Finite Group Databasebeta

← #35226583

Subgroups — #35226583

Order 20864.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782350C211yesmaximal, characteristic
2Q8×C22⋊D163C211yesmaximal, characteristic
3#23782032C211yesmaximal, characteristic, complemented
4#23782005C211yesmaximal, characteristic, complemented
5#23781995C211yesmaximal, characteristic, complemented
6#23781974C211yesmaximal, characteristic, complemented
7#23781922C211yesmaximal, characteristic, complemented
8#23781906C211yesmaximal, characteristic, complemented
9#23781874C211yesmaximal, characteristic, complemented
10#23781822C211yesmaximal, characteristic, complemented
11#23781821C211yesmaximal, characteristic, complemented
12#23781769C211yesmaximal, characteristic, complemented
13#23781736C211yesmaximal, characteristic
14#23781672C211yesmaximal, characteristic, complemented
15C2×#103×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×Dic652C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579689C2211yescharacteristic
20C2×#14579682C2211yescharacteristic
21#19104733—21no—
22#19104733C2211yescharacteristic
23#19104730—21no—
24#19104726C2211yescharacteristic
25#19104725—21no—
26#19104718—21no—
27#19104715C2211yescharacteristic
28#19104714—21no—
29#19104714C2211yescharacteristic
30#19104714C2211yescharacteristic
31#19104711—21no—
32#19104711C2211yescharacteristic
33Q8×Dic163C2211yescharacteristic
34#19104700C2211yescharacteristic
35C4×C22⋊D163—21no—
36C4×C22⋊D163C2211yescharacteristic
37#19104694—21no—
38#19104693C2211yescharacteristic
39#19104669C2211yescharacteristic
40#19104666C2211yescharacteristic
41#19104666C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic
52C22×Q8×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54C2×C4⋊C4×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#103×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61C2×Dic326C2311yescharacteristic
62Q8×D163—22no—
63Q8×D163—41no—
64C2×C22⋊D163D411yescharacteristic
65C2×C4×D163—21no—
66C2×C4×D163C2311yescharacteristic
67Dic652—41no—
68#14579689—22no—
69#14579689—41no—
70#14579682—41no—
71C2×C163⋊C8C2311yescharacteristic
72C2×C163⋊C8C2311yescharacteristic
73C4⋊Dic163—21no—
74C4⋊Dic163—21no—
75C4⋊Dic163—21no—
76C4⋊Dic163C2311yescharacteristic
77C4⋊Dic163C2311yescharacteristic
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4—21no—
84Dic163⋊C4—21no—
85Dic163⋊C4D411yescharacteristic
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163D411yescharacteristic
89(C2×D163)⋊C4—21no—
90(C2×D163)⋊C4—21no—
91(C2×D163)⋊C4—21no—
92(C2×D163)⋊C4D411yescharacteristic
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163—21no—
96C4×Dic163C2311yescharacteristic
97C2×Q8×C163—21no—
98C2×Q8×C163—22no—
99C2×Q8×C163—41no—
100C2×Q8×C163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C22×C4×C163—21no—
104C22×C4×C163C2311yescharacteristic
105C2×C8×C163—41no—
106C2×C8×C163C2311yescharacteristic
107C2×C8×C163C2311yescharacteristic
108C4⋊C4×C163—21no—
109C4⋊C4×C163C2311yescharacteristic
110C4⋊C4×C163C2311yescharacteristic
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—42no—
117C22⋊D163—22no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163C2×D411yescharacteristic
123C2×Dic163C2×D411yescharacteristic
124C2×Dic163C2×D411yescharacteristic
125C4×D163—21no—
126C4×D163—21no—
127C4×D163—42no—
128C163⋊C8—21no—
129C163⋊C8—21no—
130C23×C163C2×D411yescharacteristic
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163—21no—
134Q8×C163—42no—
135Q8×C163#4112yes—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—42no—
143C2×C4×C163—42no—
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2411yescharacteristic
146C8×C163—21no—
147C8×C163—21no—
148C22×C163—21no—
149C22×C163—21no—
150C22×C163#13712yes—
151C22×C163C22×D411yescharacteristic
152C2×D163—21no—
153C2×D163—21no—
154C2×D163—21no—
155C4×C163—21no—
156C4×C163—22no—
157C4×C163—41no—
158C4×C163—41no—
159C4×C163—41no—
160C4×C163C2×#4111yescharacteristic
161C4×C163H(5,𝔽2)11yescharacteristic
162Dic163—21no—
163Dic163—21no—
164Dic163—21no—
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169C2×C163—22no—
170C2×C163#54611yescharacteristic
171C2×C163C2×#13711yescharacteristic
172C2×C163#57811yescharacteristic
173D163—41no—
174C163#331211yescharacteristic, complemented, Sylow
175#3312—1631nomaximal, Sylow
176#552—1631no—
177D4×Q8—1631no—
178#491—1631no—
179#485—1631no—
180#483—1631no—
181#479—1631no—
182#470—1631no—
183#467—1631no—
184#462—1631no—
185#451—1631no—
186#451—1631no—
187#441—1631no—
188#435—1631no—
189#419—1631no—
190C2×#103D16311yescharacteristic, complemented
191C22×Q8—1631no—
192C22×Q8C2×D16311yescharacteristic
193C2×Q16—1631no—
194C2×Q16—1631no—
195C2×Q16—1631no—
196C2×M16—1631no—
197C22×C8C2×D16311yescharacteristic
198C4⋊Q8—3261no—
199C4⋊Q8—1631no—
200#126—3261no—
201#125—3261no—
202#125—1631no—
203#123—3261no—
204C22⋊Q8—3261no—
205C22⋊Q8—3261no—
206C22⋊Q8—1631no—
207C22⋊Q8—1631no—
208C22⋊Q8—1631no—
209C22⋊Q8—1631no—
210C4×Q8—1631no—
211C4×Q8—1631no—
212C4×D4—3261no—
213C4×D4—1631no—
214#117—3261no—
215#117—1631no—
216C2×C4⋊C4C2×D16311yescharacteristic
217#107—1631no—
218#106—1631no—
219#106—1631no—
220C4⋊C8—1631no—
221#103—1631no—
222#103—1631no—
223#103—1631no—
224#103—1631no—
225#103—1631no—
226#103—1631no—
227#103C2×D16311yescharacteristic
228#103C2×D16311yescharacteristic
229#103C2×D16311yescharacteristic
230#103C2×D16311yescharacteristic
231C22⋊C8—1631no—
232C22⋊C8—1631no—
233#97—1631no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—3261no—
238C2×Q8—3262no—
239C2×Q8—6521no—
240C2×Q8—1631no—
241C2×Q8—1631no—
242C2×Q8C22×D16311yescharacteristic
243C2×Q8C22×D16311yescharacteristic
244C2×D4—1631no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3261no—
248C22×C4—1631no—
249C22×C4C22×D16311yescharacteristic
250Q16—3262no—
251Q16—6521no—
252Q16—6521no—
253M16—6521no—
254C2×C8—41no—
255C2×C8—1631no—
256C2×C8—1631no—
257C2×C8C22×D16311yescharacteristic
258C2×C8C22×D16311yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—3261no—
261C4⋊C4—3261no—
262C4⋊C4—3261no—
263C4⋊C4—3261no—
264C4⋊C4—3261no—
265C4⋊C4—3261no—
266C4⋊C4—3261no—
267C4⋊C4—3261no—
268C4⋊C4—3261no—
269C4⋊C4—3261no—
270C4⋊C4—1631no—
271C4⋊C4—1631no—
272C4⋊C4—1631no—
273C4⋊C4C22×D16311yescharacteristic
274C4⋊C4C22×D16311yescharacteristic
275C22⋊C4—3261no—
276C22⋊C4—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—1631no—
281C22⋊C4—1631no—
282C42—3261no—
283C42—3261no—
284C42—3261no—
285C42—1631no—
286C23—1631no—
287C23D4×D16311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—42no—
292Q8—3261no—
293Q8—3261no—
294Q8—3261no—
295Q8—3261no—
296Q8—6522no—
297Q8#1457969912yes—
298D4—3262no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—6522no—
314C2×C4—1631no—
315C2×C4—1631no—
316C2×C4—1631no—
317C2×C4D4×D16311yescharacteristic
318C2×C4C23×D16311yescharacteristic
319C8—21no—
320C8—21no—
321C8—3261no—
322C8—3261no—
323C22—21no—
324C22—21no—
325C22—3261no—
326C22—3261no—
327C22—3261no—
328C22#1910476412yes—
329C22C2×D4×D16311yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—3261no—
336C4—3261no—
337C4—3261no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C4—6521no—
342C4C2×#1457969911yescharacteristic
343C4#1910478011yescharacteristic
344C2—22no—
345C2—6521no—
346C2#2378230511yescharacteristic
347C2C2×#1910476411yescharacteristic
348C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingC2×#103×C163C2
socleC22×C163C22×D4
prime radicalC2×C22⋊D163D4
hypercenterC2×#103D163
hypocenterC163#3312
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
3. #23782032C2semidirect
4. #23782005C2semidirect
5. #23781995C2semidirect
6. #23781974C2semidirect
7. #23781922C2semidirect
8. #23781906C2semidirect
9. #23781874C2semidirect
10. #23781822C2semidirect
11. #23781821C2semidirect
12. #23781769C2semidirect
14. #23781672C2semidirect
15. C2×#103×C163C2semidirect
174. C163#3312semidirect
190. C2×#103D163semidirect

Maximal subgroups

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

subgroupindex
#237823502
Q8×C22⋊D1632
#237820322
#237820052
#237819952
#237819742
#237819222
#237819062
#237818742
#237818222
#237818212
#237817692
#237817362
#237816722
C2×#103×C1632
#3312163