Finite Group Databasebeta

← #35226397

Subgroups — #35226397

Order 20864.

420 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
1Q8×C22⋊D163C211yesmaximal, characteristic, complemented
2#23782316C211yesmaximal, characteristic
3#23782014C211yesmaximal, characteristic, complemented
4#23781995C211yesmaximal, characteristic, complemented
5#23781994C211yesmaximal, characteristic, complemented
6#23781948C211yesmaximal, characteristic, complemented
7#23781897C211yesmaximal, characteristic, complemented
8#23781894C211yesmaximal, characteristic, complemented
9#23781869C211yesmaximal, characteristic, complemented
10#23781805C211yesmaximal, characteristic, complemented
11#23781804C211yesmaximal, characteristic, complemented
12#23781750C211yesmaximal, characteristic, complemented
13C8×C22⋊D163C211yesmaximal, characteristic, complemented
14#103×D163C211yesmaximal, characteristic, complemented
15C2×#103×C163C211yesmaximal, characteristic, complemented
16C2×#14579699—21no—
17C2×Q8×D163C2211yescharacteristic
18C2×#14579688C2211yescharacteristic
19C2×#14579686C2211yescharacteristic
20C2×#14579686C2211yescharacteristic
21C2×C8×D163C2211yescharacteristic
22#19104734C2211yescharacteristic
23#19104733—21no—
24#19104733C2211yescharacteristic
25#19104720—21no—
26#19104718—21no—
27#19104714—21no—
28#19104714C2211yescharacteristic
29#19104713—21no—
30#19104711—21no—
31#19104711C2211yescharacteristic
32#19104710C2211yescharacteristic
33#19104709C2211yescharacteristic
34Q8×Dic163C2211yescharacteristic
35#19104697C2211yescharacteristic
36C4×C22⋊D163—21no—
37C4×C22⋊D163C2211yescharacteristic
38C4⋊C4×D163C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41#19104666C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104657C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C22×Q8×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C2×C4⋊C4×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59#14579699—44no—
60C2×Dic326—21no—
61C2×Dic326—21no—
62C2×Dic326C2311yescharacteristic
63Q8×D163—22no—
64Q8×D163—41no—
65C2×D652—21no—
66C2×D652C2311yescharacteristic
67C2×C22⋊D163—21no—
68C2×C22⋊D163D411yescharacteristic
69C2×C4×D163—21no—
70C2×C4×D163—21no—
71C2×C4×D163—21no—
72C2×C4×D163C2311yescharacteristic
73#14579688—41no—
74#14579686—22no—
75#14579686—41no—
76C2×C163⋊C8C2311yescharacteristic
77C2×C163⋊C8C2311yescharacteristic
78C8×D163—41no—
79C4⋊Dic163—21no—
80C4⋊Dic163C2311yescharacteristic
81C4⋊Dic163C2311yescharacteristic
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4—21no—
86Dic163⋊C4—21no—
87Dic163⋊C4—21no—
88Dic163⋊C4D411yescharacteristic
89C22⋊Dic163—21no—
90C22⋊Dic163D411yescharacteristic
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4—21no—
93(C2×D163)⋊C4—21no—
94(C2×D163)⋊C4—21no—
95(C2×D163)⋊C4D411yescharacteristic
96C4×Dic163—21no—
97C4×Dic163C2311yescharacteristic
98C2×Q8×C163—21no—
99C2×Q8×C163—22no—
100C2×Q8×C163—41no—
101C2×Q8×C163C2311yescharacteristic
102C2×Q8×C163C2311yescharacteristic
103C22×C4×C163—21no—
104C22×C4×C163—21no—
105C22×C4×C163C2311yescharacteristic
106C2×C8×C163—41no—
107C2×C8×C163C2311yescharacteristic
108C2×C8×C163C2311yescharacteristic
109C4⋊C4×C163—21no—
110C4⋊C4×C163C2311yescharacteristic
111C4⋊C4×C163C2311yescharacteristic
112C22×D163—21no—
113C22×D163C2×D411yescharacteristic
114Dic326—21no—
115Dic326—21no—
116Dic326—42no—
117Dic326—42no—
118D652—21no—
119D652—21no—
120D652—42no—
121C22⋊D163—42no—
122C22⋊D163SD1614yescomplemented
123C2×Dic163—21no—
124C2×Dic163—21no—
125C2×Dic163—21no—
126C2×Dic163C2×D411yescharacteristic
127C2×Dic163C2×D411yescharacteristic
128C2×Dic163C2×D411yescharacteristic
129C4×D163—21no—
130C4×D163—21no—
131C4×D163—42no—
132C4×D163—42no—
133C4×D163—42no—
134C163⋊C8—21no—
135C163⋊C8—21no—
136C23×C163C2×D411yescharacteristic
137Q8×C163—21no—
138Q8×C163—21no—
139Q8×C163—21no—
140Q8×C163—42no—
141Q8×C163#4112yescomplemented
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—21no—
145C2×C4×C163—21no—
146C2×C4×C163—21no—
147C2×C4×C163—21no—
148C2×C4×C163—42no—
149C2×C4×C163—42no—
150C2×C4×C163C2×D411yescharacteristic
151C2×C4×C163C2411yescharacteristic
152C8×C163—21no—
153C8×C163—21no—
154C22×C163—21no—
155C22×C163—21no—
156C22×C163C2×SD1612yes—
157C22×C163C22×D411yescharacteristic
158C2×D163—21no—
159C2×D163—21no—
160C2×D163—41no—
161C2×D163—41no—
162C2×D163—41no—
163C2×D163C2×SD1612yes—
164C4×C163—21no—
165C4×C163—22no—
166C4×C163—41no—
167C4×C163—41no—
168C4×C163—41no—
169C4×C163C2×#4111yescharacteristic
170C4×C163H(5,𝔽2)11yescharacteristic
171Dic163—21no—
172Dic163—21no—
173Dic163—41no—
174Dic163—41no—
175Dic163—41no—
176Dic163C2×SD1612yes—
177C2×C163—22no—
178C2×C163#54611yescharacteristic
179C2×C163C22×SD1611yescharacteristic
180C2×C163#57811yescharacteristic
181D163—22no—
182D163—81no—
183C163#329811yescharacteristic, complemented, Sylow
184#3298—1631nomaximal, Sylow
185D4×Q8—1631no—
186#547—1631no—
187#487—1631no—
188#483—1631no—
189#483—1631no—
190#475—1631no—
191#465—1631no—
192#465—1631no—
193#461—1631no—
194#448—1631no—
195#448—1631no—
196C4×SD16—1631no—
197C8×D4—1631no—
198C2×#103—1631no—
199C2×#103D16311yescharacteristic, complemented
200C2×#41—3261no—
201C22×Q8—1631no—
202C22×Q8C2×D16311yescharacteristic
203C2×SD16—1631no—
204C2×SD16—1631no—
205C2×SD16—1631no—
206C22×C8—1631no—
207C22×C8C2×D16311yescharacteristic
208C4⋊Q8—3261no—
209C4⋊Q8—1631no—
210C4⋊Q8—1631no—
211#123—3261no—
212#123—3261no—
213C22⋊Q8—3261no—
214C22⋊Q8—3261no—
215C22⋊Q8—3261no—
216C22⋊Q8—1631no—
217C22⋊Q8—1631no—
218#121—1631no—
219#121—1631no—
220C4×Q8—1631no—
221C4×D4—3261no—
222C4×D4—1631no—
223C4×D4—1631no—
224C2×C4⋊C4—1631no—
225C2×C4⋊C4C2×D16311yescharacteristic
226#106—1631no—
227#106—1631no—
228#106—1631no—
229C4⋊C8—1631no—
230#103—1631no—
231#103—1631no—
232#103—1631no—
233#103—1631no—
234#103—1631no—
235#103C2×D16311yescharacteristic
236#103C2×D16311yescharacteristic
237#103C2×D16311yescharacteristic
238#103C2×D16311yescharacteristic
239#102—1631no—
240C22⋊C8—1631no—
241C22⋊C8—1631no—
242C4×C8—1631no—
243#41—6524no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—3261no—
248C2×Q8—3261no—
249C2×Q8—3262no—
250C2×Q8—6521no—
251C2×Q8—1631no—
252C2×Q8C22×D16311yescharacteristic
253C2×Q8C22×D16311yescharacteristic
254C2×D4—3261no—
255C2×D4—3261no—
256C2×D4—1631no—
257C2×D4—1631no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—3261no—
261C22×C4—3261no—
262C22×C4—3261no—
263C22×C4—1631no—
264C22×C4C22×D16311yescharacteristic
265SD16—3262no—
266SD16—6521no—
267SD16—6521no—
268C2×C8—41no—
269C2×C8—6521no—
270C2×C8—1631no—
271C2×C8—1631no—
272C2×C8C22×D16311yescharacteristic
273C2×C8C22×D16311yescharacteristic
274C4⋊C4—21no—
275C4⋊C4—3261no—
276C4⋊C4—3261no—
277C4⋊C4—3261no—
278C4⋊C4—3261no—
279C4⋊C4—3261no—
280C4⋊C4—3261no—
281C4⋊C4—3261no—
282C4⋊C4—1631no—
283C4⋊C4—1631no—
284C4⋊C4—1631no—
285C4⋊C4C22×D16311yescharacteristic
286C4⋊C4C22×D16311yescharacteristic
287C22⋊C4—3261no—
288C22⋊C4—3261no—
289C22⋊C4—3261no—
290C22⋊C4—3261no—
291C22⋊C4—3261no—
292C22⋊C4—1631no—
293C22⋊C4—1631no—
294C42—3261no—
295C42—1631no—
296C23—3261no—
297C23—1631no—
298C23D4×D16311yescharacteristic
299Q8—21no—
300Q8—21no—
301Q8—21no—
302Q8—42no—
303Q8—3261no—
304Q8—3261no—
305Q8—6522no—
306Q8—6522no—
307Q8#1457969912yescomplemented
308D4—3261no—
309D4—3261no—
310D4—6522no—
311D4—6522no—
312D4—1634no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—3261no—
322C2×C4—3261no—
323C2×C4—3261no—
324C2×C4—3261no—
325C2×C4—3261no—
326C2×C4—6522no—
327C2×C4—6522no—
328C2×C4—6522no—
329C2×C4—1631no—
330C2×C4—1631no—
331C2×C4—1631no—
332C2×C4D4×D16311yescharacteristic
333C2×C4C23×D16311yescharacteristic
334C8—21no—
335C8—21no—
336C8—3261no—
337C8—3261no—
338C22—21no—
339C22—21no—
340C22—3261no—
341C22—3261no—
342C22—6521no—
343C22—6521no—
344C22—6521no—
345C22—1632no—
346C22SD16×D16312yes—
347C22C2×D4×D16311yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—41no—
353C4—3261no—
354C4—3261no—
355C4—6521no—
356C4—6521no—
357C4—6521no—
358C4—1632no—
359C4C2×#1457969911yescharacteristic
360C4#1910478011yescharacteristic
361C2—22no—
362C2—13041no—
363C2—3262no—
364C2#2378230511yescharacteristic
365C2C2×SD16×D16311yescharacteristic
366C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#103×C163C2
socleC22×C163C22×D4
prime radical#23782316C2
hypercenterC2×#103D163
hypocenterC163#3298
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D163C2semidirect
3. #23782014C2semidirect
4. #23781995C2semidirect
5. #23781994C2semidirect
6. #23781948C2semidirect
7. #23781897C2semidirect
8. #23781894C2semidirect
9. #23781869C2semidirect
10. #23781805C2semidirect
11. #23781804C2semidirect
12. #23781750C2semidirect
13. C8×C22⋊D163C2semidirect
14. #103×D163C2semidirect
15. C2×#103×C163C2semidirect
122. C22⋊D163SD16semidirect
141. Q8×C163#41semidirect
183. C163#3298semidirect
199. C2×#103D163semidirect
307. Q8#14579699semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D1632
#237823162
#237820142
#237819952
#237819942
#237819482
#237818972
#237818942
#237818692
#237818052
#237818042
#237817502
C8×C22⋊D1632
#103×D1632
C2×#103×C1632
#3298163