Finite Group Databasebeta

← D3×#562 #87107

Subgroups — D3×#562 #87107

Order 384.

1152 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#562C211yesmaximal, characteristic, complemented
2#5700C211yesmaximal, characteristic, complemented
3D3×#125C211yesmaximal, characteristic, complemented
4#5684C212yesmaximal, complemented
5D3×#124C212yesmaximal, complemented
6#5667C214yesmaximal, complemented
7D3×#123C214yesmaximal, complemented
8#5647C214yesmaximal, complemented
9D3×C22⋊Q8C214yesmaximal, complemented
10#5621C214yesmaximal, complemented
11D3×#121C214yesmaximal, complemented
12C2×#562—31nomaximal, Sylow
13C2×D3×Q8C2212yes—
14C2×D3×D4C2214yes—
15C2×D3×D4C2212yes—
16C22×C4×D3C2214yes—
17C3×#125C2211yescharacteristic, complemented
18C3×#124C2212yescomplemented
19C3×#123C2214yescomplemented
20C3×C22⋊Q8C2214yescomplemented
21C3×#121C2214yescomplemented
22#947C2212yescomplemented
23#946C2214yescomplemented
24#939C2214yescomplemented
25#939C2214yescomplemented
26#938C2214yescomplemented
27#936C2212yescomplemented
28#935C2214yescomplemented
29#930C2214yescomplemented
30#929C2214yescomplemented
31#924C2214yescomplemented
32#897C2214yescomplemented
33#896C2214yescomplemented
34#896C2214yescomplemented
35#895C2214yescomplemented
36#894C2214yescomplemented
37#894C2214yescomplemented
38D3×C4⋊C4C2212yes—
39D3×C4⋊C4C2214yes—
40D3×C4⋊C4C2214yes—
41#890C2212yescomplemented
42#889C2214yescomplemented
43#886C2214yescomplemented
44#885C2214yescomplemented
45#885C2214yescomplemented
46#884C2214yescomplemented
47#884C2214yescomplemented
48#883C2214yescomplemented
49#883C2214yescomplemented
50D3×C22⋊C4C2214yes—
51D3×C22⋊C4C2214yes—
52D3×C22⋊C4C2214yes—
53#878C2214yescomplemented
54#878C2214yescomplemented
55#875C2212yescomplemented
56C42×D3C2211yescharacteristic
57#870C2211yescharacteristic, complemented
58#562—31no—
59#562—32no—
60#562—34no—
61#562—34no—
62#562—34no—
63#562D311yescharacteristic, direct product factor
64C2×#125—31no—
65C2×#124—32no—
66C2×#123—34no—
67C2×C22⋊Q8—34no—
68C2×#121—34no—
69C23×D3C2314yes—
70C2×C3×Q8C2312yes—
71C2×C3×D4C2314yes—
72C2×C3×D4C2312yes—
73C22×C3×C4C2314yes—
74C2×#67C2314yes—
75C2×#67C2314yes—
76C2×#67C2314yes—
77C22×Dic3C2314yes—
78D3×Q8—44no—
79D3×D4—48no—
80D3×D4—44no—
81C2×D12C2312yes—
82C2×D12C2314yes—
83C2×C4×D3—48no—
84C2×C4×D3C2311yescharacteristic
85C2×C4×D3C2312yes—
86C2×C4×D3C2314yes—
87C2×C4×D3C2314yes—
88C2×C4×D3C2314yes—
89C2×C4×D3C2314yes—
90C2×Dic6C2312yes—
91C2×Dic6C2314yes—
92C3×C4⋊C4C2314yes—
93C3×C4⋊C4C2312yes—
94C3×C4⋊C4C2314yes—
95C3×C22⋊C4C2314yes—
96C3×C22⋊C4C2314yes—
97C3×C22⋊C4C2314yes—
98C3×C42C2311yescharacteristic
99#217C2314yes—
100#217C2314yes—
101#217C2314yes—
102(C2×D3)⋊C4C2314yes—
103(C2×D3)⋊C4C2314yes—
104(C2×D3)⋊C4C2314yes—
105(C2×D3)⋊C4C2314yes—
106(C2×D3)⋊C4C2314yes—
107(C2×D3)⋊C4C2314yes—
108C4⋊Dic3C2312yes—
109C4⋊Dic3C2314yes—
110C4⋊Dic3C2314yes—
111Dic3⋊C4C2314yes—
112Dic3⋊C4C2314yes—
113Dic3⋊C4C2314yes—
114Dic3⋊C4C2314yes—
115Dic3⋊C4C2314yes—
116C4×Dic3C2311yescharacteristic
117C4×Dic3C2312yes—
118C22×Q8—32no—
119C22×D4—34no—
120C22×D4—32no—
121C23×C4—34no—
122#125—34no—
123#125—32no—
124#125—31no—
125#125C2×D311yescharacteristic, complemented
126#124—32no—
127#124—32no—
128#124—34no—
129#124—34no—
130#124—32no—
131#124C2×D312yescomplemented
132#123—34no—
133#123—34no—
134#123—34no—
135#123—34no—
136#123—34no—
137#123—34no—
138#123—34no—
139#123C2×D314yescomplemented
140C22⋊Q8—34no—
141C22⋊Q8—34no—
142C22⋊Q8—34no—
143C22⋊Q8—34no—
144C22⋊Q8—34no—
145C22⋊Q8—34no—
146C22⋊Q8—34no—
147C22⋊Q8C2×D314yescomplemented
148#121—34no—
149#121—34no—
150#121—34no—
151#121—34no—
152#121—34no—
153#121—34no—
154#121—34no—
155#121C2×D314yescomplemented
156C2×C4⋊C4—34no—
157C2×C4⋊C4—34no—
158C2×C4⋊C4—32no—
159C2×C22⋊C4—34no—
160C2×C22⋊C4—34no—
161C2×C22⋊C4—34no—
162C2×C42—31no—
163C23×C3C2414yes—
164C22×D3—28no—
165C22×D3—28no—
166C22×D3—28no—
167C22×D3C2414yes—
168C22×D3C2411yescharacteristic
169C3×Q8—42no—
170C3×D4—42no—
171C3×D4—44no—
172C2×C3×C4—44no—
173C2×C3×C4C2412yes—
174C2×C3×C4C2411yescharacteristic
175C2×C3×C4C2414yes—
176C2×C3×C4C2414yes—
177#67—44no—
178#67—44no—
179#67—44no—
180C2×Dic3—44no—
181C2×Dic3C2411yescharacteristic
182C2×Dic3C2412yes—
183C2×Dic3C2414yes—
184C2×Dic3C2414yes—
185D12—42no—
186D12—44no—
187C4×D3—28no—
188C4×D3—28no—
189C4×D3—24no—
190C4×D3—22no—
191C4×D3—44no—
192C4×D3—44no—
193Dic6—42no—
194Dic6—44no—
195C24—34no—
196C2×Q8—34no—
197C2×Q8—32no—
198C2×Q8—124no—
199C2×Q8C22×D312yes—
200C2×D4—34no—
201C2×D4—34no—
202C2×D4—34no—
203C2×D4—34no—
204C2×D4—32no—
205C2×D4—128no—
206C2×D4—124no—
207C2×D4C22×D314yes—
208C2×D4C22×D312yes—
209C22×C4—34no—
210C22×C4—31no—
211C22×C4—32no—
212C22×C4—34no—
213C22×C4—34no—
214C22×C4—34no—
215C22×C4—34no—
216C22×C4—128no—
217C22×C4C22×D314yes—
218C4⋊C4—34no—
219C4⋊C4—34no—
220C4⋊C4—34no—
221C4⋊C4—34no—
222C4⋊C4—34no—
223C4⋊C4—34no—
224C4⋊C4—34no—
225C4⋊C4—32no—
226C4⋊C4C22×D314yes—
227C4⋊C4C22×D314yes—
228C4⋊C4C22×D312yes—
229C22⋊C4—34no—
230C22⋊C4—34no—
231C22⋊C4—34no—
232C22⋊C4—34no—
233C22⋊C4—34no—
234C22⋊C4—34no—
235C22⋊C4—34no—
236C22⋊C4—34no—
237C22⋊C4—34no—
238C22⋊C4C22×D314yes—
239C22⋊C4C22×D314yes—
240C22⋊C4C22×D314yes—
241C42—32no—
242C42—31no—
243C42C22×D311yescharacteristic
244C22×C3—24no—
245C22×C3—24no—
246C22×C3—24no—
247C22×C3C2511yescharacteristic
248C2×D3—24no—
249C2×D3—24no—
250C2×D3—24no—
251C2×D3—416no—
252C2×D3H(5,𝔽2)14yes—
253C2×D32−1+412yes—
254C3×C4—24no—
255C3×C4—22no—
256C3×C4—24no—
257C3×C4—21no—
258Dic3—24no—
259Dic3—22no—
260Dic3—24no—
261Dic3—21no—
262C23—31no—
263C23—34no—
264C23—68no—
265C23—68no—
266C23—68no—
267C23C23×D314yes—
268Q8—42no—
269Q8—124no—
270Q8—122no—
271D4—44no—
272D4—42no—
273D4—124no—
274D4—122no—
275D4—124no—
276D4—124no—
277D4—124no—
278C2×C4—31no—
279C2×C4—32no—
280C2×C4—34no—
281C2×C4—34no—
282C2×C4—44no—
283C2×C4—68no—
284C2×C4—68no—
285C2×C4—64no—
286C2×C4—62no—
287C2×C4—124no—
288C2×C4—124no—
289C2×C4—124no—
290C2×C4C23×D311yescharacteristic
291C2×C4C23×D312yes—
292C2×C4C23×D314yes—
293C2×C4C23×D314yes—
294C2×C3—44no—
295C2×C3C2×H(5,𝔽2)12yes—
296C2×C3C2×2−1+411yescharacteristic
297D3—44no—
298D3#56214yesdirect product factor
299C22—24no—
300C22—24no—
301C22—24no—
302C22—34no—
303C22—32no—
304C22—64no—
305C22—64no—
306C22—64no—
307C22—1216no—
308C22C24×D311yescharacteristic
309C4—24no—
310C4—24no—
311C4—22no—
312C4—21no—
313C4—64no—
314C4—64no—
315C4—62no—
316C4—61no—
317C3C2×#56211yescharacteristic, complemented, Sylow
318C2—34no—
319C2—44no—
320C2—124no—
321C2D3×H(5,𝔽2)12yes—
322C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#562C2
hypercenter#562D3
hypocenterC3C2×#562
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. C3×#562C2semidirect
2. #5700C2semidirect
3. D3×#125C2semidirect
4. #5684C2semidirect
5. D3×#124C2semidirect
6. #5667C2semidirect
7. D3×#123C2semidirect
8. #5647C2semidirect
9. D3×C22⋊Q8C2semidirect
10. #5621C2semidirect
11. D3×#121C2semidirect
17. C3×#125C22semidirect
18. C3×#124C22semidirect
19. C3×#123C22semidirect
20. C3×C22⋊Q8C22semidirect
21. C3×#121C22semidirect
22. #947C22semidirect
23. #946C22semidirect
24. #939C22semidirect
25. #939C22semidirect
26. #938C22semidirect
27. #936C22semidirect
28. #935C22semidirect
29. #930C22semidirect
30. #929C22semidirect
31. #924C22semidirect
32. #897C22semidirect
33. #896C22semidirect
34. #896C22semidirect
35. #895C22semidirect
36. #894C22semidirect
37. #894C22semidirect
41. #890C22semidirect
42. #889C22semidirect
43. #886C22semidirect
44. #885C22semidirect
45. #885C22semidirect
46. #884C22semidirect
47. #884C22semidirect
48. #883C22semidirect
49. #883C22semidirect
53. #878C22semidirect
54. #878C22semidirect
55. #875C22semidirect
57. #870C22semidirect
63. #562D3direct (N × H)
125. #125C2×D3semidirect
131. #124C2×D3semidirect
139. #123C2×D3semidirect
147. C22⋊Q8C2×D3semidirect
155. #121C2×D3semidirect
298. D3#562direct (N × H)
317. C3C2×#562semidirect

Maximal subgroups

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

subgroupindex
C3×#5622
#57002
D3×#1252
#56842
D3×#1242
#56672
D3×#1232
#56472
D3×C22⋊Q82
#56212
D3×#1212
C2×#5623