Finite Group Databasebeta

← #35226598

Subgroups — #35226598

Order 20864.

380 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782383C211yesmaximal, characteristic, complemented
2#23782322C211yesmaximal, characteristic
3#23782023C211yesmaximal, characteristic, complemented
4#23782006C211yesmaximal, characteristic, complemented
5#23781987C211yesmaximal, characteristic, complemented
6#23781958C211yesmaximal, characteristic, complemented
7#23781910C211yesmaximal, characteristic, complemented
8#23781906C211yesmaximal, characteristic, complemented
9#23781886C211yesmaximal, characteristic, complemented
10#23781827C211yesmaximal, characteristic, complemented
11#23781822C211yesmaximal, characteristic, complemented
12#23781769C211yesmaximal, characteristic, complemented
13#23781736C211yesmaximal, characteristic, complemented
14#23781672C211yesmaximal, characteristic, complemented
15#416×C163C211yesmaximal, characteristic, complemented
16C2×Q8×D163C2211yescharacteristic
17C2×Dic652C2211yescharacteristic
18C2×#14579689C2211yescharacteristic
19C2×#14579687C2211yescharacteristic
20C2×#14579682C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104729—21no—
23#19104728—21no—
24#19104722—21no—
25#19104722C2211yescharacteristic
26#19104721—21no—
27#19104715C2211yescharacteristic
28#19104714—21no—
29#19104714C2211yescharacteristic
30#19104714C2211yescharacteristic
31#19104706—21no—
32#19104706C2211yescharacteristic
33Q8×Dic163C2211yescharacteristic
34C4×Dic326—21no—
35#19104700C2211yescharacteristic
36C4×C22⋊D163—21no—
37C4×C22⋊D163C2211yescharacteristic
38#19104693C2211yescharacteristic
39#19104669C2211yescharacteristic
40#19104668C2211yescharacteristic
41#19104666C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43#19104660C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104659C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104656C2211yescharacteristic
49C22⋊(C163⋊C8)C2211yescharacteristic
50(C2×D163)⋊C8C2211yescharacteristic
51#19104645C2211yescharacteristic
52C2×#41×C163C2211yescharacteristic
53C22×C8×C163C2211yescharacteristic
54#117×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#103×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63Q8×D163—22no—
64Q8×D163—41no—
65C2×C22⋊D163—21no—
66C2×C22⋊D163D411yescharacteristic
67C2×C4×D163—21no—
68C2×C4×D163C2311yescharacteristic
69Dic652—41no—
70#14579689—22no—
71#14579687—41no—
72#14579682—41no—
73C2×C163⋊C8C2311yescharacteristic
74C2×C163⋊C8C2311yescharacteristic
75C4⋊Dic163—21no—
76C4⋊Dic163—21no—
77C4⋊Dic163C2311yescharacteristic
78C4⋊Dic163C2311yescharacteristic
79Dic163⋊C4—21no—
80Dic163⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4—21no—
83Dic163⋊C4D411yescharacteristic
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C22⋊Dic163—21no—
89C22⋊Dic163D411yescharacteristic
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
97#41×C163—22no—
98#41×C163—41no—
99C2×Q8×C163C2311yescharacteristic
100C2×D4×C163—21no—
101C2×D4×C163C2311yescharacteristic
102C22×C4×C163—21no—
103C22×C4×C163C2311yescharacteristic
104C2×C8×C163—41no—
105C2×C8×C163C2311yescharacteristic
106C2×C8×C163C2311yescharacteristic
107C4⋊C4×C163C2311yescharacteristic
108C4⋊C4×C163C2311yescharacteristic
109C22⋊C4×C163—21no—
110C42×C163—21no—
111C22×D163C2×D411yescharacteristic
112Dic326—21no—
113Dic326—21no—
114Dic326—21no—
115Dic326—21no—
116Dic326—42no—
117Dic326—42no—
118C22⋊D163—41no—
119C22⋊D163—42no—
120C2×Dic163—21no—
121C2×Dic163—21no—
122C2×Dic163—21no—
123C2×Dic163—21no—
124C2×Dic163C2×D411yescharacteristic
125C2×Dic163C2×D411yescharacteristic
126C2×Dic163C2×D411yescharacteristic
127C4×D163—21no—
128C4×D163—21no—
129C4×D163—42no—
130C163⋊C8—21no—
131C163⋊C8—21no—
132C23×C163—21no—
133C23×C163C2×D411yescharacteristic
134Q8×C163—21no—
135Q8×C163#4112yes—
136D4×C163—21no—
137D4×C163—21no—
138D4×C163—42no—
139C2×C4×C163—21no—
140C2×C4×C163—21no—
141C2×C4×C163—21no—
142C2×C4×C163—21no—
143C2×C4×C163—21no—
144C2×C4×C163—42no—
145C2×C4×C163C2×D411yescharacteristic
146C2×C4×C163C2411yescharacteristic
147C8×C163—21no—
148C8×C163—21no—
149C22×C163—21no—
150C22×C163—21no—
151C22×C163—21no—
152C22×C163—41no—
153C22×C163—41no—
154C22×C163—41no—
155C22×C163C22×D411yescharacteristic
156C2×D163—21no—
157C2×D163—21no—
158C2×D163—21no—
159C4×C163—21no—
160C4×C163—22no—
161C4×C163—41no—
162C4×C163—41no—
163C4×C163C2×#4111yescharacteristic
164C4×C163H(5,𝔽2)11yescharacteristic
165Dic163—21no—
166Dic163—21no—
167Dic163—21no—
168Dic163—41no—
169Dic163—41no—
170Dic163—41no—
171Dic163—41no—
172C2×C163—41no—
173C2×C163—81no—
174C2×C163#54611yescharacteristic
175C2×C163#57511yescharacteristic
176C2×C163#57811yescharacteristic
177D163—41no—
178C163#331311yescharacteristic, complemented, Sylow
179#3313—1631nomaximal, Sylow
180#556—1631no—
181#548—1631no—
182#489—1631no—
183#485—1631no—
184#482—1631no—
185#477—1631no—
186#468—1631no—
187#467—1631no—
188#464—1631no—
189#452—1631no—
190#451—1631no—
191#441—1631no—
192#435—1631no—
193#419—1631no—
194#416D16311yescharacteristic, complemented
195C2×#41C2×D16311yescharacteristic
196C22×Q8—1631no—
197C2×Q16—1631no—
198C2×Q16—1631no—
199C2×SD16—1631no—
200C2×M16—1631no—
201C22×C8C2×D16311yescharacteristic
202C4⋊Q8—1631no—
203#126—3261no—
204#126—3261no—
205#124—3261no—
206#124—3261no—
207#124—1631no—
208C22⋊Q8—3261no—
209C22⋊Q8—1631no—
210C22⋊Q8—1631no—
211C22⋊Q8—1631no—
212#121—3261no—
213#121—1631no—
214C4×Q8—3261no—
215C4×Q8—1631no—
216C4×Q8—1631no—
217C4×D4—3261no—
218C4×D4—1631no—
219#117—1631no—
220#117C2×D16311yescharacteristic
221#107—1631no—
222#107—1631no—
223#106—1631no—
224C4⋊C8—1631no—
225#103—1631no—
226#103—1631no—
227#103—1631no—
228#103—1631no—
229#103—1631no—
230#103C2×D16311yescharacteristic
231#103C2×D16311yescharacteristic
232#102—1631no—
233#102C2×D16311yescharacteristic
234#102C2×D16311yescharacteristic
235C22⋊C8—1631no—
236C22⋊C8—1631no—
237#97—1631no—
238#41—22no—
239#41—41no—
240C2×Q8—3261no—
241C2×Q8—3261no—
242C2×Q8—3262no—
243C2×Q8—6521no—
244C2×Q8—1631no—
245C2×Q8—1631no—
246C2×Q8C22×D16311yescharacteristic
247C2×D4—21no—
248C2×D4—3261no—
249C2×D4—1631no—
250C2×D4C22×D16311yescharacteristic
251C22×C4—21no—
252C22×C4—3261no—
253C22×C4—1631no—
254C22×C4C22×D16311yescharacteristic
255Q16—3262no—
256Q16—6521no—
257SD16—6521no—
258M16—6521no—
259C2×C8—41no—
260C2×C8—1631no—
261C2×C8—1631no—
262C2×C8C22×D16311yescharacteristic
263C2×C8C22×D16311yescharacteristic
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—3261no—
277C22⋊C4—3261no—
278C22⋊C4—3261no—
279C22⋊C4—3261no—
280C22⋊C4—3261no—
281C22⋊C4—3261no—
282C22⋊C4—3261no—
283C22⋊C4—1631no—
284C22⋊C4—1631no—
285C42—21no—
286C42—3261no—
287C42—3261no—
288C42—3261no—
289C42—1631no—
290C23—21no—
291C23—1631no—
292C23D4×D16311yescharacteristic
293Q8—21no—
294Q8—3261no—
295Q8—3261no—
296Q8—3261no—
297Q8—3261no—
298Q8—6522no—
299Q8—6522no—
300Q8#1457969912yes—
301D4—21no—
302D4—21no—
303D4—42no—
304D4—6521no—
305D4—6522no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—3261no—
313C2×C4—3261no—
314C2×C4—3261no—
315C2×C4—3261no—
316C2×C4—3261no—
317C2×C4—3261no—
318C2×C4—6522no—
319C2×C4—1631no—
320C2×C4—1631no—
321C2×C4—1631no—
322C2×C4D4×D16311yescharacteristic
323C2×C4C23×D16311yescharacteristic
324C8—21no—
325C8—21no—
326C8—3261no—
327C8—3261no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—3261no—
335C22—3261no—
336C22—3261no—
337C22C2×D4×D16311yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—3261no—
343C4—3261no—
344C4—3261no—
345C4—6521no—
346C4—6521no—
347C4—6521no—
348C4—6521no—
349C4C2×#1457969911yescharacteristic
350C4#1910478011yescharacteristic
351C2—41no—
352C2—81no—
353C2—6521no—
354C2#2378230511yescharacteristic
355C2#2378246211yescharacteristic
356C2#2378247911yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#416×C163C2
socleC22×C163C22×D4
prime radical#23782322C2
hypercenter#416D163
hypocenterC163#3313
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #23782383C2semidirect
3. #23782023C2semidirect
4. #23782006C2semidirect
5. #23781987C2semidirect
6. #23781958C2semidirect
7. #23781910C2semidirect
8. #23781906C2semidirect
9. #23781886C2semidirect
10. #23781827C2semidirect
11. #23781822C2semidirect
12. #23781769C2semidirect
13. #23781736C2semidirect
14. #23781672C2semidirect
15. #416×C163C2semidirect
178. C163#3313semidirect
194. #416D163semidirect

Maximal subgroups

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

subgroupindex
#237823832
#237823222
#237820232
#237820062
#237819872
#237819582
#237819102
#237819062
#237818862
#237818272
#237818222
#237817692
#237817362
#237816722
#416×C1632
#3313163