Finite Group Databasebeta

← D3×#555 #86971

Subgroups — D3×#555 #86971

Order 384.

1240 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#555C211yesmaximal, characteristic, complemented
2#5734C211yesmaximal, characteristic, complemented
3D3×C4⋊Q8C211yesmaximal, characteristic, complemented
4D3×#124C214yesmaximal, complemented
5#5670C214yesmaximal, complemented
6#5616C214yesmaximal, complemented
7D3×#121C214yesmaximal, complemented
8#5563C212yesmaximal, complemented
9C4×D3×D4C212yesmaximal, complemented
10#5532C214yesmaximal, complemented
11D3×#117C214yesmaximal, complemented
12C2×#555—31nomaximal, Sylow
13C2×D3×Q8C2212yes—
14C2×D3×D4C2212yes—
15C2×D3×D4C2214yes—
16C22×C4×D3C2214yes—
17C3×C4⋊Q8C2211yescharacteristic, complemented
18C3×#124C2214yescomplemented
19C3×#121C2214yescomplemented
20C3×C4×D4C2212yescomplemented
21C3×#117C2214yescomplemented
22#947C2214yescomplemented
23#944C2212yescomplemented
24#939C2218yescomplemented
25#938C2214yescomplemented
26#936C2214yescomplemented
27D4×Dic3C2212yescomplemented
28#930C2214yescomplemented
29C4×#67C2214yescomplemented
30#926C2214yescomplemented
31#895C2214yescomplemented
32#893C2212yescomplemented
33#892C2214yescomplemented
34#892C2214yescomplemented
35D3×C4⋊C4C2214yes—
36D3×C4⋊C4C2212yes—
37#888C2214yescomplemented
38#885C2218yescomplemented
39#885C2218yescomplemented
40#884C2218yescomplemented
41#67⋊C4C2214yescomplemented
42D3×C22⋊C4C2214yes—
43D3×C22⋊C4C2218yes—
44#877C2218yescomplemented
45#875C2214yescomplemented
46C4×D12C2212yescomplemented
47#872C2218yescomplemented
48C42×D3C2214yes—
49C42×D3C2211yescharacteristic
50#869C2211yescharacteristic, complemented
51#555—31no—
52#555—34no—
53#555—34no—
54#555—32no—
55#555—34no—
56#555D311yescharacteristic, direct product factor
57C2×C4⋊Q8—31no—
58C2×#124—34no—
59C2×#121—34no—
60C2×C4×D4—32no—
61C2×#117—34no—
62C23×D3C2314yes—
63C2×C3×Q8C2312yes—
64C2×C3×D4C2312yes—
65C2×C3×D4C2314yes—
66C22×C3×C4C2314yes—
67C2×#67C2314yes—
68C2×#67C2318yes—
69C22×Dic3C2314yes—
70D3×Q8—44no—
71D3×D4—28no—
72D3×D4—48no—
73C2×D12C2312yes—
74C2×D12C2314yes—
75C2×C4×D3—216no—
76C2×C4×D3C2314yes—
77C2×C4×D3C2311yescharacteristic
78C2×C4×D3C2314yes—
79C2×C4×D3C2314yes—
80C2×C4×D3C2314yes—
81C2×C4×D3C2312yes—
82C2×Dic6C2312yes—
83C2×Dic6C2314yes—
84C3×C4⋊C4C2314yes—
85C3×C4⋊C4C2312yes—
86C3×C22⋊C4C2318yes—
87C3×C22⋊C4C2314yes—
88C3×C42C2311yescharacteristic
89C3×C42C2314yes—
90#217C2318yes—
91#217C2314yes—
92(C2×D3)⋊C4C2318yes—
93(C2×D3)⋊C4C2318yes—
94(C2×D3)⋊C4C2314yes—
95(C2×D3)⋊C4C2314yes—
96C4⋊Dic3C2314yes—
97C4⋊Dic3C2312yes—
98Dic3⋊C4C2314yes—
99Dic3⋊C4C2318yes—
100C4×Dic3C2318yes—
101C4×Dic3C2314yes—
102C4×Dic3C2311yescharacteristic
103C4×Dic3C2312yes—
104C22×Q8—32no—
105C22×D4—32no—
106C22×D4—34no—
107C23×C4—34no—
108C4⋊Q8—32no—
109C4⋊Q8—34no—
110C4⋊Q8—31no—
111C4⋊Q8C2×D311yescharacteristic, complemented
112#124—34no—
113#124—34no—
114#124—38no—
115#124—38no—
116#124—34no—
117#124C2×D314yescomplemented
118#121—34no—
119#121—38no—
120#121—34no—
121#121—34no—
122#121—38no—
123#121C2×D314yescomplemented
124C4×D4—32no—
125C4×D4—34no—
126C4×D4—32no—
127C4×D4—34no—
128C4×D4—32no—
129C4×D4C2×D312yescomplemented
130#117—34no—
131#117—34no—
132#117—34no—
133#117—38no—
134#117—38no—
135#117C2×D314yescomplemented
136C2×C4⋊C4—32no—
137C2×C4⋊C4—34no—
138C2×C22⋊C4—38no—
139C2×C22⋊C4—34no—
140C2×C42—34no—
141C2×C42—31no—
142C23×C3C2414yes—
143C22×D3—28no—
144C22×D3—28no—
145C22×D3—28no—
146C22×D3C2414yes—
147C22×D3C2411yescharacteristic
148C3×Q8—42no—
149C3×D4—24no—
150C3×D4—44no—
151C2×C3×C4—28no—
152C2×C3×C4C2414yes—
153C2×C3×C4C2414yes—
154C2×C3×C4C2411yescharacteristic
155C2×C3×C4C2412yes—
156#67—28no—
157#67—48no—
158C2×Dic3—28no—
159C2×Dic3C2414yes—
160C2×Dic3C2414yes—
161C2×Dic3C2411yescharacteristic
162C2×Dic3C2412yes—
163D12—24no—
164D12—44no—
165C4×D3—28no—
166C4×D3—28no—
167C4×D3—22no—
168C4×D3—28no—
169C4×D3—28no—
170C4×D3#4118yes—
171Dic6—44no—
172Dic6—42no—
173C24—34no—
174C2×Q8—34no—
175C2×Q8—32no—
176C2×Q8—124no—
177C2×Q8C22×D312yes—
178C2×D4—34no—
179C2×D4—38no—
180C2×D4—32no—
181C2×D4—34no—
182C2×D4—68no—
183C2×D4—128no—
184C2×D4C22×D312yes—
185C2×D4C22×D314yes—
186C22×C4—32no—
187C22×C4—34no—
188C22×C4—34no—
189C22×C4—31no—
190C22×C4—34no—
191C22×C4—34no—
192C22×C4—34no—
193C22×C4—616no—
194C22×C4C22×D314yes—
195C4⋊C4—34no—
196C4⋊C4—32no—
197C4⋊C4—38no—
198C4⋊C4—34no—
199C4⋊C4C22×D312yes—
200C4⋊C4C22×D314yes—
201C22⋊C4—38no—
202C22⋊C4—34no—
203C22⋊C4—38no—
204C22⋊C4—38no—
205C22⋊C4—34no—
206C22⋊C4—34no—
207C22⋊C4C22×D318yes—
208C22⋊C4C22×D314yes—
209C42—34no—
210C42—38no—
211C42—31no—
212C42—32no—
213C42C22×D314yes—
214C42C22×D311yescharacteristic
215C22×C3—24no—
216C22×C3—24no—
217C22×C3—24no—
218C22×C3C2511yescharacteristic
219C2×D3—24no—
220C2×D3—24no—
221C2×D3—24no—
222C2×D3—416no—
223C2×D3C2×#4114yes—
224C2×D3H(5,𝔽2)12yes—
225C3×C4—24no—
226C3×C4—21no—
227C3×C4—24no—
228C3×C4C2×#4114yes—
229Dic3—24no—
230Dic3—21no—
231Dic3—24no—
232Dic3C2×#4114yes—
233C23—31no—
234C23—34no—
235C23—68no—
236C23—68no—
237C23—68no—
238C23C23×D314yes—
239Q8—42no—
240Q8—124no—
241Q8—122no—
242D4—24no—
243D4—44no—
244D4—68no—
245D4—64no—
246D4—124no—
247D4—128no—
248C2×C4—28no—
249C2×C4—32no—
250C2×C4—38no—
251C2×C4—34no—
252C2×C4—31no—
253C2×C4—34no—
254C2×C4—68no—
255C2×C4—68no—
256C2×C4—68no—
257C2×C4—62no—
258C2×C4—68no—
259C2×C4—68no—
260C2×C4C23×D314yes—
261C2×C4C23×D311yescharacteristic
262C2×C4C23×D314yes—
263C2×C4C23×D312yes—
264C2×C3—44no—
265C2×C3C22×#4112yes—
266C2×C3C2×H(5,𝔽2)11yescharacteristic
267D3—44no—
268D3#55514yesdirect product factor
269C22—24no—
270C22—24no—
271C22—24no—
272C22—34no—
273C22—32no—
274C22—64no—
275C22—64no—
276C22—64no—
277C22—1216no—
278C22C24×D311yescharacteristic
279C4—24no—
280C4—24no—
281C4—21no—
282C4—34no—
283C4—64no—
284C4—64no—
285C4—61no—
286C4D3×#4114yes—
287C3C2×#55511yescharacteristic, complemented, Sylow
288C2—34no—
289C2—44no—
290C2—124no—
291C2C2×D3×#4112yes—
292C2D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#555C2
hypercenter#555D3
hypocenterC3C2×#555
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #555

Complements

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

NH ≅ G/Ntype
1. C3×#555C2semidirect
2. #5734C2semidirect
3. D3×C4⋊Q8C2semidirect
4. D3×#124C2semidirect
5. #5670C2semidirect
6. #5616C2semidirect
7. D3×#121C2semidirect
8. #5563C2semidirect
9. C4×D3×D4C2semidirect
10. #5532C2semidirect
11. D3×#117C2semidirect
17. C3×C4⋊Q8C22semidirect
18. C3×#124C22semidirect
19. C3×#121C22semidirect
20. C3×C4×D4C22semidirect
21. C3×#117C22semidirect
22. #947C22semidirect
23. #944C22semidirect
24. #939C22semidirect
25. #938C22semidirect
26. #936C22semidirect
27. D4×Dic3C22semidirect
28. #930C22semidirect
29. C4×#67C22semidirect
30. #926C22semidirect
31. #895C22semidirect
32. #893C22semidirect
33. #892C22semidirect
34. #892C22semidirect
37. #888C22semidirect
38. #885C22semidirect
39. #885C22semidirect
40. #884C22semidirect
41. #67⋊C4C22semidirect
44. #877C22semidirect
45. #875C22semidirect
46. C4×D12C22semidirect
47. #872C22semidirect
50. #869C22semidirect
56. #555D3direct (N × H)
111. C4⋊Q8C2×D3semidirect
117. #124C2×D3semidirect
123. #121C2×D3semidirect
129. C4×D4C2×D3semidirect
135. #117C2×D3semidirect
268. D3#555direct (N × H)
287. C3C2×#555semidirect

Maximal subgroups

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

subgroupindex
C3×#5552
#57342
D3×C4⋊Q82
D3×#1242
#56702
#56162
D3×#1212
#55632
C4×D3×D42
#55322
D3×#1172
C2×#5553