Finite Group Databasebeta

← #35227113

Subgroups — #35227113

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#23782386C211yesmaximal, characteristic, complemented
2#23782316C211yesmaximal, characteristic
3#23782069C211yesmaximal, characteristic, complemented
4#23781999C211yesmaximal, characteristic, complemented
5#23781992C211yesmaximal, characteristic, complemented
6#23781990C211yesmaximal, characteristic, complemented
7#23781948C211yesmaximal, characteristic, complemented
8#23781944C211yesmaximal, characteristic, complemented
9#23781914C211yesmaximal, characteristic, complemented
10#23781913C211yesmaximal, characteristic, complemented
11#23781750C211yesmaximal, characteristic, complemented
12#23781740C211yesmaximal, characteristic, complemented
13#23781706C211yesmaximal, characteristic, complemented
14#23781705C211yesmaximal, characteristic, complemented
15#484×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×#14579686C2211yescharacteristic
18C2×#14579683C2211yescharacteristic
19C2×#14579682C2211yescharacteristic
20#19104734C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104730—21no—
23#19104728—21no—
24#19104721—21no—
25#19104720—21no—
26#19104718—21no—
27#19104713—21no—
28#19104711C2211yescharacteristic
29#19104710C2211yescharacteristic
30#19104709C2211yescharacteristic
31Q8×Dic163C2211yescharacteristic
32#19104700—21no—
33#19104697C2211yescharacteristic
34C4×C22⋊D163C2211yescharacteristic
35#19104694C2211yescharacteristic
36#19104693C2211yescharacteristic
37C4⋊C4×D163C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104668C2211yescharacteristic
42#19104667C2211yescharacteristic
43#19104666C2211yescharacteristic
44#19104666C2211yescharacteristic
45Dic163⋊C8C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104657C2211yescharacteristic
48#19104657C2211yescharacteristic
49#19104655C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C22⋊Q8×C163C2211yescharacteristic
53C2×C4⋊C4×C163C2211yescharacteristic
54#107×C163C2211yescharacteristic
55#107×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59#14579699—44no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×D652—21no—
63C2×D652C2311yescharacteristic
64C2×C22⋊D163—21no—
65C2×C22⋊D163D411yescharacteristic
66C2×C4×D163—21no—
67C2×C4×D163—21no—
68C2×C4×D163C2311yescharacteristic
69#14579686—22no—
70#14579683—22no—
71#14579682—22no—
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C4⋊Dic163—21no—
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
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×Dic163—21no—
94C4×Dic163C2311yescharacteristic
95C2×Q8×C163C2311yescharacteristic
96C22×C4×C163—21no—
97C22×C4×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C4⋊C4×C163—21no—
101C4⋊C4×C163—21no—
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C22×D163—21no—
107C22×D163C2×D411yescharacteristic
108Dic326—42no—
109Dic326—42no—
110D652—21no—
111D652—21no—
112D652—42no—
113C22⋊D163—22no—
114C22⋊D163—42no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C4×D163—21no—
122C4×D163—21no—
123C4×D163—42no—
124C4×D163—42no—
125C163⋊C8—21no—
126C163⋊C8#4112yescomplemented
127C23×C163C2×D411yescharacteristic
128Q8×C163—21no—
129Q8×C163—21no—
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—21no—
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#484D16311yescharacteristic, complemented
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#428—1631no—
187C2×#41—3261no—
188C2×SD16—1631no—
189C2×M16—1631no—
190C2×M16—1631no—
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⋊Q8C2×D16311yescharacteristic
201#121—1631no—
202#121—1631no—
203C4×Q8—3261no—
204C4×Q8—1631no—
205C4×D4—1631no—
206C4×D4—1631no—
207#117—1631no—
208#117—1631no—
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#106—1631no—
219#106—1631no—
220C4⋊C8—1631no—
221#103—1631no—
222#103C2×D16311yescharacteristic
223#103C2×D16311yescharacteristic
224#102—1631no—
225#102—1631no—
226#102—1631no—
227C22⋊C8—1631no—
228C22⋊C8C2×D16311yescharacteristic
229C4×C8—1631no—
230#41—6524no—
231C2×Q8—3261no—
232C2×Q8—3261no—
233C2×Q8C22×D16311yescharacteristic
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—3262no—
245M16—3262no—
246C2×C8—1631no—
247C2×C8—1631no—
248C2×C8C22×D16311yescharacteristic
249C2×C8C22×D16311yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—21no—
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⋊C4—1631no—
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—3261no—
274C42—3261no—
275C42—3261no—
276C42—1631no—
277C23—3261no—
278C23—1631no—
279C23D4×D16311yescharacteristic
280Q8—21no—
281Q8—21no—
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—21no—
310C8—3261no—
311C8—1632no—
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—
333C4#41×D16311yescharacteristic
334C4#1910478211yescharacteristic
335C2—41no—
336C2—13041no—
337C2—6521no—
338C2#2378231311yescharacteristic
339C2AGL(1,ℤ/8)×D16311yescharacteristic
340C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#484×C163C2
socleC22×C163C22×D4
prime radical#23782316C2
hypercenter#484D163
hypocenterC163#3353
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. #23782386C2semidirect
3. #23782069C2semidirect
4. #23781999C2semidirect
5. #23781992C2semidirect
6. #23781990C2semidirect
7. #23781948C2semidirect
8. #23781944C2semidirect
9. #23781914C2semidirect
10. #23781913C2semidirect
11. #23781750C2semidirect
12. #23781740C2semidirect
13. #23781706C2semidirect
14. #23781705C2semidirect
15. #484×C163C2semidirect
126. C163⋊C8#41semidirect
170. C163#3353semidirect
176. #484D163semidirect

Maximal subgroups

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

subgroupindex
#237823862
#237823162
#237820692
#237819992
#237819922
#237819902
#237819482
#237819442
#237819142
#237819132
#237817502
#237817402
#237817062
#237817052
#484×C1632
#3353163