Finite Group Databasebeta

← #35226001

Subgroups — #35226001

Order 20864.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782286C211yesmaximal, characteristic, complemented
2#23781955C211yesmaximal, characteristic, complemented
3#23781952C211yesmaximal, characteristic, complemented
4#23781950C211yesmaximal, characteristic, complemented
5#23781947C211yesmaximal, characteristic, complemented
6#23781921C211yesmaximal, characteristic, complemented
7#23781916C211yesmaximal, characteristic
8#23781896C211yesmaximal, characteristic, complemented
9#23781892C211yesmaximal, characteristic, complemented
10#23781887C211yesmaximal, characteristic, complemented
11#23781884C211yesmaximal, characteristic, complemented
12#23781880C211yesmaximal, characteristic, complemented
13#23781877C211yesmaximal, characteristic, complemented
14#23781692C211yesmaximal, characteristic, complemented
15#542×C163C211yesmaximal, characteristic, complemented
16C2×#14579689C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18C2×#14579686C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20C2×#14579683C2211yescharacteristic
21C22×C163⋊C8C2211yescharacteristic
22#19104735C2211yescharacteristic
23#19104730—21no—
24#19104723C2211yescharacteristic
25#19104721—21no—
26#19104717—21no—
27#19104714C2211yescharacteristic
28#19104713—21no—
29#19104709C2211yescharacteristic
30C4×Dic326C2211yescharacteristic
31C4×D652C2211yescharacteristic
32#19104666C2211yescharacteristic
33#19104666C2211yescharacteristic
34#19104666C2211yescharacteristic
35#19104666C2211yescharacteristic
36C4⋊(C163⋊C8)C2211yescharacteristic
37C4⋊(C163⋊C8)C2211yescharacteristic
38C4⋊(C163⋊C8)C2211yescharacteristic
39C4⋊(C163⋊C8)C2211yescharacteristic
40#19104659C2211yescharacteristic
41#19104659C2211yescharacteristic
42#19104658C2211yescharacteristic
43#19104658C2211yescharacteristic
44#19104656C2211yescharacteristic
45#19104656C2211yescharacteristic
46#19104655C2211yescharacteristic
47#19104655C2211yescharacteristic
48C4⋊Q8×C163C2211yescharacteristic
49#126×C163—21no—
50#124×C163—21no—
51#124×C163C2211yescharacteristic
52#123×C163—21no—
53C22⋊Q8×C163—21no—
54C22⋊Q8×C163C2211yescharacteristic
55#121×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic
57C4×D4×C163C2211yescharacteristic
58#117×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C2×D652C2311yescharacteristic
62C2×C22⋊D163—21no—
63C2×C4×D163—21no—
64#14579689—41no—
65#14579687—41no—
66#14579686—41no—
67#14579684—41no—
68#14579683—41no—
69C2×C163⋊C8—41no—
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C4⋊Dic163—21no—
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79C22⋊Dic163—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83(C2×D163)⋊C4—21no—
84C4×Dic163—21no—
85C2×Q8×C163—21no—
86C2×Q8×C163C2311yescharacteristic
87C2×D4×C163—21no—
88C2×D4×C163C2311yescharacteristic
89C22×C4×C163—21no—
90C22×C4×C163C2311yescharacteristic
91C4⋊C4×C163—21no—
92C4⋊C4×C163—21no—
93C4⋊C4×C163—21no—
94C4⋊C4×C163D411yescharacteristic
95C4⋊C4×C163D411yescharacteristic
96C4⋊C4×C163C2311yescharacteristic
97C4⋊C4×C163C2311yescharacteristic
98C22⋊C4×C163—21no—
99C22⋊C4×C163—21no—
100C22⋊C4×C163—21no—
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163D411yescharacteristic
104C22⋊C4×C163D411yescharacteristic
105C42×C163—21no—
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109Dic326—21no—
110Dic326—21no—
111Dic326—81no—
112D652—21no—
113D652—21no—
114C22⋊D163—81no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C4×D163—81no—
119C163⋊C8—21no—
120C163⋊C8—21no—
121C163⋊C8—21no—
122C163⋊C8—21no—
123C23×C163—21no—
124C23×C163C2×D411yescharacteristic
125Q8×C163—21no—
126Q8×C163—21no—
127Q8×C163—81no—
128D4×C163—21no—
129D4×C163—21no—
130D4×C163—81no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—81no—
137C2×C4×C163C2×D411yescharacteristic
138C2×C4×C163C2×D411yescharacteristic
139C2×C4×C163C2×D411yescharacteristic
140C2×C4×C163C2×D411yescharacteristic
141C2×C4×C163C2×D411yescharacteristic
142C2×C4×C163C2411yescharacteristic
143C22×C163—21no—
144C22×C163—21no—
145C22×C163—21no—
146C22×C163—41no—
147C22×C163—41no—
148C22×C163—41no—
149C22×C163C22×D411yescharacteristic
150C2×D163—41no—
151C2×D163—41no—
152C2×D163—41no—
153C4×C163—21no—
154C4×C163—21no—
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—21no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163—41no—
161C4×C163H(5,𝔽2)11yescharacteristic
162C4×C1632−1+411yescharacteristic
163Dic163—41no—
164Dic163—41no—
165Dic163—41no—
166C2×C163—41no—
167C2×C163—81no—
168C2×C163#53711yescharacteristic
169C2×C163#57711yescharacteristic
170C2×C163#57711yescharacteristic
171D163—81no—
172C163#326311yescharacteristic, complemented, Sylow
173#3263—1631nomaximal, Sylow
174#542—1631no—
175#542D16311yescharacteristic, complemented
176#476—1631no—
177#476—1631no—
178#475—1631no—
179#475—1631no—
180#470—1631no—
181#469—1631no—
182#465—1631no—
183#465—1631no—
184#464—1631no—
185#464—1631no—
186#463—1631no—
187#463—1631no—
188#424—1631no—
189C2×Q16—1631no—
190C2×SD16—1631no—
191C2×SD16—1631no—
192C2×D8—1631no—
193C2×M16—1631no—
194C22×C8—1631no—
195C4⋊Q8—1631no—
196C4⋊Q8C2×D16311yescharacteristic
197#126—21no—
198#126—3261no—
199#124—21no—
200#124—3261no—
201#124—1631no—
202#124C2×D16311yescharacteristic
203#123—21no—
204#123—3261no—
205C22⋊Q8—21no—
206C22⋊Q8—3261no—
207C22⋊Q8—1631no—
208C22⋊Q8C2×D16311yescharacteristic
209#121—1631no—
210#121C2×D16311yescharacteristic
211C4×Q8—1631no—
212C4×Q8C2×D16311yescharacteristic
213C4×D4—1631no—
214C4×D4C2×D16311yescharacteristic
215#117C2×D16311yescharacteristic
216#106—1631no—
217#106—1631no—
218#106—1631no—
219#106—1631no—
220C4⋊C8—1631no—
221C4⋊C8—1631no—
222C4⋊C8—1631no—
223C4⋊C8—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#102—1631no—
229#102—1631no—
230#102—1631no—
231#102—1631no—
232C2×Q8—21no—
233C2×Q8—3261no—
234C2×Q8—1631no—
235C2×Q8C22×D16311yescharacteristic
236C2×D4—21no—
237C2×D4—3261no—
238C2×D4—1631no—
239C2×D4C22×D16311yescharacteristic
240C22×C4—21no—
241C22×C4—3261no—
242C22×C4C22×D16311yescharacteristic
243Q16—6521no—
244SD16—6521no—
245SD16—6521no—
246D8—6521no—
247M16—6521no—
248C2×C8—6521no—
249C2×C8—1631no—
250C2×C8—1631no—
251C2×C8—1631no—
252C2×C8—1631no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—3261no—
257C4⋊C4—3261no—
258C4⋊C4—3261no—
259C4⋊C4—1631no—
260C4⋊C4—1631no—
261C4⋊C4C22⋊D16311yescharacteristic
262C4⋊C4C22⋊D16311yescharacteristic
263C4⋊C4C22×D16311yescharacteristic
264C4⋊C4C22×D16311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—3261no—
271C22⋊C4—3261no—
272C22⋊C4—3261no—
273C22⋊C4—3261no—
274C22⋊C4—3261no—
275C22⋊C4C22⋊D16311yescharacteristic
276C22⋊C4C22⋊D16311yescharacteristic
277C42—21no—
278C42—3261no—
279C42C22×D16311yescharacteristic
280C42C22×D16311yescharacteristic
281C23—21no—
282C23—3261no—
283C23C2×C22⋊D16311yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—13041no—
288Q8—3261no—
289Q8—3261no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—13041no—
294D4—3261no—
295D4—3261no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—13041no—
303C2×C4—3261no—
304C2×C4—3261no—
305C2×C4—3261no—
306C2×C4C2×C22⋊D16311yescharacteristic
307C2×C4C2×C22⋊D16311yescharacteristic
308C2×C4C2×C22⋊D16311yescharacteristic
309C2×C4C2×C22⋊D16311yescharacteristic
310C2×C4C2×C22⋊D16311yescharacteristic
311C2×C4C23×D16311yescharacteristic
312C8—3261no—
313C8—3261no—
314C8—3261no—
315C8—3261no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—6521no—
323C22—6521no—
324C22—6521no—
325C22C22×C22⋊D16311yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—6521no—
335C4—6521no—
336C4—6521no—
337C4#1910478111yescharacteristic
338C4#1910478211yescharacteristic
339C2—41no—
340C2—81no—
341C2—13041no—
342C2#2378224311yescharacteristic
343C2#2378247311yescharacteristic
344C2#2378247611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#542×C163C2
socleC22×C163C22×D4
prime radical#23781916C2
hypercenter#542D163
hypocenterC163#3263
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #23782286C2semidirect
2. #23781955C2semidirect
3. #23781952C2semidirect
4. #23781950C2semidirect
5. #23781947C2semidirect
6. #23781921C2semidirect
8. #23781896C2semidirect
9. #23781892C2semidirect
10. #23781887C2semidirect
11. #23781884C2semidirect
12. #23781880C2semidirect
13. #23781877C2semidirect
14. #23781692C2semidirect
15. #542×C163C2semidirect
172. C163#3263semidirect
175. #542D163semidirect

Maximal subgroups

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

subgroupindex
#237822862
#237819552
#237819522
#237819502
#237819472
#237819212
#237819162
#237818962
#237818922
#237818872
#237818842
#237818802
#237818772
#237816922
#542×C1632
#3263163