Finite Group Databasebeta

← #35226615

Subgroups — #35226615

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#23782370C211yesmaximal, characteristic, complemented
2#23782306C211yesmaximal, characteristic
3#23782026C211yesmaximal, characteristic, complemented
4#23781980C211yesmaximal, characteristic, complemented
5#23781944C211yesmaximal, characteristic, complemented
6#23781912C211yesmaximal, characteristic, complemented
7#23781904C211yesmaximal, characteristic, complemented
8#23781890C211yesmaximal, characteristic, complemented
9#23781830C211yesmaximal, characteristic, complemented
10#23781818C211yesmaximal, characteristic, complemented
11#23781764C211yesmaximal, characteristic, complemented
12#23781733C211yesmaximal, characteristic, complemented
13#23781665C211yesmaximal, characteristic, complemented
14C2×#19104655C211yesmaximal, characteristic, complemented
15#484×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×#14579689C2211yescharacteristic, complemented
19C2×#14579688C2211yescharacteristic
20C2×#14579688C2211yescharacteristic
21C2×#14579682C2211yescharacteristic, complemented
22C22×C163⋊C8C2211yescharacteristic
23#19104734C2211yescharacteristic, complemented
24#19104725—21no—
25#19104724C2211yescharacteristic
26#19104720—21no—
27#19104715C2211yescharacteristic, complemented
28#19104714—21no—
29#19104714C2211yescharacteristic
30#19104713—21no—
31#19104710—21no—
32#19104710C2211yescharacteristic
33#19104705—21no—
34#19104700C2211yescharacteristic, complemented
35#19104697C2211yescharacteristic
36C4×C22⋊D163—21no—
37C4×C22⋊D163C2211yescharacteristic
38C4⋊C4×D163C2211yescharacteristic, complemented
39C22⋊C4×D163—21no—
40#19104668C2211yescharacteristic, complemented
41Dic163⋊C8C2211yescharacteristic, complemented
42#19104659C2211yescharacteristic, complemented
43#19104658C2211yescharacteristic, complemented
44#19104657C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49#19104655C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic, complemented
51#19104645C2211yescharacteristic, complemented
52C22⋊Q8×C163C2211yescharacteristic
53C2×C4⋊C4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic, complemented
55#107×C163C2211yescharacteristic, complemented
56#103×C163C2211yescharacteristic, complemented
57#103×C163C2211yescharacteristic, complemented
58C22⋊C8×C163C2211yescharacteristic
59C23×D163—21no—
60#14579700—22no—
61#14579700—41no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64C2×D652—22no—
65C2×D652—21no—
66C2×D652—21no—
67C2×D652—21no—
68C2×D652—41no—
69C2×D652C2311yescharacteristic
70C2×D652C2311yescharacteristic
71C2×C22⋊D163D411yescharacteristic
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74C2×C4×D163—21no—
75C2×C4×D163C2311yescharacteristic
76#14579689—41no—
77#14579688—22no—
78#14579688—41no—
79#14579682—41no—
80C2×C163⋊C8—41no—
81C2×C163⋊C8C2311yescharacteristic
82C2×C163⋊C8C2311yescharacteristic
83C4⋊Dic163—21no—
84C4⋊Dic163C2311yescharacteristic
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4—21no—
89Dic163⋊C4D411yescharacteristic
90C22⋊Dic163—21no—
91C22⋊Dic163D411yescharacteristic
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96(C2×D163)⋊C4—21no—
97(C2×D163)⋊C4D411yescharacteristic
98C4×Dic163—21no—
99C4×Dic163C2311yescharacteristic
100C2×Q8×C163C2311yescharacteristic
101C22×C4×C163—21no—
102C22×C4×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C4⋊C4×C163—21no—
106C4⋊C4×C163—21no—
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116C22×D163C2×D411yescharacteristic
117Dic326—21no—
118Dic326—21no—
119Dic326—81no—
120D652—21no—
121D652—21no—
122D652—21no—
123D652—42no—
124D652—42no—
125D652—81no—
126D652#4112yes—
127C22⋊D163—22no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163C2×D411yescharacteristic
131C2×Dic163C2×D411yescharacteristic
132C2×Dic163C2×D411yescharacteristic
133C4×D163—21no—
134C4×D163—21no—
135C4×D163—42no—
136C4×D163—81no—
137C4×D163—81no—
138C163⋊C8—21no—
139C163⋊C8—21no—
140C23×C163C2×D411yescharacteristic
141Q8×C163—21no—
142Q8×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—21no—
149C2×C4×C163—81no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C8×C163—21no—
153C8×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163#13712yes—
157C22×C163C22×D411yescharacteristic
158C2×D163—21no—
159C2×D163—22no—
160C2×D163—21no—
161C2×D163—21no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—41no—
166C2×D163—41no—
167C2×D163—84no—
168C4×C163—21no—
169C4×C163—41no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163—41no—
173C4×C163C2×#4111yescharacteristic
174C4×C163H(5,𝔽2)11yescharacteristic
175Dic163—21no—
176Dic163—21no—
177Dic163—21no—
178Dic163—41no—
179Dic163—41no—
180C2×C163—22no—
181C2×C163#54611yescharacteristic
182C2×C163C2×#13711yescharacteristic
183C2×C163#57611yescharacteristic
184D163—41no—
185D163—42no—
186D163—81no—
187C163#331411yescharacteristic, complemented, Sylow
188#3314—1631nomaximal, Sylow
189#554—1631no—
190#546—1631no—
191#489—1631no—
192#484D16311yescharacteristic, complemented
193#480—1631no—
194#474—1631no—
195#468—1631no—
196#467—1631no—
197#464—1631no—
198#452—1631no—
199#450—1631no—
200#440—1631no—
201#435—1631no—
202#417—1631no—
203C2×#102—1631no—
204C2×#41—1631no—
205C22×D4—1631no—
206C2×Q16—1631no—
207C2×SD16—1631no—
208C2×SD16—1631no—
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
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⋊C4C2×D16311yescharacteristic
229C2×C22⋊C4—3261no—
230#107—1631no—
231#107C2×D16311yescharacteristic, complemented
232#107C2×D16311yescharacteristic, complemented
233C4⋊C8—1631no—
234#103—1631no—
235#103—1631no—
236#103C2×D16311yescharacteristic, complemented
237#103C2×D16311yescharacteristic, complemented
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242#102—1631no—
243#102—1631no—
244C22⋊C8—1631no—
245C22⋊C8C2×D16311yescharacteristic
246#97—1631no—
247C24—3261no—
248#41—3262no—
249#41—6521no—
250C2×Q8—3261no—
251C2×Q8—1631no—
252C2×Q8C22×D16311yescharacteristic
253C2×D4—3261no—
254C2×D4—3262no—
255C2×D4—3261no—
256C2×D4—3261no—
257C2×D4—6521no—
258C2×D4—1631no—
259C2×D4—1631no—
260C2×D4—1631no—
261C22×C4—21no—
262C22×C4—3261no—
263C22×C4—3261no—
264C22×C4—3261no—
265C22×C4—1631no—
266C22×C4C22×D16311yescharacteristic
267Q16—6521no—
268SD16—3262no—
269SD16—6521no—
270M16—6521no—
271C2×C8—6521no—
272C2×C8—1631no—
273C2×C8—1631no—
274C2×C8C22×D16311yescharacteristic
275C2×C8C22×D16311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3261no—
279C4⋊C4—3261no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—3261no—
283C4⋊C4—1631no—
284C4⋊C4—1631no—
285C4⋊C4C22×D16311yescharacteristic
286C4⋊C4C22×D16311yescharacteristic
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—13041no—
300C23—3261no—
301C23—3261no—
302C23—6522no—
303C23—6522no—
304C23—1631no—
305C23D4×D16311yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—13041no—
309Q8—3261no—
310Q8—3261no—
311D4—13041no—
312D4—3262no—
313D4—3261no—
314D4—3261no—
315D4—3261no—
316D4—6522no—
317D4—6522no—
318D4—1632no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—81no—
326C2×C4—13041no—
327C2×C4—13041no—
328C2×C4—3261no—
329C2×C4—3261no—
330C2×C4—3261no—
331C2×C4—3261no—
332C2×C4—6522no—
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—13044no—
345C22—3261no—
346C22—3261no—
347C22—3261no—
348C22—3262no—
349C22—6521no—
350C22—6521no—
351C22—6521no—
352C22—6521no—
353C22—6521no—
354C22#1910476712yes—
355C22C2×D4×D16311yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—3261no—
362C4—3261no—
363C4—3261no—
364C4—6521no—
365C4—6521no—
366C4#41×D16311yescharacteristic
367C4#1910478111yescharacteristic
368C2—22no—
369C2—13041no—
370C2—6522no—
371C2—6521no—
372C2#2378230711yescharacteristic
373C2C2×#1910476711yescharacteristic
374C2#2378246911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#484×C163C2
socleC22×C163C22×D4
prime radical#23782306C2
hypercenter#484D163
hypocenterC163#3314
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #23782370C2semidirect
3. #23782026C2semidirect
4. #23781980C2semidirect
5. #23781944C2semidirect
6. #23781912C2semidirect
7. #23781904C2semidirect
8. #23781890C2semidirect
9. #23781830C2semidirect
10. #23781818C2semidirect
11. #23781764C2semidirect
12. #23781733C2semidirect
13. #23781665C2semidirect
14. C2×#19104655C2semidirect
15. #484×C163C2semidirect
18. C2×#14579689C22semidirect
21. C2×#14579682C22semidirect
23. #19104734C22semidirect
27. #19104715C22semidirect
34. #19104700C22semidirect
38. C4⋊C4×D163C22semidirect
40. #19104668C22semidirect
41. Dic163⋊C8C22semidirect
42. #19104659C22semidirect
43. #19104658C22semidirect
50. (C2×D163)⋊C8C22semidirect
51. #19104645C22semidirect
54. #107×C163C22semidirect
55. #107×C163C22semidirect
56. #103×C163C22semidirect
57. #103×C163C22semidirect
187. C163#3314semidirect
192. #484D163semidirect
231. #107C2×D163semidirect
232. #107C2×D163semidirect
236. #103C2×D163semidirect
237. #103C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823702
#237823062
#237820262
#237819802
#237819442
#237819122
#237819042
#237818902
#237818302
#237818182
#237817642
#237817332
#237816652
C2×#191046552
#484×C1632
#3314163