Finite Group Databasebeta

← #86257

Subgroups — #86257

Order 384.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5728C212yesmaximal, complemented
3#5725C212yesmaximal, complemented
4#5691C214yesmaximal, complemented
5#5627C214yesmaximal, complemented
6#5624C214yesmaximal, complemented
7#5580C214yesmaximal, complemented
8C4×#239C214yesmaximal, complemented
9#5524C212yesmaximal, complemented
10#5522C212yesmaximal, complemented
11#5477C211yesmaximal, characteristic, complemented
12C2×C42×D3C211yesmaximal, characteristic, complemented
13#3444—31nomaximal, Sylow
14C2×#239C2214yes—
15C2×#235C2212yes—
16C22×C4×D3C2211yescharacteristic
17C22×C4×D3C2212yes—
18C3×C4⋊Q8C2212yescomplemented
19C3×#125C2212yescomplemented
20C3×C22⋊Q8C2214yes—
21C3×C4×Q8C2214yescomplemented
22C3×#117C2212yes—
23C2×C3×C42C2211yescharacteristic
24#947C2214yescomplemented
25#946C2214yes—
26Q8×Dic3C2214yescomplemented
27#930C2214yes—
28C2×C4×Dic3C2211yescharacteristic
29C2×C4×Dic3C2212yes—
30#898C2218yescomplemented
31#897C2214yes—
32#895C2214yes—
33#895C2214yes—
34#894C2218yes—
35#893C2218yes—
36#893C2214yescomplemented
37#893C2214yescomplemented
38#892C2214yes—
39#892C2218yescomplemented
40#890C2214yescomplemented
41#887C2214yescomplemented
42#886C2214yes—
43#882C2214yes—
44#67⋊C4C2218yes—
45#875C2212yescomplemented
46#874C2211yescharacteristic, complemented
47C4×D12C2218yes—
48C4×D12C2214yescomplemented
49#872C2214yescomplemented
50C42×D3C2212yes—
51C42×D3C2212yes—
52C42×D3C2212yes—
53C42×D3C2212yes—
54C42×D3C2214yescomplemented
55#869C2211yescharacteristic, complemented
56#533—32no—
57#533D311yescharacteristic, complemented
58#532—32no—
59#532—31no—
60#529—34no—
61#529—34no—
62#529—34no—
63#525—34no—
64#525—32no—
65C4×#41—34no—
66C4×#41—32no—
67C22×C42—31no—
68C23×D3C2311yescharacteristic
69C2×C3×Q8C2314yes—
70C22×C3×C4C2312yes—
71C22×C3×C4C2311yescharacteristic
72C2×#67C2314yes—
73C22×Dic3C2312yes—
74C22×Dic3C2311yescharacteristic
75#239—216no—
76#235—28no—
77C2×D12C2318yes—
78C2×D12C2314yes—
79C2×D12C2312yes—
80C2×C4×D3—24no—
81C2×C4×D3—28no—
82C2×C4×D3C2312yes—
83C2×C4×D3C2312yes—
84C2×C4×D3C2312yes—
85C2×C4×D3C2312yes—
86C2×C4×D3C2312yes—
87C2×C4×D3C2314yes—
88C2×C4×D3C2318yes—
89C2×C4×D3C2312yes—
90C2×Dic6C2312yes—
91C3×C4⋊C4C2314yes—
92C3×C4⋊C4C2314yes—
93C3×C4⋊C4C2318yes—
94C3×C22⋊C4C2314yes—
95C3×C42C2311yescharacteristic
96C3×C42C2312yes—
97C3×C42C2311yescharacteristic
98C3×C42C2314yes—
99(C2×D3)⋊C4C2318yes—
100(C2×D3)⋊C4C2318yes—
101(C2×D3)⋊C4C2318yes—
102(C2×D3)⋊C4C2318yes—
103C4⋊Dic3C2314yes—
104C4⋊Dic3C2318yes—
105Dic3⋊C4C2318yes—
106C4×Dic3C2311yescharacteristic
107C4×Dic3C2311yescharacteristic
108C4×Dic3C2312yes—
109C4×Dic3C2312yes—
110C4×Dic3C2312yes—
111C4×Dic3C2312yes—
112C4×Dic3C2312yes—
113C4×Dic3C2318yes—
114C2×#41—34no—
115C2×#41—32no—
116C23×C4—32no—
117C23×C4—31no—
118C4⋊Q8—31no—
119C4⋊Q8C2×D312yescomplemented
120#127—31no—
121#126—38no—
122#125—34no—
123#125C2×D312yescomplemented
124#124—34no—
125#124—32no—
126#123—38no—
127#123—34no—
128C22⋊Q8—34no—
129C22⋊Q8—34no—
130C22⋊Q8C2×D314yes—
131#121—34no—
132#121—34no—
133#121—34no—
134#120—34no—
135C4×Q8—34no—
136C4×Q8—34no—
137C4×Q8C2×D314yescomplemented
138C4×D4—38no—
139C4×D4—34no—
140C4×D4—34no—
141C4×D4—38no—
142C4×D4—38no—
143C4×D4—34no—
144#117—34no—
145#117—38no—
146#117—34no—
147#117C2×D312yes—
148C2×C42—31no—
149C2×C42—32no—
150C2×C42—34no—
151C2×C42—32no—
152C2×C42—32no—
153C2×C42—32no—
154C2×C42—32no—
155C2×C42C2×D311yescharacteristic
156C23×C3C2411yescharacteristic
157C22×D3—22no—
158C22×D3—24no—
159C22×D3C2414yes—
160C22×D3C2412yes—
161C3×Q8—28no—
162C2×C3×C4—24no—
163C2×C3×C4#4114yescomplemented
164C2×C3×C4C2412yes—
165C2×C3×C4C2418yes—
166C2×C3×C4C2412yes—
167C2×C3×C4C2411yescharacteristic
168C2×C3×C4C2411yescharacteristic
169#67—28no—
170C2×Dic3—22no—
171C2×Dic3—24no—
172C2×Dic3C2412yes—
173C2×Dic3C2412yes—
174C2×Dic3C2414yes—
175C2×Dic3C2412yes—
176D12—216no—
177D12—24no—
178D12—28no—
179C4×D3—24no—
180C4×D3—24no—
181C4×D3—24no—
182C4×D3—24no—
183C4×D3—28no—
184C4×D3—24no—
185C4×D3—216no—
186C4×D3#4118yescomplemented
187Dic6—24no—
188C24—31no—
189#41—616no—
190#41—68no—
191C2×Q8—32no—
192C2×Q8C22×D314yes—
193C2×D4—34no—
194C2×D4—38no—
195C2×D4—32no—
196C2×D4—34no—
197C22×C4—32no—
198C22×C4—32no—
199C22×C4—31no—
200C22×C4—32no—
201C22×C4—32no—
202C22×C4—32no—
203C22×C4—32no—
204C22×C4—38no—
205C22×C4—32no—
206C22×C4—34no—
207C22×C4—68no—
208C22×C4—64no—
209C22×C4C22×D312yes—
210C22×C4C22×D311yescharacteristic
211C4⋊C4—34no—
212C4⋊C4—38no—
213C4⋊C4—38no—
214C4⋊C4C22×D314yes—
215C4⋊C4C22×D318yes—
216C4⋊C4C22×D314yes—
217C22⋊C4—38no—
218C22⋊C4—38no—
219C22⋊C4—38no—
220C22⋊C4—38no—
221C22⋊C4C22×D314yes—
222C42—32no—
223C42—32no—
224C42—32no—
225C42—32no—
226C42—32no—
227C42—38no—
228C42—31no—
229C42—31no—
230C42C22×D311yescharacteristic
231C42C22×D311yescharacteristic
232C42C22×D312yes—
233C42C22×D314yes—
234C22×C3—22no—
235C22×C3C2×#4112yes—
236C22×C3C2511yescharacteristic
237C2×D3—28no—
238C2×D3—22no—
239C2×D3—22no—
240C2×D3—28no—
241C2×D3—24no—
242C2×D3C2×#4114yes—
243C3×C4—22no—
244C3×C4—28no—
245C3×C4C2×#4112yes—
246C3×C4C2×#4112yes—
247C3×C4C2×#4114yes—
248Dic3—24no—
249Dic3—22no—
250Dic3—22no—
251Dic3C2×#4114yes—
252C23—32no—
253C23—34no—
254C23—64no—
255C23—62no—
256C23C23×D311yescharacteristic
257Q8—28no—
258Q8—64no—
259D4—616no—
260D4—68no—
261D4—64no—
262D4—68no—
263C2×C4—24no—
264C2×C4—32no—
265C2×C4—38no—
266C2×C4—32no—
267C2×C4—32no—
268C2×C4—34no—
269C2×C4—64no—
270C2×C4—616no—
271C2×C4—64no—
272C2×C4—64no—
273C2×C4—62no—
274C2×C4—64no—
275C2×C4—64no—
276C2×C4—68no—
277C2×C4—64no—
278C2×C4#23914yescomplemented
279C2×C4C23×D312yes—
280C2×C4C23×D318yes—
281C2×C4C23×D311yescharacteristic
282C2×C4C23×D312yes—
283C2×C4C23×D311yescharacteristic
284C2×C3—22no—
285C2×C3C22×#4111yescharacteristic
286C2×C3C22×#4112yes—
287D3—24no—
288D3—44no—
289C22—22no—
290C22—34no—
291C22—68no—
292C22—62no—
293C22—62no—
294C22—64no—
295C22—68no—
296C22C2×#23912yes—
297C22C24×D311yescharacteristic
298C4—22no—
299C4—28no—
300C4—34no—
301C4—62no—
302C4—64no—
303C4—62no—
304C4C2×#23912yes—
305C4C2×#23912yes—
306C4D3×#4114yes—
307C3#344411yescharacteristic, complemented, Sylow
308C2—22no—
309C2—64no—
310C2—124no—
311C2C22×#23911yescharacteristic
312C2C2×D3×#4112yes—
subgroupG/H
centerC2×C4C23×D3
commutator, socle, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC3×#533C2
hypercenter#533D3
hypocenterC3#3444
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5728C2semidirect
3. #5725C2semidirect
4. #5691C2semidirect
5. #5627C2semidirect
6. #5624C2semidirect
7. #5580C2semidirect
8. C4×#239C2semidirect
9. #5524C2semidirect
10. #5522C2semidirect
11. #5477C2semidirect
12. C2×C42×D3C2semidirect
18. C3×C4⋊Q8C22semidirect
19. C3×#125C22semidirect
21. C3×C4×Q8C22semidirect
24. #947C22semidirect
26. Q8×Dic3C22semidirect
30. #898C22semidirect
36. #893C22semidirect
37. #893C22semidirect
39. #892C22semidirect
40. #890C22semidirect
41. #887C22semidirect
45. #875C22semidirect
46. #874C22semidirect
48. C4×D12C22semidirect
49. #872C22semidirect
54. C42×D3C22semidirect
55. #869C22semidirect
57. #533D3semidirect
119. C4⋊Q8C2×D3semidirect
123. #125C2×D3semidirect
137. C4×Q8C2×D3semidirect
163. C2×C3×C4#41semidirect
186. C4×D3#41semidirect
278. C2×C4#239semidirect
307. C3#3444semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57282
#57252
#56912
#56272
#56242
#55802
C4×#2392
#55242
#55222
#54772
C2×C42×D32
#34443