Finite Group Databasebeta

← #35226587

Subgroups — #35226587

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#23782355C211yesmaximal, characteristic
2#23782033C211yesmaximal, characteristic, complemented
3#23782003C211yesmaximal, characteristic, complemented
4#23781993C211yesmaximal, characteristic, complemented
5#23781972C211yesmaximal, characteristic, complemented
6#23781921C211yesmaximal, characteristic, complemented
7#23781904C211yesmaximal, characteristic, complemented
8#23781872C211yesmaximal, characteristic, complemented
9#23781821C211yesmaximal, characteristic, complemented
10#23781821C211yesmaximal, characteristic, complemented
11#23781770C211yesmaximal, characteristic, complemented
12#23781737C211yesmaximal, characteristic
13#23781674C211yesmaximal, characteristic, complemented
14C2×#19104659C211yesmaximal, characteristic, complemented
15D4×Q8×C163C211yesmaximal, characteristic
16C2×#14579689C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579683C2211yescharacteristic
20C22×C163⋊C8C2211yescharacteristic
21#19104728—21no—
22#19104727C2211yescharacteristic
23#19104725—21no—
24#19104716—21no—
25#19104714C2211yescharacteristic
26#19104714C2211yescharacteristic
27C4×Dic326C2211yescharacteristic
28#19104694C2211yescharacteristic
29#19104691—21no—
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#19104659C2211yescharacteristic
42#19104659C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104658C2211yescharacteristic
45C22⋊(C163⋊C8)C2211yescharacteristic
46C22⋊(C163⋊C8)C2211yescharacteristic
47#19104646C2211yescharacteristic
48C22×Q8×C163C2211yescharacteristic
49C22×Q8×C163C2211yescharacteristic
50C4⋊Q8×C163—21no—
51C4⋊Q8×C163C2211yescharacteristic
52C22⋊Q8×C163—21no—
53C22⋊Q8×C163—21no—
54C22⋊Q8×C163C2211yescharacteristic
55C22⋊Q8×C163C2211yescharacteristic
56C4×Q8×C163C2211yescharacteristic
57C4×D4×C163—21no—
58C4×D4×C163C2211yescharacteristic
59C2×Dic326C2311yescharacteristic
60C22×Dic163—21no—
61#14579689—22no—
62#14579689—41no—
63#14579689—41no—
64#14579683—41no—
65C2×C163⋊C8—41no—
66C2×C163⋊C8C2311yescharacteristic
67C2×C163⋊C8C2311yescharacteristic
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C4⋊Dic163—21no—
71C4⋊Dic163C2311yescharacteristic
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74Dic163⋊C4—21no—
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79C22⋊Dic163—21no—
80C22⋊Dic163—21no—
81C22⋊Dic163—21no—
82C4×Dic163—21no—
83C4×Dic163—21no—
84C2×Q8×C163—21no—
85C2×Q8×C163—21no—
86C2×Q8×C163—22no—
87C2×Q8×C163—22no—
88C2×Q8×C163—41no—
89C2×Q8×C163—41no—
90C2×Q8×C163C2311yescharacteristic
91C2×Q8×C163C2311yescharacteristic
92C2×Q8×C163C2311yescharacteristic
93C2×D4×C163D411yescharacteristic
94C22×C4×C163—21no—
95C22×C4×C163—21no—
96C22×C4×C163C2311yescharacteristic
97C22×C4×C163C2311yescharacteristic
98C4⋊C4×C163—21no—
99C4⋊C4×C163—21no—
100C4⋊C4×C163—21no—
101C4⋊C4×C163—21no—
102C4⋊C4×C163—21no—
103C4⋊C4×C163D411yescharacteristic
104C4⋊C4×C163C2311yescharacteristic
105C22⋊C4×C163—21no—
106C22⋊C4×C163—21no—
107C22⋊C4×C163D411yescharacteristic
108C22⋊C4×C163D411yescharacteristic
109C42×C163—21no—
110C42×C163C2311yescharacteristic
111Dic326—21no—
112Dic326—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—21no—
117C2×Dic163—42no—
118C163⋊C8—21no—
119C163⋊C8—21no—
120C163⋊C8—21no—
121C163⋊C8—21no—
122C23×C163C2×D411yescharacteristic
123C23×C163C2×D411yescharacteristic
124Q8×C163—21no—
125Q8×C163—21no—
126Q8×C163—21no—
127Q8×C163—21no—
128Q8×C163—21no—
129Q8×C163—42no—
130Q8×C163—42no—
131Q8×C163#4112yes—
132D4×C163—22no—
133C2×C4×C163—21no—
134C2×C4×C163—21no—
135C2×C4×C163—21no—
136C2×C4×C163—21no—
137C2×C4×C163—21no—
138C2×C4×C163—21no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—42no—
142C2×C4×C163—42no—
143C2×C4×C163C2×D411yescharacteristic
144C2×C4×C163C2×D411yescharacteristic
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2×D411yescharacteristic
147C2×C4×C163C2411yescharacteristic
148C22×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—21no—
153C22×C163#13712yes—
154C22×C163C22×D411yescharacteristic
155C4×C163—21no—
156C4×C163—21no—
157C4×C163—22no—
158C4×C163—21no—
159C4×C163—21no—
160C4×C163—41no—
161C4×C163—41no—
162C4×C163—41no—
163C4×C163C2×#4111yescharacteristic
164C4×C163H(5,𝔽2)11yescharacteristic
165Dic163—41no—
166Dic163—41no—
167Dic163—41no—
168Dic163—41no—
169C2×C163—22no—
170C2×C163—41no—
171C2×C163#54611yescharacteristic
172C2×C163C2×#13711yescharacteristic
173C2×C163#57811yescharacteristic
174C163#331211yescharacteristic, complemented, Sylow
175#3312—1631nomaximal, Sylow
176#552—1631no—
177D4×Q8D16311yescharacteristic
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×#103—1631no—
191C22×Q8C2×D16311yescharacteristic
192C22×Q8C2×D16311yescharacteristic
193C2×Q16—1631no—
194C2×Q16—1631no—
195C2×Q16—1631no—
196C2×M16—1631no—
197C22×C8—1631no—
198C4⋊Q8—21no—
199C4⋊Q8C2×D16311yescharacteristic
200#126—3261no—
201#125—3261no—
202#125—1631no—
203#123—3261no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—1631no—
207C22⋊Q8—1631no—
208C22⋊Q8C2×D16311yescharacteristic
209C22⋊Q8C2×D16311yescharacteristic
210C4×Q8—1631no—
211C4×Q8C2×D16311yescharacteristic
212C4×D4—21no—
213C4×D4C2×D16311yescharacteristic
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#103—1631no—
230#103—1631no—
231C22⋊C8—1631no—
232C22⋊C8—1631no—
233#97—1631no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—1631no—
241C2×Q8C22×D16311yescharacteristic
242C2×Q8C22×D16311yescharacteristic
243C2×Q8C22×D16311yescharacteristic
244C2×D4C22⋊D16311yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—3261no—
248C22×C4C22×D16311yescharacteristic
249C22×C4C22×D16311yescharacteristic
250Q16—3262no—
251Q16—6521no—
252Q16—6521no—
253M16—6521no—
254C2×C8—6521no—
255C2×C8—1631no—
256C2×C8—1631no—
257C2×C8—1631no—
258C2×C8—1631no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
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—21no—
276C22⋊C4—21no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4C22⋊D16311yescharacteristic
281C22⋊C4C22⋊D16311yescharacteristic
282C42—21no—
283C42—3261no—
284C42—3261no—
285C42C22×D16311yescharacteristic
286C23C2×C22⋊D16311yescharacteristic
287C23C2×C22⋊D16311yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—42no—
294Q8—42no—
295Q8—3261no—
296Q8—3261no—
297Q8#1457969812yes—
298D4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—3261no—
310C2×C4—3261no—
311C2×C4—3261no—
312C2×C4—3261no—
313C2×C4—6522no—
314C2×C4C2×C22⋊D16311yescharacteristic
315C2×C4C2×C22⋊D16311yescharacteristic
316C2×C4C2×C22⋊D16311yescharacteristic
317C2×C4C2×C22⋊D16311yescharacteristic
318C2×C4C23×D16311yescharacteristic
319C8—3261no—
320C8—3261no—
321C8—3261no—
322C8—3261no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22#1910476612yes—
329C22C22×C22⋊D16311yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—22no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—6521no—
339C4—6521no—
340C4—6521no—
341C4—6521no—
342C4C2×#1457969811yescharacteristic
343C4#1910478011yescharacteristic
344C2—22no—
345C2—41no—
346C2#2378230911yescharacteristic
347C2C2×#1910476611yescharacteristic
348C2#2378248011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
FittingD4×Q8×C163C2
socleC22×C163C22×D4
prime radicalC2×D4×C163D4
hypercenterD4×Q8D163
hypocenterC163#3312
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
2. #23782033C2semidirect
3. #23782003C2semidirect
4. #23781993C2semidirect
5. #23781972C2semidirect
6. #23781921C2semidirect
7. #23781904C2semidirect
8. #23781872C2semidirect
9. #23781821C2semidirect
10. #23781821C2semidirect
11. #23781770C2semidirect
13. #23781674C2semidirect
14. C2×#19104659C2semidirect
174. C163#3312semidirect

Maximal subgroups

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

subgroupindex
#237823552
#237820332
#237820032
#237819932
#237819722
#237819212
#237819042
#237818722
#237818212
#237817702
#237817372
#237816742
C2×#191046592
D4×Q8×C1632
#3312163