Finite Group Databasebeta

← C2×#83120 #11001061

Subgroups — C2×#83120 #11001061

Order 768.

3354 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#126C211yesmaximal, characteristic, complemented
2C22×C42×D3C211yesmaximal, characteristic, complemented
3#83120C2116yesmaximal, direct product factor
4C2×#4989C213yesmaximal, complemented
5C2×#4978C213yesmaximal, complemented
6C2×#4956C213yesmaximal
7C2×#4944C213yesmaximal, complemented
8C2×#4937C211yesmaximal, characteristic, complemented
9C2×#2852—31nomaximal, Sylow
10C23×#67C2211yescharacteristic
11C23×C4×D3C2213yes—
12C2×C3×#126—28no—
13C2×C3×#126C2218yescomplemented
14C22×C3×C4⋊C4C2213yes—
15C22×C3×C22⋊C4C2213yes—
16C22×C3×C42C2211yescharacteristic
17C22×#217C2213yes—
18C22×(C2×D3)⋊C4C2213yes—
19C22×Dic3⋊C4C2213yes—
20C22×C4×Dic3C2213yes—
21C2×C42×D3—28no—
22C2×C42×D3C2218yescomplemented
23C2×C3⋊#95C2213yes—
24C2×C3⋊#95C2213yes—
25C2×C3⋊#95C2213yes—
26C2×C3⋊#95C2213yes—
27#4989C22124yescomplemented
28#4978C22124yescomplemented
29#4956C22124yes—
30#4944C22124yescomplemented
31#4937C2218yescomplemented
32C22×#126D311yescharacteristic, complemented
33C23×C42—31no—
34#2852—316no—
35C2×#388—33no—
36C2×#387—33no—
37C2×#386—33no—
38C2×#385—33no—
39C2×#383—31no—
40C25×C3C2311yescharacteristic
41C24×D3C2311yescharacteristic
42C23×C3×C4C2313yes—
43C23×C3×C4C2313yes—
44C22×#67—212no—
45C22×#67C2314yes—
46C23×Dic3C2311yescharacteristic
47C23×Dic3C2313yes—
48C23×Dic3C2313yes—
49C22×C4×D3—212no—
50C22×C4×D3—212no—
51C22×C4×D3—212no—
52C22×C4×D3C23112yes—
53C3×#126—232no—
54C2×C3×C4⋊C4—212no—
55C2×C3×C4⋊C4C23112yes—
56C2×C3×C22⋊C4—212no—
57C2×C3×C22⋊C4C23112yes—
58C2×C3×C42D418yescomplemented
59C2×C3×C42C2314yes—
60C2×#217—212no—
61C2×#217C23112yes—
62C2×(C2×D3)⋊C4—212no—
63C2×(C2×D3)⋊C4C23112yes—
64C2×Dic3⋊C4—212no—
65C2×Dic3⋊C4C23112yes—
66C2×C4×Dic3—212no—
67C2×C4×Dic3C23112yes—
68C42×D3—232no—
69C3⋊#95C23112yes—
70C3⋊#95C23112yes—
71C3⋊#95C23112yes—
72C3⋊#95C23112yes—
73C23×D4—31no—
74C24×C4—33no—
75C2×#126—28no—
76C2×#126C2×D318yescomplemented
77C22×C4⋊C4—33no—
78C22×C4⋊C4C2×D313yes—
79C22×C22⋊C4—33no—
80C22×C22⋊C4—33no—
81C22×C22⋊C4C2×D313yes—
82C22×C42—33no—
83C22×C42—38no—
84C22×C42—68no—
85C22×C42C2×D311yescharacteristic
86#388—324no—
87#387—324no—
88#386—324no—
89#385—324no—
90#383—38no—
91C2×#95—33no—
92C2×#95—33no—
93C2×#95—33no—
94C2×#95—33no—
95C24×C3—22no—
96C24×C3—26no—
97C24×C3—26no—
98C24×C3C2412yes—
99C24×C3C2411yescharacteristic
100C23×D3—22no—
101C23×D3—26no—
102C23×D3#41112yes—
103C23×D3C2412yes—
104C22×C3×C4—26no—
105C22×C3×C4—26no—
106C22×C3×C4—26no—
107C22×C3×C4—26no—
108C22×C3×C4—26no—
109C22×C3×C4C2×D4112yes—
110C22×C3×C4C2416yes—
111C22×C3×C4C2416yes—
112C2×#67—224no—
113C2×#67—416no—
114C22×Dic3—26no—
115C22×Dic3—26no—
116C22×Dic3—26no—
117C22×Dic3—26no—
118C22×Dic3—26no—
119C22×Dic3—26no—
120C22×Dic3#41112yes—
121C22×Dic3C2412yes—
122C22×Dic3C2416yes—
123C22×Dic3C2416yes—
124C2×C4×D3—296no—
125C2×C4×D3—224no—
126C2×C4×D3—224no—
127C2×C4×D3—224no—
128C3×C4⋊C4—224no—
129C3×C22⋊C4—224no—
130C3×C42C2×D4116yescomplemented
131#217—224no—
132(C2×D3)⋊C4—224no—
133Dic3⋊C4—224no—
134C4×Dic3—224no—
135C25—31no—
136C25C22×D311yescharacteristic
137C22×D4—34no—
138C22×D4—612no—
139C23×C4—33no—
140C23×C4—33no—
141C23×C4—31no—
142C23×C4—312no—
143C23×C4—612no—
144C23×C4—612no—
145C23×C4—612no—
146C23×C4C22×D313yes—
147C23×C4C22×D313yes—
148#126—232no—
149C2×C4⋊C4—212no—
150C2×C4⋊C4—312no—
151C2×C4⋊C4—612no—
152C2×C4⋊C4C22×D3112yes—
153C2×C22⋊C4—212no—
154C2×C22⋊C4—312no—
155C2×C22⋊C4—312no—
156C2×C22⋊C4—612no—
157C2×C22⋊C4—612no—
158C2×C22⋊C4C22×D3112yes—
159C2×C42—312no—
160C2×C42—612no—
161C2×C42—632no—
162C2×C42#6718yescomplemented
163C2×C42C22×D314yes—
164#95—312no—
165#95—312no—
166#95—312no—
167#95—312no—
168C23×C3—22no—
169C23×C3—26no—
170C23×C3—26no—
171C23×C3—412no—
172C23×C3—44no—
173C23×C3—412no—
174C23×C3C22×D412yes—
175C23×C3C2×#4116yes—
176C23×C3C2×#4116yes—
177C23×C3C2511yescharacteristic
178C22×D3—224no—
179C22×D3—224no—
180C22×D3—22no—
181C22×D3—26no—
182C22×D3#126116yescomplemented
183C22×D3C2×#41112yes—
184C2×C3×C4—224no—
185C2×C3×C4—26no—
186C2×C3×C4—26no—
187C2×C3×C4—26no—
188C2×C3×C4—26no—
189C2×C3×C4—26no—
190C2×C3×C4—412no—
191C2×C3×C4C22×D4112yes—
192#67—416no—
193C2×Dic3—224no—
194C2×Dic3—26no—
195C2×Dic3—26no—
196C2×Dic3—26no—
197C2×Dic3—26no—
198C2×Dic3—26no—
199C2×Dic3—26no—
200C2×Dic3—44no—
201C2×Dic3—412no—
202C2×Dic3C2×#41112yes—
203C4×D3—296no—
204C24—26no—
205C24—26no—
206C24—22no—
207C24—312no—
208C24—32no—
209C24—66no—
210C24—62no—
211C24C23×D312yes—
212C24C23×D311yescharacteristic
213C2×D4—624no—
214C2×D4—1216no—
215C22×C4—26no—
216C22×C4—26no—
217C22×C4—26no—
218C22×C4—26no—
219C22×C4—26no—
220C22×C4—312no—
221C22×C4—36no—
222C22×C4—32no—
223C22×C4—36no—
224C22×C4—624no—
225C22×C4—624no—
226C22×C4—696no—
227C22×C4—66no—
228C22×C4—66no—
229C22×C4—624no—
230C22×C4—66no—
231C22×C4—66no—
232C22×C4—66no—
233C22×C4—66no—
234C22×C4C2×#67112yes—
235C22×C4C23×D316yes—
236C22×C4C23×D316yes—
237C4⋊C4—224no—
238C4⋊C4—624no—
239C22⋊C4—224no—
240C22⋊C4—624no—
241C22⋊C4—624no—
242C42—624no—
243C42C2×#67116yescomplemented
244C22×C3—46no—
245C22×C3—42no—
246C22×C3—46no—
247C22×C3—88no—
248C22×C3#525112yes—
249C22×C3C2×#12614yes—
250C22×C3#529112yes—
251C22×C3C23×D411yescharacteristic
252C22×C3C22×#4113yes—
253C22×C3C22×#4113yes—
254C2×D3—232no—
255C2×D3—212no—
256C2×D3—212no—
257C2×D3C2×#12618yescomplemented
258C3×C4—212no—
259C3×C4—46no—
260Dic3—212no—
261Dic3—42no—
262Dic3—46no—
263C23—26no—
264C23—26no—
265C23—22no—
266C23—316no—
267C23—312no—
268C23—44no—
269C23—412no—
270C23—412no—
271C23—66no—
272C23—624no—
273C23—62no—
274C23—624no—
275C23D3×#4116yes—
276C23D3×#4116yes—
277C23C22×#6712yes—
278C23C24×D311yescharacteristic
279D4—1216no—
280C2×C4—26no—
281C2×C4—26no—
282C2×C4—224no—
283C2×C4—26no—
284C2×C4—26no—
285C2×C4—26no—
286C2×C4—312no—
287C2×C4—412no—
288C2×C4—696no—
289C2×C4—66no—
290C2×C4—66no—
291C2×C4—624no—
292C2×C4—66no—
293C2×C4—66no—
294C2×C4—66no—
295C2×C4—66no—
296C2×C4—124no—
297C2×C4—1212no—
298C2×C4C22×#67112yes—
299C2×C3—82no—
300C2×C3#285218yescomplemented
301C2×C3C2×#52513yes—
302C2×C3C22×#12611yescharacteristic
303C2×C3C2×#52913yes—
304D3—28no—
305C22—38no—
306C22—46no—
307C22—42no—
308C22—46no—
309C22—612no—
310C22—632no—
311C22—612no—
312C22—88no—
313C22D3×#12614yes—
314C22#5706112yes—
315C22#5828112yes—
316C22C2×D3×#4113yes—
317C22C2×D3×#4113yes—
318C22C23×#6711yescharacteristic
319C4—212no—
320C4—46no—
321C4—612no—
322C4—126no—
323C4—122no—
324C3C2×#285211yescharacteristic, complemented, Sylow
325C2—68no—
326C2—82no—
327C2#8312018yesdirect product factor
328C2C2×#570613yes—
329C2C2×D3×#12611yescharacteristic
330C2C2×#582813yes—
subgroupG/H
centerC24C23×D3
commutator, cosocleC23×C3C25
FrattiniC23C24×D3
FittingC22×C3×#126C2
socleC24×C3C24
prime radicalC23×#67C22
hypercenterC22×#126D3
hypocenterC3C2×#2852
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#126

Complements

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

NH ≅ G/Ntype
1. C22×C3×#126C2semidirect
2. C22×C42×D3C2semidirect
3. #83120C2direct (N × H)
4. C2×#4989C2semidirect
5. C2×#4978C2semidirect
7. C2×#4944C2semidirect
8. C2×#4937C2semidirect
13. C2×C3×#126C22semidirect
22. C2×C42×D3C22semidirect
27. #4989C22semidirect
28. #4978C22semidirect
30. #4944C22semidirect
31. #4937C22semidirect
32. C22×#126D3semidirect
58. C2×C3×C42D4semidirect
76. C2×#126C2×D3semidirect
130. C3×C42C2×D4semidirect
162. C2×C42#67semidirect
182. C22×D3#126semidirect
243. C42C2×#67semidirect
257. C2×D3C2×#126semidirect
300. C2×C3#2852semidirect
324. C3C2×#2852semidirect
327. C2#83120direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1262
C22×C42×D32
#831202
C2×#49892
C2×#49782
C2×#49562
C2×#49442
C2×#49372
C2×#28523