Finite Group Databasebeta

← #35227116

Subgroups — #35227116

Order 20864.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782383C211yesmaximal, characteristic, complemented
2#23782316C211yesmaximal, characteristic
3#23782069C211yesmaximal, characteristic, complemented
4#23782002C211yesmaximal, characteristic, complemented
5#23782000C211yesmaximal, characteristic, complemented
6#23781991C211yesmaximal, characteristic, complemented
7#23781989C211yesmaximal, characteristic, complemented
8#23781949C211yesmaximal, characteristic, complemented
9#23781944C211yesmaximal, characteristic, complemented
10#23781912C211yesmaximal, characteristic, complemented
11#23781911C211yesmaximal, characteristic, complemented
12#23781752C211yesmaximal, characteristic, complemented
13#23781741C211yesmaximal, characteristic, complemented
14#23781704C211yesmaximal, characteristic, complemented
15#428×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×#14579688C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19#19104734C2211yescharacteristic
20#19104734C2211yescharacteristic
21#19104729—21no—
22#19104728—21no—
23#19104721—21no—
24#19104720—21no—
25#19104718—21no—
26#19104715C2211yescharacteristic
27#19104714C2211yescharacteristic
28#19104713—21no—
29#19104710C2211yescharacteristic
30#19104709C2211yescharacteristic
31C4×Dic326—21no—
32#19104700C2211yescharacteristic
33#19104697C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35#19104693C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic
37#19104669C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104666C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×M16×C163C2211yescharacteristic
53#117×C163C2211yescharacteristic
54C2×C4⋊C4×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#107×C163C2211yescharacteristic
57#106×C163C2211yescharacteristic
58#106×C163C2211yescharacteristic
59#14579699—44no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63C2×D652—21no—
64C2×D652C2311yescharacteristic
65C2×C22⋊D163—21no—
66C2×C22⋊D163D411yescharacteristic
67C2×C4×D163—21no—
68C2×C4×D163—21no—
69C2×C4×D163C2311yescharacteristic
70#14579688—22no—
71#14579682—22no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C4⋊Dic163—21no—
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4D411yescharacteristic
83C22⋊Dic163—21no—
84C22⋊Dic163D411yescharacteristic
85(C2×D163)⋊C4—21no—
86(C2×D163)⋊C4—21no—
87(C2×D163)⋊C4—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4D411yescharacteristic
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163C2311yescharacteristic
94C22×C4×C163—21no—
95C22×C4×C163C2311yescharacteristic
96M16×C163—22no—
97C2×C8×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C4⋊C4×C163—21no—
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C22⋊C4×C163—21no—
105C42×C163—21no—
106C22×D163—21no—
107C22×D163C2×D411yescharacteristic
108Dic326—21no—
109Dic326—21no—
110Dic326—42no—
111Dic326—42no—
112D652—21no—
113D652—21no—
114D652—42no—
115C22⋊D163—22no—
116C22⋊D163—42no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C4×D163—21no—
124C4×D163—21no—
125C4×D163—42no—
126C4×D163—42no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C23×C163C2×D411yescharacteristic
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—42no—
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2411yescharacteristic
139C8×C163—21no—
140C8×C163#4112yescomplemented
141C22×C163—21no—
142C22×C163—21no—
143C22×C163—21no—
144C22×C163C22×D411yescharacteristic
145C2×D163—21no—
146C2×D163—21no—
147C2×D163—21no—
148C2×D163—41no—
149C2×D163—41no—
150C2×D163—41no—
151C4×C163—21no—
152C4×C163—41no—
153C4×C163—41no—
154C4×C163—41no—
155C4×C163—41no—
156C4×C163C2×#4111yescharacteristic
157C4×C1632−1+411yescharacteristic
158Dic163—21no—
159Dic163—21no—
160Dic163—41no—
161Dic163—41no—
162Dic163—41no—
163Dic163AGL(1,ℤ/8)12yes—
164C2×C163—41no—
165C2×C163#54711yescharacteristic
166C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C163#57811yescharacteristic
168D163—41no—
169D163—81no—
170C163#335311yescharacteristic, complemented, Sylow
171#3353—1631nomaximal, Sylow
172#556—1631no—
173#547—1631no—
174#500—1631no—
175#484—1631no—
176#484—1631no—
177#482—1631no—
178#482—1631no—
179#475—1631no—
180#474—1631no—
181#468—1631no—
182#468—1631no—
183C4×SD16—1631no—
184#436—1631no—
185#428—1631no—
186#428D16311yescharacteristic, complemented
187C2×#41—3261no—
188C2×SD16—1631no—
189C2×M16—1631no—
190C2×M16C2×D16311yescharacteristic
191C4⋊Q8—1631no—
192C4⋊Q8—1631no—
193#126—3261no—
194#126—3261no—
195#124—3261no—
196#123—3261no—
197#123—3261no—
198C22⋊Q8—3261no—
199C22⋊Q8—1631no—
200C22⋊Q8—1631no—
201#121—1631no—
202#121—1631no—
203C4×Q8—3261no—
204C4×Q8—1631no—
205C4×D4—1631no—
206C4×D4—1631no—
207#117—1631no—
208#117C2×D16311yescharacteristic
209C2×C4⋊C4—1631no—
210C2×C4⋊C4C2×D16311yescharacteristic
211#107—1631no—
212#107—1631no—
213#107—1631no—
214#107—1631no—
215#107C2×D16311yescharacteristic
216#107C2×D16311yescharacteristic
217#106—1631no—
218#106C2×D16311yescharacteristic
219#106C2×D16311yescharacteristic
220C4⋊C8—1631no—
221#103—1631no—
222#103—1631no—
223#103—1631no—
224#102—1631no—
225#102—1631no—
226#102—1631no—
227C22⋊C8—1631no—
228C22⋊C8—1631no—
229C4×C8—1631no—
230#41—6524no—
231C2×Q8—3261no—
232C2×Q8—3261no—
233C2×Q8—1631no—
234C2×D4—3261no—
235C2×D4—3261no—
236C2×D4—1631no—
237C2×D4—1631no—
238C22×C4—21no—
239C22×C4—3261no—
240C22×C4—3261no—
241C22×C4—1631no—
242C22×C4C22×D16311yescharacteristic
243SD16—3262no—
244M16—22no—
245M16—3262no—
246C2×C8—1631no—
247C2×C8—1631no—
248C2×C8C22×D16311yescharacteristic
249C2×C8C22×D16311yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—3261no—
252C4⋊C4—3261no—
253C4⋊C4—3261no—
254C4⋊C4—3261no—
255C4⋊C4—3261no—
256C4⋊C4—3261no—
257C4⋊C4—1631no—
258C4⋊C4—1631no—
259C4⋊C4—1631no—
260C4⋊C4C22×D16311yescharacteristic
261C4⋊C4C22×D16311yescharacteristic
262C4⋊C4C22×D16311yescharacteristic
263C4⋊C4C22×D16311yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—3261no—
266C22⋊C4—3261no—
267C22⋊C4—3261no—
268C22⋊C4—3261no—
269C22⋊C4—3261no—
270C22⋊C4—3261no—
271C22⋊C4—1631no—
272C22⋊C4—1631no—
273C42—21no—
274C42—3261no—
275C42—3261no—
276C42—1631no—
277C23—3261no—
278C23—1631no—
279C23D4×D16311yescharacteristic
280Q8—3261no—
281Q8—3261no—
282Q8—6522no—
283Q8—6522no—
284D4—3262no—
285D4—3261no—
286D4—3261no—
287D4—6522no—
288D4—6522no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—3261no—
297C2×C4—3261no—
298C2×C4—3261no—
299C2×C4—3261no—
300C2×C4—3261no—
301C2×C4—6522no—
302C2×C4—6522no—
303C2×C4—1631no—
304C2×C4—1631no—
305C2×C4—1631no—
306C2×C4D4×D16311yescharacteristic
307C2×C4C23×D16311yescharacteristic
308C8—21no—
309C8—3261no—
310C8—3261no—
311C8#1457969912yescomplemented
312C22—21no—
313C22—21no—
314C22—21no—
315C22—3261no—
316C22—3261no—
317C22—3261no—
318C22—6521no—
319C22—6521no—
320C22—6521no—
321C22C2×D4×D16311yescharacteristic
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—3261no—
328C4—3261no—
329C4—6521no—
330C4—6521no—
331C4—6521no—
332C4—1632no—
333C4C2×#1457969911yescharacteristic
334C4#1910478311yescharacteristic
335C2—41no—
336C2—13041no—
337C2—6521no—
338C2#2378231611yescharacteristic
339C2AGL(1,ℤ/8)×D16311yescharacteristic
340C2#2378248111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#428×C163C2
socleC22×C163C22×D4
prime radical#23782316C2
hypercenter#428D163
hypocenterC163#3353
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #23782383C2semidirect
3. #23782069C2semidirect
4. #23782002C2semidirect
5. #23782000C2semidirect
6. #23781991C2semidirect
7. #23781989C2semidirect
8. #23781949C2semidirect
9. #23781944C2semidirect
10. #23781912C2semidirect
11. #23781911C2semidirect
12. #23781752C2semidirect
13. #23781741C2semidirect
14. #23781704C2semidirect
15. #428×C163C2semidirect
140. C8×C163#41semidirect
170. C163#3353semidirect
186. #428D163semidirect
311. C8#14579699semidirect

Maximal subgroups

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

subgroupindex
#237823832
#237823162
#237820692
#237820022
#237820002
#237819912
#237819892
#237819492
#237819442
#237819122
#237819112
#237817522
#237817412
#237817042
#428×C1632
#3353163