Finite Group Databasebeta

← #35227273

Subgroups — #35227273

Order 20864.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23782395C211yesmaximal, characteristic, complemented
2#23782384C211yesmaximal, characteristic
3#23782023C211yesmaximal, characteristic
4#23782014C211yesmaximal, characteristic, complemented
5#23781953C211yesmaximal, characteristic
6#23781948C211yesmaximal, characteristic, complemented
7#23781947C211yesmaximal, characteristic, complemented
8#23781886C211yesmaximal, characteristic
9#23781884C211yesmaximal, characteristic
10#23781874C211yesmaximal, characteristic, complemented
11#23781793C211yesmaximal, characteristic, complemented
12#23781773C211yesmaximal, characteristic, complemented
13#23781769C211yesmaximal, characteristic, complemented
14#23781756C211yesmaximal, characteristic
15#489×C163C211yesmaximal, characteristic
16C2×Dic652C2211yescharacteristic
17C2×#14579689C2211yescharacteristic
18C2×#14579687C2211yescharacteristic
19#19104734—21no—
20#19104734C2211yescharacteristic
21#19104734C2211yescharacteristic
22#19104733—21no—
23#19104733—21no—
24#19104733C2211yescharacteristic
25#19104728—21no—
26#19104728—21no—
27#19104722C2211yescharacteristic
28#19104712—21no—
29Q8×Dic163C2211yescharacteristic
30C4×Dic326C2211yescharacteristic
31C4×Dic326C2211yescharacteristic
32C4×Dic326C2211yescharacteristic
33#19104700—21no—
34#19104700C2211yescharacteristic
35#19104700C2211yescharacteristic
36D4×Dic163C2211yescharacteristic
37#19104691—21no—
38#19104667C2211yescharacteristic
39#19104666C2211yescharacteristic
40#19104666C2211yescharacteristic
41C4⋊(C163⋊C8)C2211yescharacteristic
42Dic163⋊C8C2211yescharacteristic
43Dic163⋊C8C2211yescharacteristic
44#19104660C2211yescharacteristic
45#19104660C2211yescharacteristic
46#19104659C2211yescharacteristic
47#19104658C2211yescharacteristic
48#19104658C2211yescharacteristic
49#19104656C2211yescharacteristic
50#19104645C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C4⋊Q8×C163C2211yescharacteristic
53#124×C163C2211yescharacteristic
54#103×C163C2211yescharacteristic
55#103×C163C2211yescharacteristic
56#102×C163C2211yescharacteristic
57#102×C163C2211yescharacteristic
58#97×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326—21no—
61C2×Dic326D411yescharacteristic
62C2×Dic326C2311yescharacteristic
63C2×Dic326C2311yescharacteristic
64C22×Dic163—21no—
65Dic652—22no—
66#14579689—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⋊C4—21no—
81Dic163⋊C4—21no—
82Dic163⋊C4D411yescharacteristic
83Dic163⋊C4D411yescharacteristic
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C4×Dic163—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163—21no—
91C4×Dic163—21no—
92C4×Dic163C2311yescharacteristic
93C4×Dic163C2311yescharacteristic
94C2×Q8×C163—21no—
95C2×Q8×C163C2311yescharacteristic
96C2×D4×C163C2311yescharacteristic
97C2×C8×C163C2311yescharacteristic
98C2×C8×C163C2311yescharacteristic
99C4⋊C4×C163—21no—
100C4⋊C4×C163C2311yescharacteristic
101C4⋊C4×C163C2311yescharacteristic
102C22⋊C4×C163—21no—
103C22⋊C4×C163—21no—
104C42×C163C2311yescharacteristic
105Dic326—21no—
106Dic326—22no—
107Dic326—21no—
108Dic326—21no—
109Dic326—81no—
110Dic326—81no—
111Dic326#4112yes—
112C2×Dic163—21no—
113C2×Dic163—21no—
114C2×Dic163—21no—
115C2×Dic163—21no—
116C2×Dic163—42no—
117C2×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C2×Dic163C2×D411yescharacteristic
121C163⋊C8—21no—
122C163⋊C8—21no—
123C23×C163—21no—
124Q8×C163—21no—
125Q8×C163—21no—
126Q8×C163—81no—
127D4×C163—21no—
128D4×C163—21no—
129C2×C4×C163—21no—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163C2×D411yescharacteristic
133C2×C4×C163C2×D411yescharacteristic
134C2×C4×C163C2411yescharacteristic
135C8×C163—21no—
136C8×C163—21no—
137C22×C163—41no—
138C22×C163—41no—
139C22×C163—41no—
140C22×C163C22×D411yescharacteristic
141C4×C163—21no—
142C4×C163—21no—
143C4×C163—41no—
144C4×C163—41no—
145C4×C163—41no—
146C4×C163C2×#4111yescharacteristic
147C4×C1632−1+411yescharacteristic
148Dic163—22no—
149Dic163—21no—
150Dic163—21no—
151Dic163—21no—
152Dic163—41no—
153Dic163—41no—
154Dic163—41no—
155Dic163#13712yes—
156C2×C163—81no—
157C2×C163#54811yescharacteristic
158C2×C163C2×#13711yescharacteristic
159C2×C163#57711yescharacteristic
160C163#336911yescharacteristic, complemented, Sylow
161#3369—1631nomaximal, Sylow
162Q82—1631no—
163#556—1631no—
164#489—1631no—
165#489D16311yescharacteristic
166#487—1631no—
167#476—1631no—
168#475—1631no—
169#475—1631no—
170#464—1631no—
171#464—1631no—
172#462—1631no—
173#446—1631no—
174#441—1631no—
175#441—1631no—
176C4×SD16—1631no—
177C2×Q16—1631no—
178C2×Q16—1631no—
179C2×SD16—1631no—
180C4⋊Q8—3261no—
181C4⋊Q8—3261no—
182C4⋊Q8—3261no—
183C4⋊Q8—1631no—
184C4⋊Q8—1631no—
185C4⋊Q8—1631no—
186C4⋊Q8C2×D16311yescharacteristic
187#126—3261no—
188#126—3261no—
189#124—1631no—
190#124C2×D16311yescharacteristic
191C22⋊Q8—3261no—
192C4×Q8—3261no—
193C4×Q8—1631no—
194C4×Q8—1631no—
195C4×Q8—1631no—
196C4×Q8—1631no—
197C4×Q8—1631no—
198C4×Q8—1631no—
199C4×D4—1631no—
200#117—3261no—
201#106—1631no—
202#106—1631no—
203#106—1631no—
204C4⋊C8—1631no—
205C4⋊C8—1631no—
206C4⋊C8—1631no—
207#103—1631no—
208#103—1631no—
209#103—1631no—
210#103—1631no—
211#103—1631no—
212#103C2×D16311yescharacteristic
213#103C2×D16311yescharacteristic
214#102—1631no—
215#102C2×D16311yescharacteristic
216#102C2×D16311yescharacteristic
217#97—1631no—
218#97C2×D16311yescharacteristic
219C4×C8—1631no—
220C2×Q8—21no—
221C2×Q8—3261no—
222C2×Q8—3261no—
223C2×Q8—1631no—
224C2×Q8—1631no—
225C2×Q8—1631no—
226C2×Q8C22×D16311yescharacteristic
227C2×D4C22×D16311yescharacteristic
228C22×C4—3261no—
229Q16—3262no—
230Q16—3262no—
231SD16—3262no—
232C2×C8—1631no—
233C2×C8—1631no—
234C2×C8C22×D16311yescharacteristic
235C2×C8C22×D16311yescharacteristic
236C4⋊C4—21no—
237C4⋊C4—3261no—
238C4⋊C4—3261no—
239C4⋊C4—3261no—
240C4⋊C4—3261no—
241C4⋊C4—3261no—
242C4⋊C4—3261no—
243C4⋊C4—3261no—
244C4⋊C4—3261no—
245C4⋊C4—3261no—
246C4⋊C4—1631no—
247C4⋊C4—1631no—
248C4⋊C4—1631no—
249C4⋊C4—1631no—
250C4⋊C4—1631no—
251C4⋊C4C22×D16311yescharacteristic
252C4⋊C4C22×D16311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—3261no—
256C22⋊C4—3261no—
257C22⋊C4—3261no—
258C42—3261no—
259C42—3261no—
260C42—3261no—
261C42—3261no—
262C42—3261no—
263C42—1631no—
264C42—1631no—
265C42C22×D16311yescharacteristic
266C23—21no—
267Q8—21no—
268Q8—21no—
269Q8—81no—
270Q8—13041no—
271Q8—13041no—
272Q8—3262no—
273Q8—3261no—
274Q8—3261no—
275Q8—3261no—
276Q8—1632no—
277D4—21no—
278D4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—3261no—
283C2×C4—3261no—
284C2×C4—3261no—
285C2×C4—3261no—
286C2×C4—6522no—
287C2×C4—1631no—
288C2×C4—1631no—
289C2×C4—1631no—
290C2×C4—1631no—
291C2×C4D4×D16311yescharacteristic
292C2×C4D4×D16311yescharacteristic
293C2×C4C23×D16311yescharacteristic
294C8—21no—
295C8—21no—
296C8—3261no—
297C8—3261no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22C2×D4×D16311yescharacteristic
302C4—21no—
303C4—21no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—3261no—
308C4—3261no—
309C4—3261no—
310C4—3262no—
311C4—6521no—
312C4—6521no—
313C4—6521no—
314C4—1632no—
315C4#41×D16311yescharacteristic
316C4#1910478211yescharacteristic
317C2—81no—
318C2#2378232311yescharacteristic
319C2#137×D16311yescharacteristic
320C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#489×C163C2
socleC22×C163C22×D4
prime radicalC2×D4×C163C23
hypercenter#489D163
hypocenterC163#3369
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #23782395C2semidirect
4. #23782014C2semidirect
6. #23781948C2semidirect
7. #23781947C2semidirect
10. #23781874C2semidirect
11. #23781793C2semidirect
12. #23781773C2semidirect
13. #23781769C2semidirect
160. C163#3369semidirect

Maximal subgroups

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

subgroupindex
#237823952
#237823842
#237820232
#237820142
#237819532
#237819482
#237819472
#237818862
#237818842
#237818742
#237817932
#237817732
#237817692
#237817562
#489×C1632
#3369163