C163C2semidirect
20. #19104735C22semidirect
22. #19104723C22semidirect
28. C4×Dic326C22semidirect
29. C4×D652C22semidirect
30. #19104668C22semidirect
31. #19104668C22semidirect
32. #19104666C22semidirect
33. #19104666C22semidirect
34. C4⋊(C163⋊C8)C22semidirect
35. C4⋊(C163⋊C8)C22semidirect
36. #19104658C22semidirect
37. #19104658C22semidirect
42. #19104655C22semidirect
43. #19104655C22semidirect
46. #19104646C22semidirect
47. C4×C163⋊C8C22semidirect
181. C163#3118semidirect
184. #518D163semidirect

Maximal subgroups

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

subgroupindex
#237822862
#237820062
#237819892
#237819552
#237819452
#237818802
#237818162
#237817992
#237817752
#237817652
C4×#145796872
C4×#145796842
#237817262
#518×C1632
#3118163