Finite Group Databasebeta

← #344×D163 #35214276

Subgroups — #344×D163 #35214276

Order 20864.

388 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×M16×D163C211yesmaximal, characteristic
2C2×M16×D163C211yesmaximal, characteristic
3#117×D163C211yesmaximal, characteristic
4#23781437C211yesmaximal, characteristic, complemented
5#23781436C211yesmaximal, characteristic, complemented
6#23781435C211yesmaximal, characteristic, complemented
7#344×C163C211yesmaximal, characteristic, complemented
8C22×C4×D163C2211yescharacteristic
9C2×#14579683C2211yescharacteristic
10C2×#14579683C2211yescharacteristic
11C2×#14579682—21no—
12C2×#14579682C412yescomplemented
13M16×D163—21no—
14M16×D163—21no—
15M16×D163—21no—
16M16×D163—21no—
17M16×D163—21no—
18M16×D163—21no—
19M16×D163C412yescomplemented
20M16×D163C412yescomplemented
21C2×C8×D163—21no—
22C2×C8×D163C412yescomplemented
23#19104694C2211yescharacteristic
24#19104693—21no—
25#19104692—21no—
26#19104691C412yes—
27C4⋊C4×D163—21no—
28C22⋊C4×D163C412yes—
29C42×D163—21no—
30C2×M16×C163C2211yescharacteristic
31C2×M16×C163C2211yescharacteristic
32#117×C163C2211yescharacteristic
33C23×D163C2×C411yescharacteristic
34C22×Dic163C2×C411yescharacteristic
35C2×C4×D163—22no—
36C2×C4×D163C2×C411yescharacteristic
37C2×C4×D163C2×C411yescharacteristic
38C2×C4×D163C2×C411yescharacteristic
39C2×C4×D163C2×C411yescharacteristic
40C2×C4×D163D411yescharacteristic
41C2×C4×D163D411yescharacteristic
42C2×C4×D163D411yescharacteristic
43C2×C4×D163D411yescharacteristic
44C2×C4×D163D411yescharacteristic
45C2×C4×D163D411yescharacteristic
46C2×C4×D163Q811yescharacteristic
47C2×C4×D163Q811yescharacteristic
48#14579683—21no—
49#14579683—21no—
50#14579683—21no—
51#14579683C2×C412yescomplemented
52#14579682—22no—
53#14579682—22no—
54#14579682—22no—
55#14579682—22no—
56C2×C163⋊C8—21no—
57C2×C163⋊C8C2×C412yescomplemented
58C8×D163—22no—
59C8×D163—22no—
60C8×D163—22no—
61C8×D163—22no—
62C4⋊Dic163—21no—
63Dic163⋊C4—22no—
64C22⋊Dic163C2×C412yes—
65(C2×D163)⋊C4—22no—
66C4×Dic163—22no—
67C4×Dic163—21no—
68C22×C4×C163C2311yescharacteristic
69M16×C163—21no—
70M16×C163—21no—
71M16×C163—21no—
72M16×C163C2×C412yescomplemented
73C2×C8×C163—21no—
74C2×C8×C163C2×C412yescomplemented
75C4⋊C4×C163—21no—
76C22⋊C4×C163C2×C412yes—
77C42×C163—21no—
78C22×D163—42no—
79C22×D163C4211yescharacteristic
80C22×D163C4211yescharacteristic
81C22×D163C22⋊C411yescharacteristic
82C22×D163C22⋊C411yescharacteristic
83C22×D163C4⋊C411yescharacteristic
84C22×D163C4⋊C411yescharacteristic
85C2×Dic163—22no—
86C2×Dic163C4211yescharacteristic
87C2×Dic163C4211yescharacteristic
88C2×Dic163C22⋊C411yescharacteristic
89C2×Dic163C22⋊C411yescharacteristic
90C2×Dic163C4⋊C411yescharacteristic
91C2×Dic163C4⋊C411yescharacteristic
92C4×D163—44no—
93C4×D163C22⋊C411yescharacteristic
94C4×D163C22⋊C411yescharacteristic
95C4×D163C22⋊C411yescharacteristic
96C4×D163C22⋊C411yescharacteristic
97C4×D163C22⋊C411yescharacteristic
98C4×D163C22⋊C411yescharacteristic
99C4×D163C22⋊C411yescharacteristic
100C4×D163C22⋊C411yescharacteristic
101C4×D163C4⋊C411yescharacteristic
102C4×D163C4⋊C411yescharacteristic
103C4×D163C4⋊C411yescharacteristic
104C4×D163C4⋊C411yescharacteristic
105C4×D163C4⋊C411yescharacteristic
106C4×D163C4⋊C411yescharacteristic
107C4×D163C4⋊C411yescharacteristic
108C4×D163C4⋊C411yescharacteristic
109C163⋊C8—22no—
110C163⋊C8—22no—
111C23×C163C22×C411yescharacteristic
112C2×C4×C163—22no—
113C2×C4×C163C22×C411yescharacteristic
114C2×C4×C163C22×C411yescharacteristic
115C2×C4×C163C2×D411yescharacteristic
116C2×C4×C163C2×D411yescharacteristic
117C2×C4×C163C2×D411yescharacteristic
118C2×C4×C163C2×Q811yescharacteristic
119C8×C163—22no—
120C8×C163—22no—
121C22×C163—41no—
122C22×C163C2×C4211yescharacteristic
123C22×C163C2×C22⋊C411yescharacteristic
124C22×C163C2×C4⋊C411yescharacteristic
125C2×D163—22no—
126C2×D163—22no—
127C2×D163—22no—
128C2×D163—41no—
129C2×D163—41no—
130C2×D163—41no—
131C2×D163#9511yescharacteristic
132C2×D163#9511yescharacteristic
133C2×D163#9511yescharacteristic
134C2×D163#9511yescharacteristic
135C4×C163—42no—
136C4×C163C2×C22⋊C411yescharacteristic
137C4×C163C2×C22⋊C411yescharacteristic
138C4×C163C2×C4⋊C411yescharacteristic
139C4×C163C2×C4⋊C411yescharacteristic
140Dic163—42no—
141Dic163#9511yescharacteristic
142Dic163#9511yescharacteristic
143Dic163#9511yescharacteristic
144Dic163#9511yescharacteristic
145C2×C163—21no—
146C2×C163—21no—
147C2×C163—21no—
148C2×C163C2×#9511yescharacteristic
149D163—21no—
150D163—21no—
151D163—21no—
152D163#34412yesdirect product factor
153C163C2×#34411yescharacteristic, complemented, Sylow
154C2×#344—1631nomaximal, Sylow
155C22×M16—1631no—
156C22×M16—1631no—
157C2×#117—1631no—
158#344—1631no—
159#344—1631no—
160#344—1631no—
161#344D16311yescharacteristic, direct product factor
162C23×C4—1631no—
163C2×M16—3261no—
164C2×M16—3261no—
165C2×M16—3261no—
166C2×M16—3261no—
167C2×M16—3261no—
168C2×M16—3261no—
169C2×M16—3261no—
170C2×M16—1631no—
171C2×M16—1632no—
172C2×M16—1632no—
173C2×M16—1631no—
174C2×M16—1632no—
175C2×M16C2×D16311yescharacteristic
176C2×M16C2×D16311yescharacteristic
177C22×C8—3261no—
178C22×C8—1632no—
179#117—3261no—
180#117—3261no—
181#117—1631no—
182#117—1632no—
183#117C2×D16311yescharacteristic
184C2×C4⋊C4—3261no—
185C2×C22⋊C4—1632no—
186C2×C42—3261no—
187C24—1631no—
188C22×C4—3262no—
189C22×C4—1631no—
190C22×C4—1631no—
191C22×C4—1631no—
192C22×C4—1631no—
193C22×C4—1631no—
194C22×C4—1631no—
195C22×C4—1631no—
196C22×C4—1631no—
197C22×C4—1631no—
198C22×C4—1631no—
199C22×C4—1631no—
200C22×C4—1631no—
201C22×C4—1631no—
202C22×C4C22×D16311yescharacteristic
203M16—21no—
204M16—21no—
205M16—21no—
206M16—3262no—
207M16—3262no—
208M16—3262no—
209M16—3262no—
210M16—3261no—
211M16—3261no—
212M16—3261no—
213M16—1632no—
214M16C4×D16312yescomplemented
215C2×C8—21no—
216C2×C8—3262no—
217C2×C8—3262no—
218C2×C8—3262no—
219C2×C8—3262no—
220C2×C8—3261no—
221C2×C8—1632no—
222C2×C8C4×D16312yescomplemented
223C4⋊C4—21no—
224C4⋊C4—3262no—
225C4⋊C4—3261no—
226C22⋊C4—3262no—
227C22⋊C4—1632no—
228C22⋊C4C4×D16312yes—
229C42—21no—
230C42—3262no—
231C42—3261no—
232C23—6522no—
233C23—1631no—
234C23—1631no—
235C23—1631no—
236C23—1631no—
237C23—1631no—
238C23—1631no—
239C23C2×C4×D16311yescharacteristic
240C2×C4—22no—
241C2×C4—3262no—
242C2×C4—6524no—
243C2×C4—1631no—
244C2×C4—1631no—
245C2×C4—1631no—
246C2×C4—1631no—
247C2×C4—1631no—
248C2×C4—1631no—
249C2×C4—1631no—
250C2×C4—1631no—
251C2×C4—1631no—
252C2×C4—1631no—
253C2×C4—1631no—
254C2×C4—1631no—
255C2×C4—1631no—
256C2×C4—1631no—
257C2×C4—1631no—
258C2×C4—1631no—
259C2×C4—1631no—
260C2×C4—1631no—
261C2×C4—1631no—
262C2×C4—1631no—
263C2×C4—1631no—
264C2×C4—1631no—
265C2×C4C2×C4×D16311yescharacteristic
266C2×C4C2×C4×D16311yescharacteristic
267C2×C4D4×D16311yescharacteristic
268C2×C4D4×D16311yescharacteristic
269C2×C4D4×D16311yescharacteristic
270C2×C4Q8×D16311yescharacteristic
271C8—22no—
272C8—22no—
273C8—3262no—
274C8—3262no—
275C22—41no—
276C22—3262no—
277C22—3262no—
278C22—3262no—
279C22—6521no—
280C22—6521no—
281C22—6521no—
282C22—1631no—
283C22—1631no—
284C22—1631no—
285C22—1631no—
286C22C42×D16311yescharacteristic
287C22C22⋊C4×D16311yescharacteristic
288C22C4⋊C4×D16311yescharacteristic
289C4—42no—
290C4—6522no—
291C4—1631no—
292C4—1631no—
293C4—1631no—
294C4—1631no—
295C4C22⋊C4×D16311yescharacteristic
296C4C22⋊C4×D16311yescharacteristic
297C4C4⋊C4×D16311yescharacteristic
298C4C4⋊C4×D16311yescharacteristic
299C2—21no—
300C2—21no—
301C2—21no—
302C2—3261no—
303C2—3261no—
304C2—3261no—
305C2—1632no—
306C2#95×D16311yescharacteristic
subgroupG/H
centerC4C22⋊C4×D163
commutatorC22×C163C2×C42
FrattiniC22×C4C22×D163
Fitting#344×C163C2
socleC2×C163C2×#95
prime radicalC23×D163C2×C4
cosocleC22×C4×C163C23
hypercenter#344D163
hypocenterC163C2×#344
LCSG ▷ C22×C163 ▷ C2×C163 ▷ C163
UCSC1 ◁ C4 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. #23781437C2semidirect
5. #23781436C2semidirect
6. #23781435C2semidirect
7. #344×C163C2semidirect
12. C2×#14579682C4semidirect
19. M16×D163C4semidirect
20. M16×D163C4semidirect
22. C2×C8×D163C4semidirect
51. #14579683C2×C4semidirect
57. C2×C163⋊C8C2×C4semidirect
72. M16×C163C2×C4semidirect
74. C2×C8×C163C2×C4semidirect
152. D163#344direct (N × H)
153. C163C2×#344semidirect
161. #344D163direct (N × H)
214. M16C4×D163semidirect
222. C2×C8C4×D163semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1632
#117×D1632
#237814372
#237814362
#237814352
#344×C1632
C2×#344163