Finite Group Databasebeta

← #35224683

Subgroups — #35224683

Order 20864.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782212C211yesmaximal, characteristic, complemented
2C4×#14579698C211yesmaximal, characteristic, complemented
3#23781991C211yesmaximal, characteristic, complemented
4#23781987C211yesmaximal, characteristic, complemented
5#23781949C211yesmaximal, characteristic, complemented
6#23781943C211yesmaximal, characteristic, complemented
7#23781864C211yesmaximal, characteristic, complemented
8#23781858C211yesmaximal, characteristic, complemented
9#23781811C211yesmaximal, characteristic, complemented
10#23781807C211yesmaximal, characteristic, complemented
11#23781775C211yesmaximal, characteristic, complemented
12#23781767C211yesmaximal, characteristic, complemented
13#23781765C211yesmaximal, characteristic, complemented
14#23781720C211yesmaximal, characteristic, complemented
15#442×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×#14579698C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579684C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104731C2211yescharacteristic
23#19104721—21no—
24#19104710C2211yescharacteristic
25#19104708—21no—
26#19104706C2211yescharacteristic
27C4×Dic326C2211yescharacteristic
28#19104700C2211yescharacteristic
29C4×D652C2211yescharacteristic
30D4×Dic163C2211yescharacteristic
31C4×C22⋊D163—21no—
32C4×C22⋊D163—21no—
33(C22⋊D163)⋊C4—21no—
34#19104693C2211yescharacteristic
35#19104691—21no—
36C2×C4×Dic163—21no—
37C42×D163C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104668C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104658C2211yescharacteristic
45#19104657C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104655C2211yescharacteristic
49(C2×D163)⋊C8C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104646C2211yescharacteristic
52C2×D8×C163C2211yescharacteristic
53C4×D4×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55#106×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59#14579699—44no—
60#14579698—22no—
61#14579698—22no—
62C2×Dic326—21no—
63C2×Dic326C2311yescharacteristic
64C2×D652—21no—
65C2×D652C2311yescharacteristic
66C2×C22⋊D163—21no—
67C2×C22⋊D163—21no—
68C2×C22⋊D163—21no—
69C2×C22⋊D163—21no—
70C22×Dic163—21no—
71C22×Dic163—21no—
72C2×C4×D163—21no—
73C2×C4×D163D411yescharacteristic
74C2×C4×D163D411yescharacteristic
75C2×C4×D163C2311yescharacteristic
76#14579688—22no—
77#14579687—22no—
78#14579684—22no—
79C2×C163⋊C8C2311yescharacteristic
80C2×C163⋊C8C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82C4⋊Dic163C2311yescharacteristic
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88(C2×D163)⋊C4—21no—
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91C4×Dic163—21no—
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163D411yescharacteristic
95C4×Dic163D411yescharacteristic
96C4×Dic163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C2×D4×C163C2311yescharacteristic
99C22×C4×C163—21no—
100C22×C4×C163—21no—
101D8×C163—22no—
102C2×C8×C163C2311yescharacteristic
103C2×C8×C163C2311yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C4⋊C4×C163C2311yescharacteristic
106C22⋊C4×C163—21no—
107C22⋊C4×C163—21no—
108C42×C163C2311yescharacteristic
109C22×D163—21no—
110C22×D163C2×D411yescharacteristic
111Dic326—21no—
112Dic326—42no—
113Dic326#4112yescomplemented
114D652—21no—
115D652—21no—
116D652—42no—
117C22⋊D163—24no—
118C22⋊D163—42no—
119C22⋊D163—42no—
120C22⋊D163—42no—
121C2×Dic163—24no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—42no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—21no—
130C4×D163—21no—
131C4×D163—22no—
132C4×D163—41no—
133C4×D163—42no—
134C163⋊C8—21no—
135C163⋊C8—21no—
136C23×C163—21no—
137C23×C163—21no—
138D4×C163—21no—
139D4×C163—21no—
140D4×C163—21no—
141D4×C163#4112yescomplemented
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—42no—
145C2×C4×C163—42no—
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2411yescharacteristic
149C8×C163—21no—
150C8×C163—21no—
151C22×C163—22no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163—41no—
156C22×C163—41no—
157C22×C163C22×D411yescharacteristic
158C2×D163—21no—
159C2×D163—21no—
160C2×D163—21no—
161C2×D163—41no—
162C2×D163—41no—
163C2×D163—41no—
164C4×C163—21no—
165C4×C163—21no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163C2×#4111yescharacteristic
169C4×C163C2×#4111yescharacteristic
170Dic163—21no—
171Dic163—22no—
172Dic163—21no—
173Dic163—41no—
174Dic163—41no—
175Dic163AGL(1,ℤ/8)12yes—
176C2×C163—42no—
177C2×C163—81no—
178C2×C163#52511yescharacteristic
179C2×C163C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C163#57511yescharacteristic
181D163—41no—
182D163—81no—
183C163#311511yescharacteristic, complemented, Sylow
184#3115—1631nomaximal, Sylow
185#532—1631no—
186C4×#41—1631no—
187#482—1631no—
188#482—1631no—
189#475—1631no—
190#474—1631no—
191#460—1631no—
192#459—1631no—
193#449—1631no—
194#449—1631no—
195#442—1631no—
196#442D16311yescharacteristic, complemented
197#440—1631no—
198#440—1631no—
199#432—1631no—
200C2×#41—3261no—
201C2×#41—1631no—
202C2×SD16—1631no—
203C2×SD16—1631no—
204C2×D8—1631no—
205C2×D8C2×D16311yescharacteristic
206C4⋊Q8—1631no—
207#127—1631no—
208#124—3261no—
209#121—3261no—
210#121—1631no—
211#121—1631no—
212C4×Q8—1631no—
213C4×Q8—1631no—
214C4×D4—3261no—
215C4×D4—3261no—
216C4×D4—3261no—
217C4×D4—1631no—
218C4×D4—1631no—
219C4×D4C2×D16311yescharacteristic
220C4×D4C2×D16311yescharacteristic
221#117—3261no—
222#117—1631no—
223C2×C42—3261no—
224C2×C42—1631no—
225#107—1631no—
226#107—1631no—
227#106—1631no—
228#106C2×D16311yescharacteristic
229C4⋊C8—1631no—
230C4⋊C8—1631no—
231#103—1631no—
232#103—1631no—
233#102—1631no—
234#102—1631no—
235#102—1631no—
236#102—1631no—
237#102C2×D16311yescharacteristic
238#102C2×D16311yescharacteristic
239C22⋊C8—1631no—
240C22⋊C8—1631no—
241#97—1631no—
242#97C2×D16311yescharacteristic
243#41—3262no—
244#41—3262no—
245#41—6524no—
246C2×Q8—3261no—
247C2×Q8—1631no—
248C2×D4—3261no—
249C2×D4—3261no—
250C2×D4—3261no—
251C2×D4—3261no—
252C2×D4—3261no—
253C2×D4—1631no—
254C2×D4C22×D16311yescharacteristic
255C2×D4C22×D16311yescharacteristic
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—3261no—
259C22×C4—3261no—
260C22×C4—3261no—
261C22×C4—1631no—
262C22×C4—1631no—
263C22×C4—1631no—
264SD16—3262no—
265SD16—3262no—
266D8—22no—
267D8—3262no—
268C2×C8—1631no—
269C2×C8—1631no—
270C2×C8C22×D16311yescharacteristic
271C2×C8C22×D16311yescharacteristic
272C4⋊C4—3261no—
273C4⋊C4—3261no—
274C4⋊C4—3261no—
275C4⋊C4—1631no—
276C4⋊C4—1631no—
277C4⋊C4C22×D16311yescharacteristic
278C4⋊C4C22×D16311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—3261no—
284C22⋊C4—3261no—
285C22⋊C4—3261no—
286C42—3261no—
287C42—3261no—
288C42—3261no—
289C42—1631no—
290C42—1631no—
291C42—1631no—
292C42C22×D16311yescharacteristic
293C23—21no—
294C23—21no—
295C23—3261no—
296C23—1631no—
297Q8—3261no—
298Q8—6522no—
299Q8—1632no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—3264no—
304D4—3261no—
305D4—3261no—
306D4—6522no—
307D4—6522no—
308D4—6522no—
309D4—6522no—
310D4#1457969912yescomplemented
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—3262no—
316C2×C4—3264no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4—3261no—
320C2×C4—3261no—
321C2×C4—3261no—
322C2×C4—6521no—
323C2×C4—6522no—
324C2×C4—6522no—
325C2×C4—1631no—
326C2×C4—1631no—
327C2×C4—1631no—
328C2×C4D4×D16311yescharacteristic
329C2×C4D4×D16311yescharacteristic
330C2×C4C23×D16311yescharacteristic
331C8—21no—
332C8—21no—
333C8—3261no—
334C8—3261no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—3261no—
342C22—3261no—
343C22—3261no—
344C22—6521no—
345C22—6521no—
346C22—6521no—
347C22C2×D4×D16311yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—41no—
352C4—3261no—
353C4—3262no—
354C4—3261no—
355C4—6521no—
356C4—6521no—
357C4—1632no—
358C4#41×D16311yescharacteristic
359C4C2×#1457969911yescharacteristic
360C2—42no—
361C2—81no—
362C2—13041no—
363C2—6521no—
364C2#2378217411yescharacteristic
365C2AGL(1,ℤ/8)×D16311yescharacteristic
366C2#2378246111yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#442×C163C2
socleC22×C163C22×D4
hypercenter#442D163
hypocenterC163#3115
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #23782212C2semidirect
2. C4×#14579698C2semidirect
3. #23781991C2semidirect
4. #23781987C2semidirect
5. #23781949C2semidirect
6. #23781943C2semidirect
7. #23781864C2semidirect
8. #23781858C2semidirect
9. #23781811C2semidirect
10. #23781807C2semidirect
11. #23781775C2semidirect
12. #23781767C2semidirect
13. #23781765C2semidirect
14. #23781720C2semidirect
15. #442×C163C2semidirect
113. Dic326#41semidirect
141. D4×C163#41semidirect
183. C163#3115semidirect
196. #442D163semidirect
310. D4#14579699semidirect

Maximal subgroups

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

subgroupindex
#237822122
C4×#145796982
#237819912
#237819872
#237819492
#237819432
#237818642
#237818582
#237818112
#237818072
#237817752
#237817672
#237817652
#237817202
#442×C1632
#3115163