Finite Group Databasebeta

← #35226438

Subgroups — #35226438

Order 20864.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782361C211yesmaximal, characteristic, complemented
2#23782019C211yesmaximal, characteristic, complemented
3#23782003C211yesmaximal, characteristic, complemented
4#23781981C211yesmaximal, characteristic, complemented
5#23781972C211yesmaximal, characteristic, complemented
6#23781899C211yesmaximal, characteristic, complemented
7#23781896C211yesmaximal, characteristic, complemented
8#23781877C211yesmaximal, characteristic, complemented
9#23781810C211yesmaximal, characteristic, complemented
10#23781804C211yesmaximal, characteristic, complemented
11C4×#14579686C211yesmaximal, characteristic, complemented
12D4×C163⋊C8C211yesmaximal, characteristic, complemented
13#23781661C211yesmaximal, characteristic, complemented
14C2×#19104659C211yesmaximal, characteristic, complemented
15#548×C163C211yesmaximal, characteristic, complemented
16C2×#14579686C2211yescharacteristic
17C2×#14579686C2211yescharacteristic
18C2×#14579684C2211yescharacteristic
19C22×C163⋊C8C2211yescharacteristic
20C22×C163⋊C8C2211yescharacteristic
21#19104730—21no—
22#19104727C2211yescharacteristic
23#19104717—21no—
24#19104709C2211yescharacteristic
25#19104709C2211yescharacteristic
26#19104707—21no—
27C4×D652C2211yescharacteristic
28(C22⋊D163)⋊C4—21no—
29#19104694C2211yescharacteristic
30C2×C4⋊Dic163C2211yescharacteristic
31#19104668C2211yescharacteristic
32#19104666C2211yescharacteristic
33#19104666C2211yescharacteristic
34C4⋊(C163⋊C8)C2211yescharacteristic
35#19104659C2211yescharacteristic
36#19104659C2211yescharacteristic
37#19104659C2211yescharacteristic
38#19104659C2211yescharacteristic
39#19104659C2211yescharacteristic
40#19104659C2211yescharacteristic
41#19104656C2211yescharacteristic
42#19104656C2211yescharacteristic
43#19104656C2211yescharacteristic
44#19104655C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic
47C4×C163⋊C8C2211yescharacteristic
48C2×#41×C163C2211yescharacteristic
49C22×Q8×C163C2211yescharacteristic
50#124×C163—21no—
51#124×C163C2211yescharacteristic
52C22⋊Q8×C163—21no—
53C22⋊Q8×C163C2211yescharacteristic
54#121×C163—21no—
55#121×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic
57C4×D4×C163—21no—
58C4×D4×C163C2211yescharacteristic
59C2×D652C2311yescharacteristic
60C2×C22⋊D163—21no—
61C2×C22⋊D163—21no—
62C22×Dic163—21no—
63C2×C4×D163—21no—
64#14579686—22no—
65#14579686—41no—
66#14579684—41no—
67C2×C163⋊C8—41no—
68C2×C163⋊C8—41no—
69C2×C163⋊C8C2311yescharacteristic
70C2×C163⋊C8C2311yescharacteristic
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C4⋊Dic163—21no—
74C4⋊Dic163C2311yescharacteristic
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—
83C4×Dic163—21no—
84#41×C163—22no—
85#41×C163—41no—
86C2×Q8×C163—21no—
87C2×Q8×C163—22no—
88C2×Q8×C163—41no—
89C2×Q8×C163C2311yescharacteristic
90C2×Q8×C163C2311yescharacteristic
91C2×D4×C163—21no—
92C2×D4×C163—21no—
93C2×D4×C163D411yescharacteristic
94C2×D4×C163C2311yescharacteristic
95C22×C4×C163—21no—
96C22×C4×C163—21no—
97C22×C4×C163C2311yescharacteristic
98C22×C4×C163C2311yescharacteristic
99C4⋊C4×C163—21no—
100C4⋊C4×C163—21no—
101C4⋊C4×C163D411yescharacteristic
102C4⋊C4×C163C2311yescharacteristic
103C22⋊C4×C163—21no—
104C22⋊C4×C163—21no—
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C22⋊C4×C163D411yescharacteristic
108C22⋊C4×C163D411yescharacteristic
109C42×C163—21no—
110C42×C163C2311yescharacteristic
111C22×D163—21no—
112D652—21no—
113D652—21no—
114C22⋊D163—42no—
115C22⋊D163—81no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—81no—
120C4×D163—81no—
121C163⋊C8—21no—
122C163⋊C8—21no—
123C163⋊C8—21no—
124C163⋊C8—21no—
125C23×C163—21no—
126C23×C163C2×D411yescharacteristic
127C23×C163C2×D411yescharacteristic
128Q8×C163—21no—
129Q8×C163—21no—
130Q8×C163—21no—
131Q8×C163—42no—
132Q8×C163#4112yes—
133D4×C163—22no—
134D4×C163—21no—
135D4×C163—21no—
136D4×C163—42no—
137D4×C163—42no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—42no—
146C2×C4×C163—42no—
147C2×C4×C163C2×D411yescharacteristic
148C2×C4×C163C2×D411yescharacteristic
149C2×C4×C163C2×D411yescharacteristic
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C22×C163—21no—
153C22×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163—21no—
157C22×C163—41no—
158C22×C163—41no—
159C22×C163—41no—
160C22×C163#13512yes—
161C22×C163C22×D411yescharacteristic
162C2×D163—41no—
163C2×D163—41no—
164C2×D163—41no—
165C4×C163—21no—
166C4×C163—21no—
167C4×C163—22no—
168C4×C163—21no—
169C4×C163—21no—
170C4×C163—41no—
171C4×C163—41no—
172C4×C163C2×#4111yescharacteristic
173C4×C163H(5,𝔽2)11yescharacteristic
174Dic163—41no—
175Dic163—41no—
176Dic163—41no—
177C2×C163—22no—
178C2×C163—41no—
179C2×C163—81no—
180C2×C163#54611yescharacteristic
181C2×C163C2×#13511yescharacteristic
182C2×C163#57711yescharacteristic
183D163—81no—
184C163#330111yescharacteristic, complemented, Sylow
185#3301—1631nomaximal, Sylow
186#553—1631no—
187#548D16311yescharacteristic, complemented
188#488—1631no—
189#485—1631no—
190#481—1631no—
191#479—1631no—
192#466—1631no—
193#465—1631no—
194#463—1631no—
195#449—1631no—
196#448—1631no—
197C4×SD16—1631no—
198C8×D4—1631no—
199#416—1631no—
200C2×#103—1631no—
201C2×#41C2×D16311yescharacteristic
202C22×Q8C2×D16311yescharacteristic
203C2×SD16—1631no—
204C2×SD16—1631no—
205C2×D8—1631no—
206C22×C8—1631no—
207C22×C8—1631no—
208#126—3261no—
209#125—1631no—
210#124—21no—
211#124C2×D16311yescharacteristic
212#123—3261no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D16311yescharacteristic
215#121—21no—
216#121—3261no—
217#121—1631no—
218#121—1631no—
219#121C2×D16311yescharacteristic
220C4×Q8C2×D16311yescharacteristic
221C4×D4—21no—
222C4×D4—3261no—
223C4×D4—1631no—
224C4×D4C2×D16311yescharacteristic
225#117—1631no—
226C2×C4⋊C4—1631no—
227#107—1631no—
228#106—1631no—
229#106—1631no—
230C4⋊C8—1631no—
231#103—1631no—
232#103—1631no—
233#103—1631no—
234#103—1631no—
235#103—1631no—
236#103—1631no—
237#102—1631no—
238#102—1631no—
239#102—1631no—
240#102—1631no—
241C22⋊C8—1631no—
242C22⋊C8—1631no—
243C4×C8—1631no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D16311yescharacteristic
250C2×Q8C22×D16311yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—3261no—
254C2×D4—3261no—
255C2×D4—1631no—
256C2×D4C22⋊D16311yescharacteristic
257C2×D4C22×D16311yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3261no—
261C22×C4—3261no—
262C22×C4C22×D16311yescharacteristic
263C22×C4C22×D16311yescharacteristic
264SD16—3262no—
265SD16—6521no—
266D8—6521no—
267C2×C8—6521no—
268C2×C8—6521no—
269C2×C8—1631no—
270C2×C8—1631no—
271C2×C8—1631no—
272C2×C8—1631no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—3261no—
276C4⋊C4—3261no—
277C4⋊C4—3261no—
278C4⋊C4—1631no—
279C4⋊C4—1631no—
280C4⋊C4—1631no—
281C4⋊C4C22⋊D16311yescharacteristic
282C4⋊C4C22×D16311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4C22⋊D16311yescharacteristic
292C22⋊C4C22⋊D16311yescharacteristic
293C42—21no—
294C42—3261no—
295C42C22×D16311yescharacteristic
296C23—21no—
297C23—3261no—
298C23C2×C22⋊D16311yescharacteristic
299C23C2×C22⋊D16311yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1457969812yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—13041no—
311D4—3261no—
312D4—3261no—
313D4—6522no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—13041no—
324C2×C4—13041no—
325C2×C4—3261no—
326C2×C4—3261no—
327C2×C4—3261no—
328C2×C4C2×C22⋊D16311yescharacteristic
329C2×C4C2×C22⋊D16311yescharacteristic
330C2×C4C2×C22⋊D16311yescharacteristic
331C2×C4C2×C22⋊D16311yescharacteristic
332C2×C4C23×D16311yescharacteristic
333C8—3261no—
334C8—3261no—
335C8—3261no—
336C8—3261no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—6521no—
346C22—6521no—
347C22—6521no—
348C22#1910475512yes—
349C22C22×C22⋊D16311yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—6521no—
358C4—6521no—
359C4—6521no—
360C4C2×#1457969811yescharacteristic
361C4#1910478011yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—13041no—
366C2#2378230911yescharacteristic
367C2C2×#1910475511yescharacteristic
368C2#2378247311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#548×C163C2
socleC22×C163C22×D4
hypercenter#548D163
hypocenterC163#3301
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #23782361C2semidirect
2. #23782019C2semidirect
3. #23782003C2semidirect
4. #23781981C2semidirect
5. #23781972C2semidirect
6. #23781899C2semidirect
7. #23781896C2semidirect
8. #23781877C2semidirect
9. #23781810C2semidirect
10. #23781804C2semidirect
11. C4×#14579686C2semidirect
12. D4×C163⋊C8C2semidirect
13. #23781661C2semidirect
14. C2×#19104659C2semidirect
15. #548×C163C2semidirect
184. C163#3301semidirect
187. #548D163semidirect

Maximal subgroups

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

subgroupindex
#237823612
#237820192
#237820032
#237819812
#237819722
#237818992
#237818962
#237818772
#237818102
#237818042
C4×#145796862
D4×C163⋊C82
#237816612
C2×#191046592
#548×C1632
#3301163