Finite Group Databasebeta

← #35224896

Subgroups — #35224896

Order 20864.

372 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#23782292C211yesmaximal, characteristic, complemented
2#23782240C211yesmaximal, characteristic
3#23782033C211yesmaximal, characteristic, complemented
4#23782032C211yesmaximal, characteristic, complemented
5#23782027C211yesmaximal, characteristic, complemented
6#23782026C211yesmaximal, characteristic, complemented
7#23782021C211yesmaximal, characteristic, complemented
8#23782020C211yesmaximal, characteristic, complemented
9#23782014C211yesmaximal, characteristic, complemented
10#23781672C211yesmaximal, characteristic, complemented
11#23781665C211yesmaximal, characteristic
12#23781660C211yesmaximal, characteristic
13#103×D163C211yesmaximal, characteristic, complemented
14#23781619C211yesmaximal, characteristic, complemented
15#487×C163C211yesmaximal, characteristic, complemented
16C2×#14579700C2211yescharacteristic
17C2×Q8×D163C2211yescharacteristic
18C2×#14579682C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20#19104734C2211yescharacteristic
21#19104733—21no—
22#19104733C2211yescharacteristic
23#19104727C2211yescharacteristic
24#19104726C2211yescharacteristic
25#19104725—21no—
26#19104724C2211yescharacteristic
27#19104723C2211yescharacteristic
28#19104720—21no—
29#19104720—21no—
30#19104713—21no—
31#19104713—21no—
32#19104713—21no—
33#19104711—21no—
34#19104693C2211yescharacteristic
35#19104692C2211yescharacteristic
36C4⋊C4×D163C2211yescharacteristic
37#19104660C2211yescharacteristic
38#19104660C2211yescharacteristic
39#19104659C2211yescharacteristic
40#19104659C2211yescharacteristic
41#19104659C2211yescharacteristic
42#19104659C2211yescharacteristic
43#19104658C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104657C2211yescharacteristic
47#19104655C2211yescharacteristic
48#19104655C2211yescharacteristic
49#19104646C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C4⋊Q8×C163C2211yescharacteristic
53C4⋊Q8×C163C2211yescharacteristic
54#103×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C4×C8×C163C2211yescharacteristic
59#14579700—22no—
60#14579700—22no—
61C2×Dic326—21no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64Q8×D163—22no—
65Q8×D163—22no—
66C2×D652—21no—
67C2×D652C2311yescharacteristic
68C2×C4×D163—21no—
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163C2311yescharacteristic
72#14579682—22no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C8×D163—22no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4D411yescharacteristic
87Dic163⋊C4D411yescharacteristic
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
94(C2×D163)⋊C4D411yescharacteristic
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C2×Q8×C163—21no—
98C2×Q8×C163C2311yescharacteristic
99C2×Q8×C163C2311yescharacteristic
100C2×C8×C163C2311yescharacteristic
101C2×C8×C163C2311yescharacteristic
102C4⋊C4×C163—21no—
103C4⋊C4×C163—21no—
104C4⋊C4×C163—21no—
105C4⋊C4×C163C2311yescharacteristic
106C4⋊C4×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163C2×D411yescharacteristic
110Dic326—21no—
111Dic326—21no—
112Dic326—42no—
113Dic326—81no—
114D652—21no—
115D652—21no—
116D652—42no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C4×D163—42no—
124C4×D163—42no—
125C4×D163—81no—
126C4×D163#4112yes—
127C4×D163#4112yes—
128C163⋊C8—21no—
129C163⋊C8—21no—
130Q8×C163—21no—
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163—21no—
134Q8×C163—81no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2411yescharacteristic
142C8×C163—21no—
143C8×C163—21no—
144C22×C163C22×D411yescharacteristic
145C2×D163—21no—
146C2×D163—21no—
147C2×D163—41no—
148C2×D163—41no—
149C2×D163—41no—
150C2×D163#12412yes—
151C4×C163—21no—
152C4×C163—21no—
153C4×C163—41no—
154C4×C163—41no—
155C4×C163—41no—
156C4×C163—41no—
157C4×C163C2×#4111yescharacteristic
158C4×C163C2×#4111yescharacteristic
159Dic163—21no—
160Dic163—21no—
161Dic163—41no—
162Dic163—41no—
163Dic163—41no—
164Dic163#12412yes—
165C2×C163C2×#12411yescharacteristic
166C2×C163#57711yescharacteristic
167C2×C163#57811yescharacteristic
168D163—22no—
169D163—81no—
170C163#313711yescharacteristic, complemented, Sylow
171#3137—1631nomaximal, Sylow
172#544—1631no—
173#536—1631no—
174#491—1631no—
175#491—1631no—
176#489—1631no—
177#489—1631no—
178#488—1631no—
179#488—1631no—
180#487—1631no—
181#487D16311yescharacteristic, complemented
182#419—1631no—
183#417—1631no—
184#416—1631no—
185C2×#103—1631no—
186#405—1631no—
187C2×#41—1631no—
188C22×Q8—1631no—
189C2×M16—1631no—
190C22×C8—1631no—
191C4⋊Q8—3261no—
192C4⋊Q8—1631no—
193C4⋊Q8—1631no—
194C4⋊Q8C2×D16311yescharacteristic
195C4⋊Q8C2×D16311yescharacteristic
196#125—3261no—
197#125—1631no—
198#125—1631no—
199#124—1631no—
200#124—1631no—
201#123—3261no—
202#123—3261no—
203C22⋊Q8—3261no—
204C22⋊Q8—3261no—
205C22⋊Q8—3261no—
206C22⋊Q8—3261no—
207#117—1631no—
208#117—1631no—
209C2×C4⋊C4—1631no—
210#103—1631no—
211#103—1631no—
212#103—1631no—
213#103—1631no—
214#103—1631no—
215#103—1631no—
216#103—1631no—
217#103—1631no—
218#103C2×D16311yescharacteristic
219#103C2×D16311yescharacteristic
220#103C2×D16311yescharacteristic
221#103C2×D16311yescharacteristic
222#102—1631no—
223#102—1631no—
224#102—1631no—
225#102—1631no—
226#97—1631no—
227#97—1631no—
228C4×C8—1631no—
229C4×C8C2×D16311yescharacteristic
230#41—3262no—
231#41—3262no—
232C2×Q8—21no—
233C2×Q8—3261no—
234C2×Q8—3261no—
235C2×Q8—3262no—
236C2×Q8—3262no—
237C2×Q8—1631no—
238C2×Q8C22×D16311yescharacteristic
239C2×Q8C22×D16311yescharacteristic
240C2×D4—3261no—
241C2×D4—1631no—
242C22×C4—3261no—
243C22×C4—3261no—
244C22×C4—3261no—
245C22×C4—1631no—
246M16—3262no—
247C2×C8—3262no—
248C2×C8—1631no—
249C2×C8—1631no—
250C2×C8C22×D16311yescharacteristic
251C2×C8C22×D16311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—3261no—
256C4⋊C4—3261no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—3261no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—1631no—
264C4⋊C4—1631no—
265C4⋊C4—1631no—
266C4⋊C4—1631no—
267C4⋊C4C22×D16311yescharacteristic
268C4⋊C4C22×D16311yescharacteristic
269C22⋊C4—3261no—
270C22⋊C4—3261no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—1631no—
275C22⋊C4—1631no—
276C42—3261no—
277C42—1631no—
278C42C22×D16311yescharacteristic
279C23—3261no—
280C23—1631no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—81no—
286Q8—13041no—
287Q8—3261no—
288Q8—3261no—
289Q8—6522no—
290D4—3261no—
291D4—3261no—
292D4—6522no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—13041no—
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×C4—1632no—
307C2×C4—1632no—
308C2×C4D4×D16311yescharacteristic
309C2×C4D4×D16311yescharacteristic
310C2×C4C23×D16311yescharacteristic
311C8—21no—
312C8—21no—
313C8—3261no—
314C8—3261no—
315C22—3261no—
316C22—3261no—
317C22—6521no—
318C22—6521no—
319C22—6521no—
320C22—1632no—
321C22C2×D4×D16311yescharacteristic
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—3261no—
329C4—3261no—
330C4—6521no—
331C4—6521no—
332C4—6521no—
333C4—1632no—
334C4#41×D16311yescharacteristic
335C4#41×D16311yescharacteristic
336C2—13041no—
337C2—3262no—
338C2#124×D16311yescharacteristic
339C2#2378247111yescharacteristic
340C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#487×C163C2
socleC22×C163C22×D4
prime radicalC2×#14579700C22
hypercenter#487D163
hypocenterC163#3137
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #23782292C2semidirect
3. #23782033C2semidirect
4. #23782032C2semidirect
5. #23782027C2semidirect
6. #23782026C2semidirect
7. #23782021C2semidirect
8. #23782020C2semidirect
9. #23782014C2semidirect
10. #23781672C2semidirect
13. #103×D163C2semidirect
14. #23781619C2semidirect
15. #487×C163C2semidirect
170. C163#3137semidirect
181. #487D163semidirect

Maximal subgroups

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

subgroupindex
#237822922
#237822402
#237820332
#237820322
#237820272
#237820262
#237820212
#237820202
#237820142
#237816722
#237816652
#237816602
#103×D1632
#237816192
#487×C1632
#3137163