Finite Group Databasebeta

← #35227299

Subgroups — #35227299

Order 20864.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782401C211yesmaximal, characteristic
2#23782392C211yesmaximal, characteristic, complemented
3#23782023C211yesmaximal, characteristic, complemented
4#23782018C211yesmaximal, characteristic, complemented
5#23781969C211yesmaximal, characteristic, complemented
6#23781967C211yesmaximal, characteristic, complemented
7#23781953C211yesmaximal, characteristic, complemented
8#23781886C211yesmaximal, characteristic, complemented
9#23781886C211yesmaximal, characteristic, complemented
10#23781862C211yesmaximal, characteristic
11#23781793C211yesmaximal, characteristic, complemented
12#23781773C211yesmaximal, characteristic, complemented
13#23781762C211yesmaximal, characteristic, complemented
14#23781756C211yesmaximal, characteristic, complemented
15#490×C163C211yesmaximal, characteristic
16C2×Dic652C2211yescharacteristic
17C2×#14579687C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19#19104734—21no—
20#19104734C2211yescharacteristic
21#19104726C2211yescharacteristic
22#19104725—21no—
23#19104725—21no—
24#19104722—21no—
25#19104722C2211yescharacteristic
26#19104722C2211yescharacteristic
27#19104716—21no—
28#19104716—21no—
29C4×Dic326C2211yescharacteristic
30C4×Dic326C2211yescharacteristic
31C4×Dic326C2211yescharacteristic
32#19104700—21no—
33#19104700C2211yescharacteristic
34#19104700C2211yescharacteristic
35D4×Dic163—21no—
36D4×Dic163C2211yescharacteristic
37D4×Dic163C2211yescharacteristic
38#19104668C2211yescharacteristic
39#19104667C2211yescharacteristic
40#19104666C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43Dic163⋊C8C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104660C2211yescharacteristic
46#19104658C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104656C2211yescharacteristic
49#19104656C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52#127×C163C2211yescharacteristic
53#125×C163C2211yescharacteristic
54#102×C163C2211yescharacteristic
55#102×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326D411yescharacteristic
61C2×Dic326C2311yescharacteristic
62C2×Dic326C2311yescharacteristic
63C22×Dic163—21no—
64C22×Dic163—21no—
65Dic652—22no—
66#14579687—22no—
67#14579687—22no—
68C2×C163⋊C8C2311yescharacteristic
69C2×C163⋊C8C2311yescharacteristic
70C4⋊Dic163—21no—
71C4⋊Dic163D411yescharacteristic
72C4⋊Dic163C2311yescharacteristic
73C4⋊Dic163C2311yescharacteristic
74Dic163⋊C4—21no—
75Dic163⋊C4—21no—
76Dic163⋊C4—21no—
77Dic163⋊C4—21no—
78Dic163⋊C4—21no—
79Dic163⋊C4—21no—
80Dic163⋊C4D411yescharacteristic
81Dic163⋊C4D411yescharacteristic
82C22⋊Dic163—21no—
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163C2311yescharacteristic
93C4×Dic163C2311yescharacteristic
94C2×D4×C163—21no—
95C2×D4×C163—21no—
96C2×D4×C163C2311yescharacteristic
97C2×D4×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C4⋊C4×C163—21no—
101C4⋊C4×C163—21no—
102C4⋊C4×C163C2311yescharacteristic
103C4⋊C4×C163C2311yescharacteristic
104C42×C163C2311yescharacteristic
105Dic326—21no—
106Dic326—21no—
107Dic326—21no—
108Dic326—41no—
109Dic326—42no—
110Dic326#4112yes—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—81no—
116C2×Dic163—81no—
117C2×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C163⋊C8—21no—
122C163⋊C8—21no—
123C23×C163—21no—
124C23×C163—21no—
125D4×C163—21no—
126D4×C163—21no—
127D4×C163—21no—
128D4×C163—21no—
129D4×C163—42no—
130D4×C163—81no—
131C2×C4×C163—21no—
132C2×C4×C163—21no—
133C2×C4×C163C2×D411yescharacteristic
134C2×C4×C163C2×D411yescharacteristic
135C2×C4×C163C2411yescharacteristic
136C8×C163—21no—
137C8×C163—21no—
138C22×C163—41no—
139C22×C163—41no—
140C22×C163—41no—
141C22×C163—41no—
142C22×C163—41no—
143C22×C163—41no—
144C22×C163C22×D411yescharacteristic
145C4×C163—21no—
146C4×C163—21no—
147C4×C163—41no—
148C4×C163—41no—
149C4×C163C2×#4111yescharacteristic
150C4×C1632−1+411yescharacteristic
151Dic163—22no—
152Dic163—21no—
153Dic163—21no—
154Dic163—21no—
155Dic163—21no—
156Dic163—41no—
157Dic163—41no—
158Dic163—41no—
159C2×C163—81no—
160C2×C163—81no—
161C2×C163#54811yescharacteristic
162C2×C163#57511yescharacteristic
163C2×C163#57711yescharacteristic
164C163#337111yescharacteristic, complemented, Sylow
165#3371—1631nomaximal, Sylow
166#559—1631no—
167#557—1631no—
168#490D16311yescharacteristic
169#489—1631no—
170#488—1631no—
171#479—1631no—
172#478—1631no—
173#476—1631no—
174#464—1631no—
175#464—1631no—
176#460—1631no—
177#446—1631no—
178#441—1631no—
179#440—1631no—
180C4×SD16—1631no—
181C2×Q16—1631no—
182C2×SD16—1631no—
183C2×SD16—1631no—
184C4⋊Q8—3261no—
185C4⋊Q8—1631no—
186#127C2×D16311yescharacteristic
187#125—3261no—
188#125—3261no—
189#125—1631no—
190#125C2×D16311yescharacteristic
191#124—3261no—
192#124—1631no—
193#124—1631no—
194#123—3261no—
195#123—3261no—
196C4×Q8—3261no—
197C4×Q8—1631no—
198C4×Q8—1631no—
199C4×Q8—1631no—
200C4×Q8—1631no—
201C4×Q8—1631no—
202C4×D4—3261no—
203C4×D4—1631no—
204C4×D4—1631no—
205#107—1631no—
206#106—1631no—
207#106—1631no—
208C4⋊C8—1631no—
209C4⋊C8—1631no—
210C4⋊C8—1631no—
211#103—1631no—
212#103—1631no—
213#103—1631no—
214#103—1631no—
215#102—1631no—
216#102—1631no—
217#102C2×D16311yescharacteristic
218#102C2×D16311yescharacteristic
219#102C2×D16311yescharacteristic
220#102C2×D16311yescharacteristic
221#97—1631no—
222#97C2×D16311yescharacteristic
223C4×C8—1631no—
224C2×Q8—3261no—
225C2×Q8—1631no—
226C2×Q8—1631no—
227C2×Q8—1631no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4C22×D16311yescharacteristic
231C2×D4C22×D16311yescharacteristic
232C22×C4—3261no—
233C22×C4—3261no—
234Q16—3262no—
235SD16—3262no—
236SD16—3262no—
237C2×C8—1631no—
238C2×C8—1631no—
239C2×C8C22×D16311yescharacteristic
240C2×C8C22×D16311yescharacteristic
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—3261no—
244C4⋊C4—3261no—
245C4⋊C4—3261no—
246C4⋊C4—3261no—
247C4⋊C4—3261no—
248C4⋊C4—3261no—
249C4⋊C4—3261no—
250C4⋊C4—1631no—
251C4⋊C4—1631no—
252C4⋊C4—1631no—
253C4⋊C4—1631no—
254C4⋊C4—1631no—
255C4⋊C4C22×D16311yescharacteristic
256C4⋊C4C22×D16311yescharacteristic
257C22⋊C4—3261no—
258C22⋊C4—3261no—
259C22⋊C4—3261no—
260C22⋊C4—3261no—
261C22⋊C4—3261no—
262C22⋊C4—3261no—
263C42—3261no—
264C42—3261no—
265C42—3261no—
266C42—3261no—
267C42—1631no—
268C42—1631no—
269C42C22×D16311yescharacteristic
270C23—21no—
271C23—21no—
272Q8—3261no—
273Q8—3261no—
274Q8—3261no—
275Q8—6521no—
276Q8—6522no—
277Q8—1632no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—21no—
282D4—42no—
283D4—81no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—13041no—
287C2×C4—13041no—
288C2×C4—3261no—
289C2×C4—3261no—
290C2×C4—3261no—
291C2×C4—3261no—
292C2×C4—1631no—
293C2×C4—1631no—
294C2×C4—1631no—
295C2×C4—1631no—
296C2×C4D4×D16311yescharacteristic
297C2×C4D4×D16311yescharacteristic
298C2×C4C23×D16311yescharacteristic
299C8—21no—
300C8—21no—
301C8—3261no—
302C8—3261no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×D4×D16311yescharacteristic
310C4—21no—
311C4—21no—
312C4—41no—
313C4—41no—
314C4—3261no—
315C4—3261no—
316C4—3261no—
317C4—3261no—
318C4—3262no—
319C4—6521no—
320C4—6521no—
321C4—6521no—
322C4#41×D16311yescharacteristic
323C4#1910478211yescharacteristic
324C2—81no—
325C2—81no—
326C2#2378232311yescharacteristic
327C2#2378246111yescharacteristic
328C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#490×C163C2
socleC22×C163C22×D4
prime radical#127×C163C22
hypercenter#490D163
hypocenterC163#3371
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
2. #23782392C2semidirect
3. #23782023C2semidirect
4. #23782018C2semidirect
5. #23781969C2semidirect
6. #23781967C2semidirect
7. #23781953C2semidirect
8. #23781886C2semidirect
9. #23781886C2semidirect
11. #23781793C2semidirect
12. #23781773C2semidirect
13. #23781762C2semidirect
14. #23781756C2semidirect
164. C163#3371semidirect

Maximal subgroups

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

subgroupindex
#237824012
#237823922
#237820232
#237820182
#237819692
#237819672
#237819532
#237818862
#237818622
#237817932
#237817732
#237817622
#237817562
#490×C1632
#3371163