Finite Group Databasebeta

← #35226373

Subgroups — #35226373

Order 20864.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782355C211yesmaximal, characteristic, complemented
2#23782311C211yesmaximal, characteristic
3#23782020C211yesmaximal, characteristic, complemented
4#23781996C211yesmaximal, characteristic, complemented
5#23781990C211yesmaximal, characteristic, complemented
6#23781966C211yesmaximal, characteristic, complemented
7#23781906C211yesmaximal, characteristic, complemented
8#23781894C211yesmaximal, characteristic, complemented
9#23781888C211yesmaximal, characteristic, complemented
10#23781829C211yesmaximal, characteristic, complemented
11#23781818C211yesmaximal, characteristic, complemented
12C4×#14579688C211yesmaximal, characteristic, complemented
13#23781662C211yesmaximal, characteristic, complemented
14C2×#19104657C211yesmaximal, characteristic, complemented
15C8×D4×C163C211yesmaximal, characteristic, complemented
16C2×#14579699C2211yescharacteristic
17C22×D652C2211yescharacteristic
18C2×Dic652C2211yescharacteristic, complemented
19C2×#14579688C2211yescharacteristic
20C2×#14579688C2211yescharacteristic
21#19104728—21no—
22#19104727C2211yescharacteristic, complemented
23#19104725—21no—
24#19104723C2211yescharacteristic
25#19104720—21no—
26#19104716—21no—
27#19104714C2211yescharacteristic
28#19104714C2211yescharacteristic
29#19104714C2211yescharacteristic, complemented
30#19104709C2211yescharacteristic
31#19104708—21no—
32#19104703—21no—
33C4×Dic326C2211yescharacteristic, complemented
34C4×D652C2211yescharacteristic
35#19104694C2211yescharacteristic, complemented
36#19104691—21no—
37C2×C4⋊Dic163C2211yescharacteristic
38C22⋊C4×D163—21no—
39#19104669C2211yescharacteristic, complemented
40#19104667C2211yescharacteristic, complemented
41#19104667C2211yescharacteristic, complemented
42#19104660C2211yescharacteristic, complemented
43#19104660C2211yescharacteristic, complemented
44#19104660C2211yescharacteristic, complemented
45#19104660C2211yescharacteristic, complemented
46#19104657C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104657C2211yescharacteristic
50#19104657C2211yescharacteristic
51#19104657C2211yescharacteristic
52C22×C8×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic, complemented
54C4×D4×C163C2211yescharacteristic
55C4⋊C8×C163C2211yescharacteristic, complemented
56C22⋊C8×C163C2211yescharacteristic
57C22⋊C8×C163C2211yescharacteristic, complemented
58C4×C8×C163C2211yescharacteristic, complemented
59C23×D163—21no—
60#14579699—22no—
61#14579699—41no—
62C2×Dic326C2311yescharacteristic
63C2×Dic326C2311yescharacteristic
64C2×D652—22no—
65C2×D652—21no—
66C2×D652—41no—
67C2×D652C2311yescharacteristic
68C2×D652C2311yescharacteristic
69C2×C22⋊D163—21no—
70C2×C22⋊D163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73C2×C4×D163—21no—
74Dic652—41no—
75#14579688—22no—
76#14579688—41no—
77C4⋊Dic163—21no—
78C4⋊Dic163C2311yescharacteristic
79C4⋊Dic163C2311yescharacteristic
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163—21no—
90C22⋊Dic163—21no—
91C22⋊Dic163—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4—21no—
96C4×Dic163—21no—
97C4×Dic163—21no—
98C2×D4×C163D411yescharacteristic
99C22×C4×C163C2311yescharacteristic
100C22×C4×C163C2311yescharacteristic
101C2×C8×C163—41no—
102C2×C8×C163—41no—
103C2×C8×C163C2311yescharacteristic
104C2×C8×C163C2311yescharacteristic
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163D411yescharacteristic
108C22⋊C4×C163D411yescharacteristic
109C22⋊C4×C163D411yescharacteristic
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112C22×D163—21no—
113C22×D163—42no—
114C22×D163—42no—
115C22×D163—81no—
116Dic326—21no—
117Dic326—21no—
118Dic326—21no—
119Dic326—21no—
120D652—21no—
121D652—21no—
122D652—21no—
123D652—42no—
124D652#4112yes—
125C22⋊D163—42no—
126C22⋊D163—81no—
127C2×Dic163—21no—
128C2×Dic163—21no—
129C2×Dic163—21no—
130C2×Dic163—21no—
131C2×Dic163—21no—
132C2×Dic163—21no—
133C2×Dic163—42no—
134C4×D163—42no—
135C4×D163—81no—
136C23×C163C2×D411yescharacteristic
137C23×C163C2×D411yescharacteristic
138D4×C163—22no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C8×C163—21no—
149C8×C163—21no—
150C8×C163—21no—
151C8×C163—21no—
152C22×C163—21no—
153C22×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163#13512yes—
158C22×C163C22×D411yescharacteristic
159C2×D163—22no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C2×D163—84no—
166C4×C163—21no—
167C4×C163—21no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163C2×#4111yescharacteristic
171C4×C163H(5,𝔽2)11yescharacteristic
172Dic163—41no—
173Dic163—41no—
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177Dic163—41no—
178C2×C163—22no—
179C2×C163—41no—
180C2×C163#54611yescharacteristic
181C2×C163C2×#13511yescharacteristic
182C2×C163#57711yescharacteristic
183D163—42no—
184D163—81no—
185C163#329611yescharacteristic, complemented, Sylow
186#3296—1631nomaximal, Sylow
187#552—1631no—
188#546—1631no—
189#488—1631no—
190#483—1631no—
191#482—1631no—
192#478—1631no—
193#467—1631no—
194#465—1631no—
195#464—1631no—
196#452—1631no—
197#450—1631no—
198C4×SD16—1631no—
199C8×D4D16311yescharacteristic, complemented
200#416—1631no—
201C2×#102—1631no—
202C2×#41—1631no—
203C22×D4—1631no—
204C2×Q16—1631no—
205C2×SD16—1631no—
206C2×SD16—1631no—
207C22×C8C2×D16311yescharacteristic
208C22×C8C2×D16311yescharacteristic, complemented
209#126—3261no—
210#125—3261no—
211#125—1631no—
212#124—1631no—
213#123—3261no—
214#123—3261no—
215C22⋊Q8—1631no—
216C22⋊Q8—1631no—
217C22⋊Q8—1631no—
218#121—3261no—
219#121—1631no—
220#120—3261no—
221C4×Q8—1631no—
222C4×D4—1631no—
223C4×D4C2×D16311yescharacteristic
224#117—3261no—
225#117—1631no—
226C2×C4⋊C4—1631no—
227C2×C22⋊C4—3261no—
228#107—1631no—
229#106—1631no—
230#106—1631no—
231C4⋊C8C2×D16311yescharacteristic, complemented
232#103—1631no—
233#103—1631no—
234#103—1631no—
235#103—1631no—
236#102—1631no—
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#102—1631no—
241#102—1631no—
242C22⋊C8C2×D16311yescharacteristic
243C22⋊C8C2×D16311yescharacteristic, complemented
244C4×C8C2×D16311yescharacteristic, complemented
245C24—3261no—
246#41—3262no—
247#41—6521no—
248C2×Q8—1631no—
249C2×Q8—1631no—
250C2×D4—3262no—
251C2×D4—3261no—
252C2×D4—3261no—
253C2×D4—3261no—
254C2×D4—6521no—
255C2×D4—1631no—
256C2×D4—1631no—
257C2×D4D65211yescharacteristic
258C22×C4—3261no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4C22×D16311yescharacteristic
262C22×C4C22×D16311yescharacteristic
263Q16—6521no—
264SD16—3262no—
265SD16—6521no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8C22×D16311yescharacteristic
269C2×C8C22×D16311yescharacteristic
270C2×C8C22×D16311yescharacteristic
271C2×C8C22×D16311yescharacteristic
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—3261no—
275C4⋊C4—3261no—
276C4⋊C4—3261no—
277C4⋊C4—3261no—
278C4⋊C4—3261no—
279C4⋊C4—1631no—
280C4⋊C4—1631no—
281C4⋊C4—1631no—
282C4⋊C4—1631no—
283C4⋊C4D65211yescharacteristic
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C22⋊C4—3261no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4D65211yescharacteristic
293C22⋊C4D65211yescharacteristic
294C42—3261no—
295C42—3261no—
296C42C22×D16311yescharacteristic
297C23—13041no—
298C23—3261no—
299C23—3261no—
300C23—6522no—
301C23—6522no—
302C23C2×D65211yescharacteristic
303C23C2×D65211yescharacteristic
304Q8—3261no—
305Q8—3261no—
306Q8—3261no—
307Q8—3261no—
308D4—22no—
309D4—13041no—
310D4—3261no—
311D4—3261no—
312D4—3261no—
313D4—6522no—
314D4—6522no—
315D4—1632no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—13041no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—6522no—
328C2×C4—6522no—
329C2×C4C2×D65211yescharacteristic
330C2×C4C2×D65211yescharacteristic
331C2×C4C2×D65211yescharacteristic
332C2×C4C2×D65211yescharacteristic
333C2×C4C23×D16311yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—13044no—
344C22—3262no—
345C22—6521no—
346C22—6521no—
347C22—6521no—
348C22—6521no—
349C22—6521no—
350C22#1910475712yes—
351C22C22×D65211yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—6521no—
357C4—6521no—
358C4—6521no—
359C4—6521no—
360C4—6521no—
361C4—6521no—
362C4#41×D16311yescharacteristic
363C4#1910478011yescharacteristic
364C2—22no—
365C2—41no—
366C2—13041no—
367C2—6522no—
368C2#2378231011yescharacteristic
369C2C2×#1910475711yescharacteristic
370C2#2378247711yescharacteristic
subgroupG/H
centerC22C22×D652
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC8×D4×C163C2
socleC22×C163C22×D4
prime radical#23782311C2
hypercenterC8×D4D163
hypocenterC163#3296
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #23782355C2semidirect
3. #23782020C2semidirect
4. #23781996C2semidirect
5. #23781990C2semidirect
6. #23781966C2semidirect
7. #23781906C2semidirect
8. #23781894C2semidirect
9. #23781888C2semidirect
10. #23781829C2semidirect
11. #23781818C2semidirect
12. C4×#14579688C2semidirect
13. #23781662C2semidirect
14. C2×#19104657C2semidirect
15. C8×D4×C163C2semidirect
18. C2×Dic652C22semidirect
22. #19104727C22semidirect
29. #19104714C22semidirect
33. C4×Dic326C22semidirect
35. #19104694C22semidirect
39. #19104669C22semidirect
40. #19104667C22semidirect
41. #19104667C22semidirect
42. #19104660C22semidirect
43. #19104660C22semidirect
44. #19104660C22semidirect
45. #19104660C22semidirect
53. C22×C8×C163C22semidirect
55. C4⋊C8×C163C22semidirect
57. C22⋊C8×C163C22semidirect
58. C4×C8×C163C22semidirect
185. C163#3296semidirect
199. C8×D4D163semidirect
208. C22×C8C2×D163semidirect
231. C4⋊C8C2×D163semidirect
243. C22⋊C8C2×D163semidirect
244. C4×C8C2×D163semidirect

Maximal subgroups

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

subgroupindex
#237823552
#237823112
#237820202
#237819962
#237819902
#237819662
#237819062
#237818942
#237818882
#237818292
#237818182
C4×#145796882
#237816622
C2×#191046572
C8×D4×C1632
#3296163