Finite Group Databasebeta

← #35224735

Subgroups — #35224735

Order 20864.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782270C211yesmaximal, characteristic, complemented
2#23781983C211yesmaximal, characteristic, complemented
3#23781978C211yesmaximal, characteristic, complemented
4#23781972C211yesmaximal, characteristic, complemented
5#23781965C211yesmaximal, characteristic, complemented
6#23781865C211yesmaximal, characteristic, complemented
7#23781860C211yesmaximal, characteristic, complemented
8#23781810C211yesmaximal, characteristic, complemented
9#23781810C211yesmaximal, characteristic, complemented
10#23781775C211yesmaximal, characteristic, complemented
11#23781763C211yesmaximal, characteristic, complemented
12C4×#14579686C211yesmaximal, characteristic, complemented
13C4×#14579684C211yesmaximal, characteristic, complemented
14#23781726C211yesmaximal, characteristic, complemented
15#520×C163C211yesmaximal, characteristic, complemented
16C2×#14579686C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C2×#14579684C2211yescharacteristic
20#19104732C2211yescharacteristic
21#19104727C2211yescharacteristic
22#19104720—21no—
23#19104720—21no—
24#19104709C2211yescharacteristic
25#19104709C2211yescharacteristic
26#19104708—21no—
27#19104708—21no—
28C4×D652C2211yescharacteristic
29C4×D652C2211yescharacteristic
30#19104668C2211yescharacteristic
31#19104668C2211yescharacteristic
32#19104666C2211yescharacteristic
33#19104666C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35C4⋊(C163⋊C8)C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104656C2211yescharacteristic
39#19104656C2211yescharacteristic
40#19104655C2211yescharacteristic
41#19104655C2211yescharacteristic
42#19104655C2211yescharacteristic
43#19104655C2211yescharacteristic
44C22⋊(C163⋊C8)C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46#19104646C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48C2×#41×C163C2211yescharacteristic
49C4×Q8×C163C2211yescharacteristic
50C4×Q8×C163C2211yescharacteristic
51C4×D4×C163—21no—
52C4×D4×C163—21no—
53C4×D4×C163C2211yescharacteristic
54C4×D4×C163C2211yescharacteristic
55#117×C163—21no—
56#117×C163C2211yescharacteristic
57C2×C4⋊C4×C163—21no—
58C2×C4⋊C4×C163C2211yescharacteristic
59C2×D652—21no—
60C2×D652—21no—
61C2×D652C2311yescharacteristic
62C2×D652C2311yescharacteristic
63C2×C22⋊D163—21no—
64C2×C22⋊D163—21no—
65C2×C4×D163—21no—
66C2×C4×D163—21no—
67#14579686—22no—
68#14579686—22no—
69#14579684—22no—
70#14579684—22no—
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C4⋊Dic163C2311yescharacteristic
76C4⋊Dic163C2311yescharacteristic
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79(C2×D163)⋊C4—21no—
80(C2×D163)⋊C4—21no—
81(C2×D163)⋊C4—21no—
82(C2×D163)⋊C4—21no—
83#41×C163—22no—
84#41×C163—22no—
85C2×Q8×C163C2311yescharacteristic
86C2×D4×C163—21no—
87C2×D4×C163C2311yescharacteristic
88C22×C4×C163—21no—
89C22×C4×C163—21no—
90C22×C4×C163—21no—
91C22×C4×C163—21no—
92C22×C4×C163C2311yescharacteristic
93C4⋊C4×C163—21no—
94C4⋊C4×C163—21no—
95C4⋊C4×C163—21no—
96C4⋊C4×C163D411yescharacteristic
97C4⋊C4×C163D411yescharacteristic
98C4⋊C4×C163C2311yescharacteristic
99C4⋊C4×C163C2311yescharacteristic
100C22⋊C4×C163—21no—
101C22⋊C4×C163—21no—
102C22⋊C4×C163D411yescharacteristic
103C22⋊C4×C163D411yescharacteristic
104C42×C163—21no—
105C42×C163—21no—
106C42×C163C2311yescharacteristic
107C42×C163C2311yescharacteristic
108C22×D163—21no—
109C22×D163—21no—
110D652—21no—
111D652—21no—
112D652—21no—
113D652—21no—
114D652—81no—
115D652—81no—
116C22⋊D163—81no—
117C22⋊D163—81no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C4×D163—81no—
121C4×D163—81no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124C163⋊C8—21no—
125C163⋊C8—21no—
126C23×C163—21no—
127C23×C163C2×D411yescharacteristic
128Q8×C163—21no—
129Q8×C163#4112yes—
130D4×C163—24no—
131D4×C163—21no—
132D4×C163#4112yes—
133C2×C4×C163—24no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—81no—
140C2×C4×C163—81no—
141C2×C4×C163—81no—
142C2×C4×C163C2×D411yescharacteristic
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C22×C163—21no—
149C22×C163—22no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163C22×D411yescharacteristic
155C2×D163—41no—
156C2×D163—41no—
157C2×D163—41no—
158C2×D163—41no—
159C2×D163—41no—
160C2×D163—41no—
161C4×C163—21no—
162C4×C163—22no—
163C4×C163—21no—
164C4×C163—21no—
165C4×C163—21no—
166C4×C163—21no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163C2×#4111yescharacteristic
170C4×C163C2×#4111yescharacteristic
171Dic163—41no—
172Dic163—41no—
173C2×C163—41no—
174C2×C163—42no—
175C2×C163#52511yescharacteristic
176C2×C163#57611yescharacteristic
177C2×C163#57711yescharacteristic
178D163—81no—
179D163—81no—
180C163#311911yescharacteristic, complemented, Sylow
181#3119—1631nomaximal, Sylow
182#540—1631no—
183#520D16311yescharacteristic, complemented
184#481—1631no—
185#480—1631no—
186#479—1631no—
187#478—1631no—
188#460—1631no—
189#459—1631no—
190#449—1631no—
191#449—1631no—
192#442—1631no—
193#440—1631no—
194C4×SD16—1631no—
195C4×D8—1631no—
196#433—1631no—
197C2×#41C2×D16311yescharacteristic
198C2×SD16—1631no—
199C2×SD16—1631no—
200C2×D8—1631no—
201C2×D8—1631no—
202#127—1631no—
203#125—1631no—
204#123—3261no—
205#123—3261no—
206#121—3261no—
207#121—3261no—
208#121—1631no—
209#121—1631no—
210C4×Q8C2×D16311yescharacteristic
211C4×Q8C2×D16311yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—1631no—
215C4×D4—1631no—
216C4×D4C2×D16311yescharacteristic
217C4×D4C2×D16311yescharacteristic
218#117—21no—
219#117C2×D16311yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D16311yescharacteristic
222#107—1631no—
223#107—1631no—
224#106—1631no—
225#106—1631no—
226C4⋊C8—1631no—
227C4⋊C8—1631no—
228#103—1631no—
229#103—1631no—
230#102—1631no—
231#102—1631no—
232#102—1631no—
233#102—1631no—
234#102—1631no—
235#102—1631no—
236C22⋊C8—1631no—
237C22⋊C8—1631no—
238#97—1631no—
239C4×C8—1631no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D16311yescharacteristic
243C2×D4—21no—
244C2×D4—3261no—
245C2×D4—3261no—
246C2×D4—3261no—
247C2×D4—3261no—
248C2×D4—1631no—
249C2×D4—1631no—
250C2×D4C22×D16311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—3261no—
256C22×C4—3261no—
257C22×C4C22×D16311yescharacteristic
258SD16—3262no—
259SD16—3262no—
260D8—3262no—
261D8—3262no—
262C2×C8—1631no—
263C2×C8—1631no—
264C2×C8—1631no—
265C2×C8—1631no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—3261no—
270C4⋊C4—3261no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4C22⋊D16311yescharacteristic
274C4⋊C4C22⋊D16311yescharacteristic
275C4⋊C4C22×D16311yescharacteristic
276C4⋊C4C22×D16311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4C22⋊D16311yescharacteristic
284C22⋊C4C22⋊D16311yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D16311yescharacteristic
288C42C22×D16311yescharacteristic
289C23—21no—
290C23—3261no—
291C23—3261no—
292C23C2×C22⋊D16311yescharacteristic
293Q8—21no—
294Q8#1457969912yes—
295D4—24no—
296D4—21no—
297D4—13041no—
298D4—13041no—
299D4—13041no—
300D4—13041no—
301D4—3261no—
302D4—3261no—
303D4—3261no—
304D4—3261no—
305D4#1457969912yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—13041no—
316C2×C4—13041no—
317C2×C4—3261no—
318C2×C4—3261no—
319C2×C4C2×C22⋊D16311yescharacteristic
320C2×C4C2×C22⋊D16311yescharacteristic
321C2×C4C2×C22⋊D16311yescharacteristic
322C2×C4C2×C22⋊D16311yescharacteristic
323C2×C4C2×C22⋊D16311yescharacteristic
324C2×C4C23×D16311yescharacteristic
325C8—3261no—
326C8—3261no—
327C8—3261no—
328C8—3261no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—6521no—
336C22—6521no—
337C22—6521no—
338C22—6521no—
339C22—6521no—
340C22—6521no—
341C22C22×C22⋊D16311yescharacteristic
342C4—21no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—6521no—
351C4—6521no—
352C4C2×#1457969911yescharacteristic
353C4C2×#1457969911yescharacteristic
354C2—41no—
355C2—42no—
356C2—13041no—
357C2—13041no—
358C2#2378217611yescharacteristic
359C2#2378246811yescharacteristic
360C2#2378247311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#520×C163C2
socleC22×C163C22×D4
hypercenter#520D163
hypocenterC163#3119
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #23782270C2semidirect
2. #23781983C2semidirect
3. #23781978C2semidirect
4. #23781972C2semidirect
5. #23781965C2semidirect
6. #23781865C2semidirect
7. #23781860C2semidirect
8. #23781810C2semidirect
9. #23781810C2semidirect
10. #23781775C2semidirect
11. #23781763C2semidirect
12. C4×#14579686C2semidirect
13. C4×#14579684C2semidirect
14. #23781726C2semidirect
15. #520×C163C2semidirect
180. C163#3119semidirect
183. #520D163semidirect

Maximal subgroups

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

subgroupindex
#237822702
#237819832
#237819782
#237819722
#237819652
#237818652
#237818602
#237818102
#237817752
#237817632
C4×#145796862
C4×#145796842
#237817262
#520×C1632
#3119163