Finite Group Databasebeta

← #35227180

Subgroups — #35227180

Order 20864.

350 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#23782384C211yesmaximal, characteristic, complemented
2#23782374C211yesmaximal, characteristic, complemented
3#23782019C211yesmaximal, characteristic, complemented
4#23782018C211yesmaximal, characteristic, complemented
5#23782014C211yesmaximal, characteristic, complemented
6#23781974C211yesmaximal, characteristic, complemented
7#23781973C211yesmaximal, characteristic, complemented
8#23781953C211yesmaximal, characteristic, complemented
9#23781878C211yesmaximal, characteristic
10#23781868C211yesmaximal, characteristic, complemented
11#23781789C211yesmaximal, characteristic, complemented
12SD16×Dic163C211yesmaximal, characteristic, complemented
13C4×#14579687C211yesmaximal, characteristic, complemented
14D8×Dic163C211yesmaximal, characteristic
15#463×C163C211yesmaximal, characteristic
16C2×#14579687C2211yescharacteristic
17#19104734C2211yescharacteristic
18#19104733C2211yescharacteristic
19#19104728—21no—
20#19104728—21no—
21#19104727C2211yescharacteristic
22#19104726C2211yescharacteristic
23#19104722C2211yescharacteristic
24#19104714—21no—
25#19104712—21no—
26#19104712—21no—
27#19104712—21no—
28Q8×Dic163C2211yescharacteristic
29C4×Dic326C2211yescharacteristic
30C4×Dic326C2211yescharacteristic
31D4×Dic163C2211yescharacteristic
32D4×Dic163C2211yescharacteristic
33D4×Dic163C2211yescharacteristic
34#19104691—21no—
35C2×Dic163⋊C4—21no—
36#19104669C2211yescharacteristic
37#19104667C2211yescharacteristic
38#19104666C2211yescharacteristic
39Dic163⋊C8C2211yescharacteristic
40Dic163⋊C8C2211yescharacteristic
41#19104660C2211yescharacteristic
42#19104660C2211yescharacteristic
43#19104659C2211yescharacteristic
44#19104659C2211yescharacteristic
45#19104658C2211yescharacteristic
46#19104656C2211yescharacteristic
47#19104656C2211yescharacteristic
48#19104656C2211yescharacteristic
49C4×C163⋊C8C2211yescharacteristic
50C8×Dic163C2211yescharacteristic
51C8×Dic163C2211yescharacteristic
52C2×SD16×C163C2211yescharacteristic
53C2×D8×C163C2211yescharacteristic
54#124×C163C2211yescharacteristic
55C4×D4×C163C2211yescharacteristic
56C4⋊C8×C163C2211yescharacteristic
57#103×C163C2211yescharacteristic
58#102×C163C2211yescharacteristic
59C2×Dic326—21no—
60C2×Dic326D411yescharacteristic
61C2×Dic326C2311yescharacteristic
62C22×Dic163—21no—
63C22×Dic163—21no—
64C22×Dic163—21no—
65#14579687—22no—
66C2×C163⋊C8C2311yescharacteristic
67C2×C163⋊C8C2311yescharacteristic
68C4⋊Dic163—21no—
69C4⋊Dic163—21no—
70C4⋊Dic163D411yescharacteristic
71C4⋊Dic163C2311yescharacteristic
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⋊C4D411yescharacteristic
82Dic163⋊C4D411yescharacteristic
83C22⋊Dic163—21no—
84C22⋊Dic163—21no—
85C22⋊Dic163—21no—
86C22⋊Dic163—21no—
87C22⋊Dic163—21no—
88C4×Dic163—21no—
89C4×Dic163—21no—
90C4×Dic163C2311yescharacteristic
91C4×Dic163C2311yescharacteristic
92C2×Q8×C163C2311yescharacteristic
93C2×D4×C163C2311yescharacteristic
94C2×D4×C163C2311yescharacteristic
95C22×C4×C163—21no—
96SD16×C163—22no—
97D8×C163—22no—
98C2×C8×C163C2311yescharacteristic
99C2×C8×C163C2311yescharacteristic
100C4⋊C4×C163C2311yescharacteristic
101C22⋊C4×C163—21no—
102C22⋊C4×C163—21no—
103C22⋊C4×C163—21no—
104C42×C163C2311yescharacteristic
105Dic326—22no—
106Dic326—21no—
107Dic326—21no—
108Dic326—81no—
109C2×Dic163—21no—
110C2×Dic163—21no—
111C2×Dic163—21no—
112C2×Dic163—21no—
113C2×Dic163—42no—
114C2×Dic163—42no—
115C2×Dic163—81no—
116C2×Dic163C2×D411yescharacteristic
117C2×Dic163C2×D411yescharacteristic
118C2×Dic163C2×D411yescharacteristic
119C2×Dic163C2×D411yescharacteristic
120C163⋊C8—21no—
121C163⋊C8—21no—
122C23×C163—21no—
123C23×C163—21no—
124Q8×C163—21no—
125Q8×C163—21no—
126D4×C163—21no—
127D4×C163—21no—
128D4×C163—21no—
129D4×C163#4112yes—
130C2×C4×C163—21no—
131C2×C4×C163—21no—
132C2×C4×C163—81no—
133C2×C4×C163C2×D411yescharacteristic
134C2×C4×C163C2×D411yescharacteristic
135C2×C4×C163C2411yescharacteristic
136C8×C163—21no—
137C8×C163—21no—
138C22×C163—22no—
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—21no—
152Dic163—21no—
153Dic163—21no—
154Dic163—41no—
155Dic163—41no—
156Dic163—41no—
157Dic163—41no—
158Dic163#13512yes—
159C2×C163—42no—
160C2×C163—81no—
161C2×C163#54811yescharacteristic
162C2×C163C2×#13511yescharacteristic
163C2×C163#57711yescharacteristic
164C163#336011yescharacteristic, complemented, Sylow
165#3360—1631nomaximal, Sylow
166#556—1631no—
167#554—1631no—
168#488—1631no—
169#488—1631no—
170#487—1631no—
171#479—1631no—
172#479—1631no—
173#476—1631no—
174#463—1631no—
175#463D16311yescharacteristic
176#461—1631no—
177C8×Q8—1631no—
178C4×SD16—1631no—
179C4×SD16—1631no—
180C4×D8—1631no—
181C2×SD16—1631no—
182C2×SD16C2×D16311yescharacteristic
183C2×D8C2×D16311yescharacteristic
184C4⋊Q8—1631no—
185C4⋊Q8—1631no—
186#126—3261no—
187#126—3261no—
188#125—1631no—
189#125—1631no—
190#124—1631no—
191#124C2×D16311yescharacteristic
192C22⋊Q8—3261no—
193C22⋊Q8—3261no—
194C22⋊Q8—3261no—
195C22⋊Q8—3261no—
196C4×Q8—1631no—
197C4×Q8—1631no—
198C4×Q8—1631no—
199C4×D4—1631no—
200C4×D4—1631no—
201C4×D4—1631no—
202C4×D4C2×D16311yescharacteristic
203#117—3261no—
204C2×C4⋊C4—3261no—
205#107—1631no—
206#106—1631no—
207#106—1631no—
208C4⋊C8—1631no—
209C4⋊C8—1631no—
210C4⋊C8C2×D16311yescharacteristic
211#103—1631no—
212#103—1631no—
213#103—1631no—
214#103—1631no—
215#103—1631no—
216#103C2×D16311yescharacteristic
217#102—1631no—
218#102—1631no—
219#102—1631no—
220#102C2×D16311yescharacteristic
221C4×C8—1631no—
222C4×C8—1631no—
223C4×C8—1631no—
224C2×Q8—3261no—
225C2×Q8—1631no—
226C2×Q8—1631no—
227C2×Q8C22×D16311yescharacteristic
228C2×D4C22×D16311yescharacteristic
229C2×D4C22×D16311yescharacteristic
230C22×C4—21no—
231C22×C4—3261no—
232C22×C4—3261no—
233C22×C4—3261no—
234SD16—22no—
235SD16—3262no—
236D8—22no—
237C2×C8—1631no—
238C2×C8—1631no—
239C2×C8C22×D16311yescharacteristic
240C2×C8C22×D16311yescharacteristic
241C4⋊C4—3261no—
242C4⋊C4—3261no—
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⋊C4—1631no—
256C4⋊C4C22×D16311yescharacteristic
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—3261no—
261C22⋊C4—3261no—
262C22⋊C4—3261no—
263C22⋊C4—3261no—
264C22⋊C4—3261no—
265C42—3261no—
266C42—3261no—
267C42—1631no—
268C42—1631no—
269C42C22×D16311yescharacteristic
270C23—21no—
271C23—21no—
272Q8—21no—
273Q8—21no—
274Q8—13041no—
275Q8—3262no—
276Q8—3261no—
277Q8—3261no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4#1457969812yes—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—13041no—
286C2×C4—3261no—
287C2×C4—3261no—
288C2×C4—3261no—
289C2×C4—3261no—
290C2×C4—6522no—
291C2×C4—6522no—
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—22no—
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—6521no—
318C4—6521no—
319C4—6521no—
320C4—6521no—
321C4—1632no—
322C4C2×#1457969811yescharacteristic
323C4#1910478211yescharacteristic
324C2—42no—
325C2—81no—
326C2#2378232111yescharacteristic
327C2#135×D16311yescharacteristic
328C2#2378247511yescharacteristic
subgroupG/H
centerC22C2×D4×D163
commutator, cosocleC2×C4×C163C24
FrattiniC2×C4C23×D163
Fitting#463×C163C2
socleC22×C163C22×D4
prime radicalC2×D8×C163C22
hypercenter#463D163
hypocenterC163#3360
LCSG ▷ C2×C4×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #23782384C2semidirect
2. #23782374C2semidirect
3. #23782019C2semidirect
4. #23782018C2semidirect
5. #23782014C2semidirect
6. #23781974C2semidirect
7. #23781973C2semidirect
8. #23781953C2semidirect
10. #23781868C2semidirect
11. #23781789C2semidirect
12. SD16×Dic163C2semidirect
13. C4×#14579687C2semidirect
164. C163#3360semidirect

Maximal subgroups

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

subgroupindex
#237823842
#237823742
#237820192
#237820182
#237820142
#237819742
#237819732
#237819532
#237818782
#237818682
#237817892
SD16×Dic1632
C4×#145796872
D8×Dic1632
#463×C1632
#3360163