Finite Group Databasebeta

← #35226580

Subgroups — #35226580

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#23782349C211yesmaximal, characteristic
2Q8×C22⋊D163C211yesmaximal, characteristic
3#23782032C211yesmaximal, characteristic, complemented
4#23782007C211yesmaximal, characteristic, complemented
5#23781994C211yesmaximal, characteristic, complemented
6#23781974C211yesmaximal, characteristic, complemented
7#23781921C211yesmaximal, characteristic, complemented
8#23781903C211yesmaximal, characteristic, complemented
9#23781873C211yesmaximal, characteristic, complemented
10#23781819C211yesmaximal, characteristic, complemented
11#23781772C211yesmaximal, characteristic, complemented
12#23781739C211yesmaximal, characteristic
13#23781674C211yesmaximal, characteristic, complemented
14#103×D163C211yesmaximal, characteristic, complemented
15#451×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579683C2211yescharacteristic
19C2×C8×D163C2211yescharacteristic
20#19104733—21no—
21#19104733C2211yescharacteristic
22#19104728—21no—
23#19104726C2211yescharacteristic
24#19104725—21no—
25#19104717—21no—
26#19104714—21no—
27#19104714C2211yescharacteristic
28#19104711—21no—
29#19104711C2211yescharacteristic
30#19104711C2211yescharacteristic
31Q8×Dic163C2211yescharacteristic
32Q8×Dic163C2211yescharacteristic
33C4×C22⋊D163—21no—
34C4×C22⋊D163C2211yescharacteristic
35#19104694C2211yescharacteristic
36#19104693—21no—
37C4⋊C4×D163C2211yescharacteristic
38#19104669C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41Dic163⋊C8C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104659C2211yescharacteristic
48#19104659C2211yescharacteristic
49#19104658C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic
52C22×Q8×C163C2211yescharacteristic
53C2×Q16×C163C2211yescharacteristic
54C2×Q16×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58C22⋊C8×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326C2311yescharacteristic
61Q8×D163—22no—
62Q8×D163—41no—
63C2×C22⋊D163D411yescharacteristic
64C2×C4×D163—21no—
65C2×C4×D163—21no—
66C2×C4×D163C2311yescharacteristic
67#14579689—41no—
68#14579683—41no—
69C2×C163⋊C8C2311yescharacteristic
70C2×C163⋊C8C2311yescharacteristic
71C8×D163—41no—
72C4⋊Dic163—21no—
73C4⋊Dic163—21no—
74C4⋊Dic163—21no—
75C4⋊Dic163C2311yescharacteristic
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)⋊C4D411yescharacteristic
92C4×Dic163—21no—
93C4×Dic163—21no—
94C4×Dic163—21no—
95C4×Dic163C2311yescharacteristic
96C2×Q8×C163—21no—
97C2×Q8×C163—22no—
98C2×Q8×C163—41no—
99C2×Q8×C163C2311yescharacteristic
100C2×Q8×C163C2311yescharacteristic
101C2×Q8×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C22×C4×C163C2311yescharacteristic
104Q16×C163—22no—
105Q16×C163—41no—
106C2×C8×C163C2311yescharacteristic
107C2×C8×C163C2311yescharacteristic
108C4⋊C4×C163—21no—
109C4⋊C4×C163C2311yescharacteristic
110C22⋊C4×C163—21no—
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—42no—
115C22⋊D163—22no—
116C2×Dic163—21no—
117C2×Dic163—21no—
118C2×Dic163—21no—
119C2×Dic163—21no—
120C2×Dic163C2×D411yescharacteristic
121C2×Dic163C2×D411yescharacteristic
122C2×Dic163C2×D411yescharacteristic
123C4×D163—21no—
124C4×D163—21no—
125C4×D163—42no—
126C4×D163—42no—
127C163⋊C8—21no—
128C163⋊C8—21no—
129C23×C163C2×D411yescharacteristic
130Q8×C163—21no—
131Q8×C163—21no—
132Q8×C163—21no—
133Q8×C163—21no—
134Q8×C163—21no—
135Q8×C163—42no—
136Q8×C163#4112yes—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
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—21no—
151C22×C163C22×D411yescharacteristic
152C2×D163—21no—
153C2×D163—21no—
154C2×D163#13712yes—
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—41no—
170C2×C163#54611yescharacteristic
171C2×C163C2×#13711yescharacteristic
172C2×C163#57811yescharacteristic
173D163—22no—
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#451D16311yescharacteristic, complemented
187#441—1631no—
188#435—1631no—
189#419—1631no—
190C2×#103—1631no—
191C22×Q8—1631no—
192C22×Q8C2×D16311yescharacteristic
193C2×Q16—1631no—
194C2×Q16C2×D16311yescharacteristic
195C2×Q16C2×D16311yescharacteristic
196C2×M16—1631no—
197C22×C8—1631no—
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⋊Q8C2×D16311yescharacteristic
210C4×Q8—1631no—
211C4×Q8—1631no—
212C4×D4—3261no—
213C4×D4—1631no—
214#117—3261no—
215#117—1631no—
216C2×C4⋊C4—1631no—
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#103—1631no—
228#103—1631no—
229#103C2×D16311yescharacteristic
230#103C2×D16311yescharacteristic
231C22⋊C8—1631no—
232C22⋊C8C2×D16311yescharacteristic
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×Q8C22×D16311yescharacteristic
242C2×Q8C22×D16311yescharacteristic
243C2×Q8C22×D16311yescharacteristic
244C2×D4—1631no—
245C22×C4—21no—
246C22×C4—3261no—
247C22×C4—3261no—
248C22×C4—1631no—
249C22×C4C22×D16311yescharacteristic
250Q16—22no—
251Q16—41no—
252Q16—6521no—
253M16—6521no—
254C2×C8—6521no—
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⋊C4—1631no—
274C4⋊C4C22×D16311yescharacteristic
275C22⋊C4—21no—
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—21no—
292Q8—21no—
293Q8—42no—
294Q8—3261no—
295Q8—3261no—
296Q8—6522no—
297Q8#1457969812yes—
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—3261no—
307C2×C4—3261no—
308C2×C4—3261no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—6522no—
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—21no—
326C22—3261no—
327C22—3261no—
328C22—1632no—
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×#1457969811yescharacteristic
343C4#1910478011yescharacteristic
344C2—41no—
345C2—3262no—
346C2#2378230211yescharacteristic
347C2#137×D16311yescharacteristic
348C2#2378247811yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#451×C163C2
socleC22×C163C22×D4
prime radicalC2×C22⋊D163D4
hypercenter#451D163
hypocenterC163#3312
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #23782032C2semidirect
4. #23782007C2semidirect
5. #23781994C2semidirect
6. #23781974C2semidirect
7. #23781921C2semidirect
8. #23781903C2semidirect
9. #23781873C2semidirect
10. #23781819C2semidirect
11. #23781772C2semidirect
13. #23781674C2semidirect
14. #103×D163C2semidirect
15. #451×C163C2semidirect
174. C163#3312semidirect
186. #451D163semidirect

Maximal subgroups

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

subgroupindex
#237823492
Q8×C22⋊D1632
#237820322
#237820072
#237819942
#237819742
#237819212
#237819032
#237818732
#237818192
#237817722
#237817392
#237816742
#103×D1632
#451×C1632
#3312163