Finite Group Databasebeta

← #35226610

Subgroups — #35226610

Order 20864.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782373C211yesmaximal, characteristic, complemented
2#23782303C211yesmaximal, characteristic
3#23782026C211yesmaximal, characteristic, complemented
4#23781999C211yesmaximal, characteristic, complemented
5#23781979C211yesmaximal, characteristic, complemented
6#23781943C211yesmaximal, characteristic, complemented
7#23781913C211yesmaximal, characteristic, complemented
8#23781903C211yesmaximal, characteristic, complemented
9#23781889C211yesmaximal, characteristic, complemented
10#23781814C211yesmaximal, characteristic, complemented
11#23781766C211yesmaximal, characteristic, complemented
12#23781739C211yesmaximal, characteristic, complemented
13#23781666C211yesmaximal, characteristic, complemented
14#102×D163C211yesmaximal, characteristic, complemented
15#452×C163C211yesmaximal, characteristic, complemented
16C2×D4×D163C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579683C2211yescharacteristic, complemented
19C2×C8×D163C2211yescharacteristic
20#19104734C2211yescharacteristic, complemented
21#19104725—21no—
22#19104724C2211yescharacteristic
23#19104718—21no—
24#19104715—21no—
25#19104712—21no—
26#19104711C2211yescharacteristic
27#19104711C2211yescharacteristic
28#19104709C2211yescharacteristic
29#19104707—21no—
30#19104703—21no—
31Q8×Dic163C2211yescharacteristic, complemented
32D4×Dic163C2211yescharacteristic
33C4×C22⋊D163C2211yescharacteristic
34(C22⋊D163)⋊C4—21no—
35C2×C4⋊Dic163C2211yescharacteristic, complemented
36C4⋊C4×D163C2211yescharacteristic
37C2×(C2×D163)⋊C4—21no—
38#19104669C2211yescharacteristic, complemented
39#19104669C2211yescharacteristic, complemented
40#19104668C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104659C2211yescharacteristic, complemented
44#19104659C2211yescharacteristic, complemented
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104655C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic, complemented
52C2×#41×C163C2211yescharacteristic
53C2×Q16×C163C2211yescharacteristic, complemented
54C2×SD16×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic, complemented
56#103×C163C2211yescharacteristic, complemented
57#102×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60C2×Dic326—21no—
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C22⋊D163—21no—
64C2×C22⋊D163D411yescharacteristic
65D4×D163—22no—
66D4×D163—41no—
67C22×Dic163—21no—
68C22×Dic163—21no—
69C2×C4×D163—21no—
70C2×C4×D163C2311yescharacteristic
71#14579686—41no—
72#14579683—41no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C8×D163—41no—
76C4⋊Dic163—21no—
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4D411yescharacteristic
86C22⋊Dic163—21no—
87C22⋊Dic163D411yescharacteristic
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4D411yescharacteristic
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96#41×C163—22no—
97#41×C163—41no—
98C2×Q8×C163C2311yescharacteristic
99C2×Q8×C163C2311yescharacteristic
100C2×D4×C163—21no—
101C2×D4×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C22×C4×C163C2311yescharacteristic
104Q16×C163—41no—
105SD16×C163—22no—
106C2×C8×C163C2311yescharacteristic
107C2×C8×C163C2311yescharacteristic
108C4⋊C4×C163—21no—
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163—42no—
113C22×D163—42no—
114C22×D163—81no—
115C22×D163C2×D411yescharacteristic
116Dic326—81no—
117D652—21no—
118D652—21no—
119C22⋊D163—22no—
120C22⋊D163—42no—
121C22⋊D163—81no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—81no—
126C2×Dic163—81no—
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C2×Dic163C2×D411yescharacteristic
130C4×D163—21no—
131C4×D163—21no—
132C4×D163—81no—
133C163⋊C8—21no—
134C163⋊C8—21no—
135C23×C163—21no—
136C23×C163C2×D411yescharacteristic
137Q8×C163—21no—
138Q8×C163—21no—
139Q8×C163—21no—
140Q8×C163—21no—
141D4×C163—21no—
142D4×C163—42no—
143D4×C163#4112yes—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—42no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C8×C163—21no—
153C8×C163—21no—
154C22×C163—21no—
155C22×C163—22no—
156C22×C163—21no—
157C22×C163—21no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163C22×D411yescharacteristic
161C2×D163—21no—
162C2×D163—21no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—84no—
167C2×D163#13712yes—
168C4×C163—21no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163C2×#4111yescharacteristic
173C4×C163H(5,𝔽2)11yescharacteristic
174Dic163—21no—
175Dic163—21no—
176Dic163—21no—
177Dic163—41no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—42no—
181C2×C163—41no—
182C2×C163#54611yescharacteristic
183C2×C163C2×#13711yescharacteristic
184C2×C163#57611yescharacteristic
185D163—22no—
186D163—81no—
187C163#331411yescharacteristic, complemented, Sylow
188#3314—1631nomaximal, Sylow
189#554—1631no—
190#546—1631no—
191#489—1631no—
192#484—1631no—
193#480—1631no—
194#474—1631no—
195#468—1631no—
196#467—1631no—
197#464—1631no—
198#452D16311yescharacteristic, complemented
199#450—1631no—
200#440—1631no—
201#435—1631no—
202#417—1631no—
203C2×#102—1631no—
204C2×#41C2×D16311yescharacteristic
205C22×D4—1631no—
206C2×Q16C2×D16311yescharacteristic, complemented
207C2×SD16—1631no—
208C2×SD16C2×D16311yescharacteristic
209C2×M16—1631no—
210C22×C8—1631no—
211C4⋊Q8—1631no—
212#125—3261no—
213#124—1631no—
214#123—3261no—
215C22⋊Q8—3261no—
216C22⋊Q8—3261no—
217C22⋊Q8—1631no—
218C22⋊Q8—1631no—
219C22⋊Q8C2×D16311yescharacteristic, complemented
220#121—3261no—
221#121—1631no—
222#120—3261no—
223C4×Q8—1631no—
224C4×D4—3261no—
225C4×D4—1631no—
226C4×D4—1631no—
227C2×C4⋊C4—1631no—
228C2×C4⋊C4—1631no—
229C2×C22⋊C4—3261no—
230#107—1631no—
231#107—1631no—
232#107—1631no—
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#103C2×D16311yescharacteristic, complemented
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102C2×D16311yescharacteristic
244C22⋊C8—1631no—
245C22⋊C8C2×D16311yescharacteristic, complemented
246#97—1631no—
247C24—3261no—
248#41—22no—
249#41—41no—
250C2×Q8—3261no—
251C2×Q8C22×D16311yescharacteristic
252C2×Q8C22×D16311yescharacteristic
253C2×D4—21no—
254C2×D4—3261no—
255C2×D4—3262no—
256C2×D4—3261no—
257C2×D4—6521no—
258C2×D4—1631no—
259C2×D4—1631no—
260C2×D4C22×D16311yescharacteristic
261C22×C4—21no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—1631no—
266C22×C4C22×D16311yescharacteristic
267Q16—41no—
268SD16—22no—
269SD16—6521no—
270M16—6521no—
271C2×C8—6521no—
272C2×C8—1631no—
273C2×C8—1631no—
274C2×C8C22×D16311yescharacteristic
275C2×C8C22×D16311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—3261no—
278C4⋊C4—3261no—
279C4⋊C4—3261no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—1631no—
284C4⋊C4—1631no—
285C4⋊C4—1631no—
286C4⋊C4—1631no—
287C4⋊C4C22×D16311yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—3261no—
293C22⋊C4—3261no—
294C22⋊C4—3261no—
295C22⋊C4—1631no—
296C22⋊C4—1631no—
297C42—3261no—
298C42—1631no—
299C23—21no—
300C23—13041no—
301C23—3261no—
302C23—6522no—
303C23—6522no—
304C23—1631no—
305C23D4×D16311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—13041no—
311D4—21no—
312D4—42no—
313D4—13041no—
314D4—3262no—
315D4—3261no—
316D4—3261no—
317D4—6522no—
318D4#1457969812yes—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—13041no—
326C2×C4—13041no—
327C2×C4—13041no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—3261no—
333C2×C4—1631no—
334C2×C4—1631no—
335C2×C4—1631no—
336C2×C4D4×D16311yescharacteristic
337C2×C4C23×D16311yescharacteristic
338C8—21no—
339C8—21no—
340C8—3261no—
341C8—3261no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—13044no—
349C22—3261no—
350C22—3261no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22—1632no—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—3261no—
361C4—3261no—
362C4—3261no—
363C4—6521no—
364C4—6521no—
365C4—6521no—
366C4C2×#1457969811yescharacteristic
367C4#1910478011yescharacteristic
368C2—41no—
369C2—42no—
370C2—13041no—
371C2—3262no—
372C2#2378230211yescharacteristic
373C2#137×D16311yescharacteristic
374C2#2378246711yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#452×C163C2
socleC22×C163C22×D4
prime radical#23782303C2
hypercenter#452D163
hypocenterC163#3314
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #23782373C2semidirect
3. #23782026C2semidirect
4. #23781999C2semidirect
5. #23781979C2semidirect
6. #23781943C2semidirect
7. #23781913C2semidirect
8. #23781903C2semidirect
9. #23781889C2semidirect
10. #23781814C2semidirect
11. #23781766C2semidirect
12. #23781739C2semidirect
13. #23781666C2semidirect
14. #102×D163C2semidirect
15. #452×C163C2semidirect
18. C2×#14579683C22semidirect
20. #19104734C22semidirect
31. Q8×Dic163C22semidirect
35. C2×C4⋊Dic163C22semidirect
38. #19104669C22semidirect
39. #19104669C22semidirect
40. #19104668C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104659C22semidirect
43. #19104659C22semidirect
44. #19104659C22semidirect
51. #19104645C22semidirect
53. C2×Q16×C163C22semidirect
55. C22⋊Q8×C163C22semidirect
56. #103×C163C22semidirect
58. C22⋊C8×C163C22semidirect
187. C163#3314semidirect
198. #452D163semidirect
206. C2×Q16C2×D163semidirect
219. C22⋊Q8C2×D163semidirect
237. #103C2×D163semidirect
245. C22⋊C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823732
#237823032
#237820262
#237819992
#237819792
#237819432
#237819132
#237819032
#237818892
#237818142
#237817662
#237817392
#237816662
#102×D1632
#452×C1632
#3314163