Finite Group Databasebeta

← C2×#23782321 #35227999

Subgroups — C2×#23782321 #35227999

Order 20864.

1620 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14579698C211yesmaximal, characteristic, complemented
2C23×Dic326C211yesmaximal, characteristic, complemented
3#23782321C2116yesmaximal, direct product factor
4C2×#19104722C211yesmaximal, characteristic, complemented
5C2×#19104721C212yesmaximal, complemented
6C2×#19104715C211yesmaximal, characteristic, complemented
7C2×#19104712C212yesmaximal, complemented
8C2×#19104707C212yesmaximal, complemented
9C2×#19104700C211yesmaximal, characteristic, complemented
10C2×D4×Dic163C212yesmaximal, complemented
11C2×C4×C22⋊D163C211yesmaximal, characteristic, complemented
12C2×#121×C163C211yesmaximal, characteristic, complemented
13C2×#14579698—28no—
14C2×#14579698C2218yescomplemented
15C22×Dic326—28no—
16C22×Dic326C2212yes—
17C22×Dic326C2211yescharacteristic
18C22×Dic326C2211yescharacteristic
19C22×Dic326C2218yescomplemented
20C22×C22⋊D163C2211yescharacteristic
21C22×C22⋊D163C2212yes—
22C23×Dic163C2212yes—
23C23×Dic163C2212yes—
24C22×C4×D163C2211yescharacteristic
25#19104722C2218yescomplemented
26#19104721C22116yescomplemented
27#19104715C2218yescomplemented
28#19104712C22116yescomplemented
29#19104707C22116yescomplemented
30#19104700C2218yescomplemented
31D4×Dic163C22116yescomplemented
32C4×C22⋊D163C2218yescomplemented
33C2×C4⋊Dic163C2212yes—
34C2×Dic163⋊C4C2212yes—
35C2×Dic163⋊C4C2211yescharacteristic
36C2×C22⋊Dic163C2212yes—
37C2×C22⋊Dic163C2212yes—
38C2×C22⋊Dic163C2211yescharacteristic
39C2×(C2×D163)⋊C4C2212yes—
40C2×(C2×D163)⋊C4C2211yescharacteristic
41C2×C4×Dic163C2212yes—
42C2×C4×Dic163C2211yescharacteristic
43C22×D4×C163C2212yes—
44C22×D4×C163C2211yescharacteristic
45C23×C4×C163C2211yescharacteristic
46#121×C163C2218yescomplemented
47C2×C4⋊C4×C163C2211yescharacteristic
48C2×C22⋊C4×C163C2212yes—
49C23×D163C2311yescharacteristic
50#14579698—232no—
51C2×Dic326—232no—
52C2×Dic326—28no—
53C2×Dic326—24no—
54C2×Dic326D418yescomplemented
55C2×Dic326C2318yes—
56C2×Dic326C2314yes—
57C2×Dic326C2314yes—
58C2×C22⋊D163—28no—
59C2×C22⋊D163—24no—
60C2×C22⋊D163C2318yes—
61C2×C22⋊D163C2314yes—
62C22×Dic163—28no—
63C22×Dic163—28no—
64C22×Dic163C2311yescharacteristic
65C22×Dic163C2312yes—
66C22×Dic163C2318yes—
67C22×Dic163C2318yes—
68C22×Dic163C2311yescharacteristic
69C22×Dic163C2311yescharacteristic
70C22×Dic163C2312yes—
71C2×C4×D163—24no—
72C2×C4×D163C2314yes—
73C4⋊Dic163C2318yes—
74Dic163⋊C4C2314yes—
75Dic163⋊C4C2318yes—
76C22⋊Dic163C2314yes—
77C22⋊Dic163C2318yes—
78C22⋊Dic163C2318yes—
79(C2×D163)⋊C4C2314yes—
80(C2×D163)⋊C4C2318yes—
81C4×Dic163C2314yes—
82C4×Dic163C2318yes—
83C24×C163C2312yes—
84C24×C163C2311yescharacteristic
85C2×D4×C163—28no—
86C2×D4×C163—24no—
87C2×D4×C163C2314yes—
88C2×D4×C163C2318yes—
89C22×C4×C163—24no—
90C22×C4×C163C2311yescharacteristic
91C22×C4×C163C2314yes—
92C22×C4×C163C2312yes—
93C22×C4×C163C2311yescharacteristic
94C4⋊C4×C163C2314yes—
95C22⋊C4×C163C2318yes—
96C22×D163—22no—
97C22×D163—22no—
98C22×D163—22no—
99C22×D163C2412yes—
100Dic326—28no—
101Dic326—216no—
102Dic326C2×D4116yescomplemented
103C22⋊D163—28no—
104C22⋊D163—216no—
105C2×Dic163—22no—
106C2×Dic163—24no—
107C2×Dic163—22no—
108C2×Dic163—22no—
109C2×Dic163—216no—
110C2×Dic163—216no—
111C2×Dic163C2×D418yes—
112C2×Dic163C2412yes—
113C2×Dic163C2412yes—
114C2×Dic163C2414yes—
115C2×Dic163C2412yes—
116C2×Dic163C2414yes—
117C4×D163—28no—
118C23×C163—22no—
119C23×C163—22no—
120C23×C163—24no—
121C23×C163—24no—
122C23×C163—24no—
123C23×C163#4114yescomplemented
124C23×C163C2414yes—
125C23×C163C2412yes—
126C23×C163C2411yescharacteristic
127D4×C163—216no—
128D4×C163—28no—
129C2×C4×C163—22no—
130C2×C4×C163—24no—
131C2×C4×C163—28no—
132C2×C4×C163C2×D414yes—
133C2×C4×C163C2412yes—
134C2×C4×C163C2414yes—
135C2×C4×C163C2412yes—
136C22×C163—28no—
137C22×C163—22no—
138C22×C163—22no—
139C22×C163—24no—
140C22×C163—24no—
141C22×C163—24no—
142C22×C163—48no—
143C22×C163C22×D412yes—
144C22×C163C2×#4112yes—
145C22×C163C2×#4114yescomplemented
146C22×C1632−1+412yes—
147C22×C163C2511yescharacteristic
148C2×D163—22no—
149C2×D163—22no—
150C2×D163—22no—
151C2×D163—44no—
152C4×C163—22no—
153C4×C163—24no—
154C4×C163C22×D414yes—
155Dic163—22no—
156Dic163—24no—
157Dic163—22no—
158Dic163—22no—
159Dic163C22×D418yes—
160C2×C163—24no—
161C2×C163—44no—
162C2×C163#54814yescomplemented
163C2×C163C23×D411yescharacteristic
164C2×C163C22×#4111yescharacteristic
165C2×C163C2×2−1+411yescharacteristic
166D163—42no—
167C163C2×#54811yescharacteristic, complemented, Sylow
168C2×#548—1631nomaximal, Sylow
169C22×#41—1631no—
170C23×Q8—1631no—
171#548—16316no—
172C2×#124—1631no—
173C2×#124—1632no—
174C2×C22⋊Q8—1631no—
175C2×C22⋊Q8—1632no—
176C2×#121—1632no—
177C2×#121D16311yescharacteristic, complemented
178C2×C4×Q8—1631no—
179C2×C4×D4—1632no—
180C2×C4×D4—1631no—
181C2×#41—3268no—
182C2×#41—1638no—
183C22×Q8—3268no—
184C22×Q8—1631no—
185C22×Q8—1632no—
186C22×Q8—1638no—
187C22×Q8—1631no—
188C22×D4—1631no—
189C22×D4—1632no—
190C22×D4C2×D16312yes—
191C22×D4C2×D16311yescharacteristic
192C23×C4—1632no—
193C23×C4—1632no—
194C23×C4—1631no—
195C23×C4C2×D16311yescharacteristic
196#124—1638no—
197#124—16316no—
198C22⋊Q8—1638no—
199C22⋊Q8—16316no—
200#121—16316no—
201#121C2×D16318yescomplemented
202C4×Q8—1638no—
203C4×D4—16316no—
204C4×D4—1638no—
205C2×C4⋊C4—1631no—
206C2×C4⋊C4—1632no—
207C2×C4⋊C4—1632no—
208C2×C4⋊C4C2×D16311yescharacteristic
209C2×C22⋊C4—1631no—
210C2×C22⋊C4—1632no—
211C2×C22⋊C4—1632no—
212C2×C22⋊C4—1631no—
213C2×C22⋊C4—1632no—
214C2×C22⋊C4C2×D16312yes—
215C2×C42—1631no—
216C2×C42—1632no—
217C24—1631no—
218C24C22×D16312yes—
219C24C22×D16311yescharacteristic
220#41—32632no—
221C2×Q8—32632no—
222C2×Q8—3268no—
223C2×Q8—3264no—
224C2×Q8—1638no—
225C2×Q8—1634no—
226C2×Q8—1638no—
227C2×Q8—1634no—
228C2×D4—28no—
229C2×D4—24no—
230C2×D4—3264no—
231C2×D4—3268no—
232C2×D4—1634no—
233C2×D4—1638no—
234C2×D4C22×D16318yes—
235C2×D4C22×D16314yes—
236C22×C4—24no—
237C22×C4—3268no—
238C22×C4—3268no—
239C22×C4—3264no—
240C22×C4—1638no—
241C22×C4—1638no—
242C22×C4—1631no—
243C22×C4—1631no—
244C22×C4—1632no—
245C22×C4—1631no—
246C22×C4—1634no—
247C22×C4—1632no—
248C22×C4C22×D16314yes—
249C22×C4C22×D16312yes—
250C22×C4C22×D16311yescharacteristic
251C22×C4C22×D16311yescharacteristic
252C4⋊C4—1634no—
253C4⋊C4—1638no—
254C4⋊C4—1638no—
255C4⋊C4C22×D16314yes—
256C22⋊C4—1634no—
257C22⋊C4—1638no—
258C22⋊C4—1638no—
259C22⋊C4—1634no—
260C22⋊C4—1638no—
261C22⋊C4C22×D16318yes—
262C42—1634no—
263C42—1638no—
264C23—22no—
265C23—22no—
266C23—24no—
267C23—24no—
268C23—24no—
269C23—3262no—
270C23—3262no—
271C23—3262no—
272C23—1632no—
273C23#1457969814yescomplemented
274C23C23×D16314yes—
275C23C23×D16312yes—
276C23C23×D16311yescharacteristic
277Q8—32616no—
278Q8—3268no—
279Q8—16316no—
280D4—216no—
281D4—28no—
282D4—3268no—
283D4—32616no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—28no—
287C2×C4—3262no—
288C2×C4—3264no—
289C2×C4—3262no—
290C2×C4—3262no—
291C2×C4—32616no—
292C2×C4—32616no—
293C2×C4—3268no—
294C2×C4—1632no—
295C2×C4—1632no—
296C2×C4—1634no—
297C2×C4—1632no—
298C2×C4—1638no—
299C2×C4—1634no—
300C2×C4D4×D16314yes—
301C2×C4C23×D16314yes—
302C2×C4C23×D16312yes—
303C2×C4C23×D16312yes—
304C22—22no—
305C22—22no—
306C22—28no—
307C22—24no—
308C22—24no—
309C22—24no—
310C22—48no—
311C22—3262no—
312C22—3262no—
313C22—3262no—
314C22—6524no—
315C22C2×D4×D16312yes—
316C22C2×#1457969812yes—
317C22C2×#1457969814yescomplemented
318C22#1910478212yes—
319C22C24×D16311yescharacteristic
320C4—22no—
321C4—24no—
322C4—3262no—
323C4—3264no—
324C4—3262no—
325C4—3262no—
326C4—1638no—
327C4C2×D4×D16314yes—
328C2—24no—
329C2—44no—
330C2—6522no—
331C2#2378232114yesdirect product factor
332C2C22×D4×D16311yescharacteristic
333C2C22×#1457969811yescharacteristic
334C2C2×#1910478211yescharacteristic
subgroupG/H
centerC23C23×D163
commutator, cosocleC22×C163C25
FrattiniC22C24×D163
FittingC2×#121×C163C2
socleC23×C163C24
hypercenterC2×#121D163
hypocenterC163C2×#548
LCSG ▷ C22×C163 ▷ C163
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C22×#14579698C2semidirect
2. C23×Dic326C2semidirect
3. #23782321C2direct (N × H)
4. C2×#19104722C2semidirect
5. C2×#19104721C2semidirect
6. C2×#19104715C2semidirect
7. C2×#19104712C2semidirect
8. C2×#19104707C2semidirect
9. C2×#19104700C2semidirect
10. C2×D4×Dic163C2semidirect
11. C2×C4×C22⋊D163C2semidirect
12. C2×#121×C163C2semidirect
14. C2×#14579698C22semidirect
19. C22×Dic326C22semidirect
25. #19104722C22semidirect
26. #19104721C22semidirect
27. #19104715C22semidirect
28. #19104712C22semidirect
29. #19104707C22semidirect
30. #19104700C22semidirect
31. D4×Dic163C22semidirect
32. C4×C22⋊D163C22semidirect
46. #121×C163C22semidirect
54. C2×Dic326D4semidirect
102. Dic326C2×D4semidirect
123. C23×C163#41semidirect
145. C22×C163C2×#41semidirect
162. C2×C163#548semidirect
167. C163C2×#548semidirect
177. C2×#121D163semidirect
201. #121C2×D163semidirect
273. C23#14579698semidirect
317. C22C2×#14579698semidirect
331. C2#23782321direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145796982
C23×Dic3262
#237823212
C2×#191047222
C2×#191047212
C2×#191047152
C2×#191047122
C2×#191047072
C2×#191047002
C2×D4×Dic1632
C2×C4×C22⋊D1632
C2×#121×C1632
C2×#548163