Finite Group Databasebeta

← C2×D3×#124 #86068

Subgroups — C2×D3×#124 #86068

Order 384.

1880 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×D3×Q8C211yesmaximal, characteristic, complemented
2C22×D3×D4C211yesmaximal, characteristic
3C2×C3×#124C211yesmaximal, characteristic, complemented
4C2×#947C211yesmaximal, characteristic, complemented
5C2×#936C211yesmaximal, characteristic, complemented
6D3×#124C2116yesmaximal, direct product factor
7C2×#885C214yesmaximal, complemented
8C2×D3×C22⋊C4C214yesmaximal, complemented
9C2×#875C211yesmaximal, characteristic, complemented
10C2×C42×D3C211yesmaximal, characteristic, complemented
11C22×#124—31nomaximal, Sylow
12C24×D3C2212yes—
13C22×C3×Q8C2211yescharacteristic, complemented
14C22×C3×D4C2211yescharacteristic
15C22×#67C2212yes—
16C2×D3×Q8—28no—
17C2×D3×Q8C2218yescomplemented
18C2×D3×D4—28no—
19C2×D3×D4C2218yes—
20C22×D12C2211yescharacteristic
21C22×C4×D3C2212yes—
22C22×C4×D3C2212yes—
23C22×C4×D3C2211yescharacteristic
24C22×Dic6C2211yescharacteristic, complemented
25C22×Dic6C2212yescomplemented
26C3×#124C2218yescomplemented
27C2×C3×C22⋊C4C2214yescomplemented
28C2×C3×C42C2211yescharacteristic, complemented
29C2×#217C2214yescomplemented
30#947C2218yescomplemented
31#936C2218yescomplemented
32C2×(C2×D3)⋊C4C2214yescomplemented
33C2×(C2×D3)⋊C4C2214yescomplemented
34C2×C4×Dic3C2212yescomplemented
35C2×C4×Dic3C2211yescharacteristic, complemented
36#885C22132yescomplemented
37D3×C22⋊C4C22132yescomplemented
38#875C2218yescomplemented
39C42×D3C2218yescomplemented
40C23×Q8—31no—
41C23×D4—31no—
42C2×#124—31no—
43C2×#124—31no—
44C2×#124—316no—
45C2×#124—34no—
46C2×#124—31no—
47C2×#124D311yescharacteristic, direct product factor
48C22×C22⋊C4—34no—
49C22×C42—31no—
50C24×C3C2312yes—
51C23×D3—28no—
52C23×D3—216no—
53C23×D3C2318yes—
54C23×D3C2312yes—
55C23×D3C2311yescharacteristic
56C2×C3×Q8—24no—
57C2×C3×Q8C2314yescomplemented
58C2×C3×D4—24no—
59C2×C3×D4C2314yes—
60C22×C3×C4C2312yes—
61C22×C3×C4C2312yes—
62C22×C3×C4C2311yescharacteristic
63C2×#67—28no—
64C2×#67C2318yes—
65C22×Dic3C2312yes—
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68D3×Q8—232no—
69D3×D4—232no—
70C2×D12—24no—
71C2×D12C2314yes—
72C2×C4×D3—28no—
73C2×C4×D3—28no—
74C2×C4×D3D418yes—
75C2×C4×D3C2318yes—
76C2×C4×D3C2318yes—
77C2×C4×D3C2314yes—
78C2×Dic6—24no—
79C2×Dic6—28no—
80C2×Dic6C2314yescomplemented
81C2×Dic6C2318yescomplemented
82C3×C22⋊C4C23116yescomplemented
83C3×C42C2314yescomplemented
84#217C23116yescomplemented
85(C2×D3)⋊C4C23116yescomplemented
86(C2×D3)⋊C4C23116yescomplemented
87C4×Dic3C2318yescomplemented
88C4×Dic3C2314yescomplemented
89C25—32no—
90C22×Q8—38no—
91C22×Q8—32no—
92C22×Q8—31no—
93C22×Q8—68no—
94C22×Q8C2×D311yescharacteristic, complemented
95C22×D4—32no—
96C22×D4—38no—
97C22×D4—31no—
98C22×D4—68no—
99C22×D4C2×D311yescharacteristic
100C23×C4—31no—
101C23×C4—32no—
102C23×C4—32no—
103#124—38no—
104#124—38no—
105#124—332no—
106#124—38no—
107#124C2×D318yesdirect product factor
108C2×C22⋊C4—34no—
109C2×C22⋊C4—34no—
110C2×C22⋊C4—34no—
111C2×C22⋊C4—332no—
112C2×C22⋊C4C2×D314yescomplemented
113C2×C42—32no—
114C2×C42—31no—
115C2×C42—38no—
116C2×C42C2×D311yescharacteristic, complemented
117C23×C3—24no—
118C23×C3—28no—
119C23×C3C2414yes—
120C23×C3C2411yescharacteristic
121C22×D3—24no—
122C22×D3—28no—
123C22×D3—216no—
124C22×D3—232no—
125C22×D3—432no—
126C22×D3C2×D414yes—
127C22×D3#4118yes—
128C22×D3C2414yes—
129C22×D3C2412yes—
130C3×Q8—28no—
131C3×D4—28no—
132C2×C3×C4—24no—
133C2×C3×C4—24no—
134C2×C3×C4C2×D414yes—
135C2×C3×C4C2414yes—
136C2×C3×C4C2414yes—
137C2×C3×C4C2412yes—
138#67—216no—
139C2×Dic3—24no—
140C2×Dic3—24no—
141C2×Dic3C2×D414yes—
142C2×Dic3C2414yes—
143C2×Dic3C2414yes—
144C2×Dic3C2412yes—
145D12—28no—
146C4×D3—216no—
147C4×D3—216no—
148C4×D3C2×D4116yes—
149Dic6—28no—
150Dic6—216no—
151C24—31no—
152C24—38no—
153C24—32no—
154C24—68no—
155C24—616no—
156C24C22×D312yes—
157C2×Q8—24no—
158C2×Q8—38no—
159C2×Q8—34no—
160C2×Q8—68no—
161C2×Q8—632no—
162C2×Q8—64no—
163C2×Q8C22×D314yescomplemented
164C2×D4—24no—
165C2×D4—38no—
166C2×D4—34no—
167C2×D4—632no—
168C2×D4—68no—
169C2×D4—64no—
170C2×D4C22×D314yes—
171C22×C4—31no—
172C22×C4—38no—
173C22×C4—34no—
174C22×C4—32no—
175C22×C4—32no—
176C22×C4—38no—
177C22×C4—38no—
178C22×C4—68no—
179C22×C4—68no—
180C22×C4C22×D312yes—
181C22×C4C22×D312yes—
182C22×C4C22×D311yescharacteristic
183C22⋊C4—316no—
184C22⋊C4—316no—
185C22⋊C4—316no—
186C22⋊C4C22×D3116yescomplemented
187C42—38no—
188C42—34no—
189C42C22×D314yescomplemented
190C22×C3—24no—
191C22×C3—28no—
192C22×C3—48no—
193C22×C3C22×D412yes—
194C22×C3C2×#4114yes—
195C22×C3C2511yescharacteristic
196C2×D3—24no—
197C2×D3—28no—
198C2×D3—48no—
199C2×D3—432no—
200C2×D3#124116yesdirect product factor
201C2×D3C22×D414yes—
202C2×D3C2×#4118yes—
203C3×C4—24no—
204C3×C4—24no—
205C3×C4C22×D414yes—
206Dic3—24no—
207Dic3—24no—
208Dic3C22×D414yes—
209C23—24no—
210C23—28no—
211C23—34no—
212C23—38no—
213C23—32no—
214C23—34no—
215C23—616no—
216C23—632no—
217C23—64no—
218C23—68no—
219C23—1232no—
220C23C23×D314yes—
221C23C23×D311yescharacteristic
222Q8—28no—
223Q8—616no—
224Q8—68no—
225D4—28no—
226D4—616no—
227D4—68no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—34no—
231C2×C4—32no—
232C2×C4—316no—
233C2×C4—34no—
234C2×C4—34no—
235C2×C4—616no—
236C2×C4—616no—
237C2×C4—64no—
238C2×C4—64no—
239C2×C4D3×D414yes—
240C2×C4C23×D314yes—
241C2×C4C23×D314yes—
242C2×C4C23×D312yes—
243C2×C3—44no—
244C2×C3C2×#12414yescomplemented
245C2×C3C23×D411yescharacteristic
246C2×C3C22×#4112yes—
247D3—44no—
248D3C2×#12418yesdirect product factor
249C22—24no—
250C22—28no—
251C22—316no—
252C22—34no—
253C22—38no—
254C22—48no—
255C22—64no—
256C22—68no—
257C22—1232no—
258C22—128no—
259C22C2×D3×D412yes—
260C22D3×#4114yes—
261C22C24×D311yescharacteristic
262C4—24no—
263C4—24no—
264C4—34no—
265C4—64no—
266C4—64no—
267C4C2×D3×D414yes—
268C3C22×#12411yescharacteristic, complemented, Sylow
269C2—38no—
270C2—44no—
271C2—124no—
272C2D3×#12414yesdirect product factor
273C2C22×D3×D411yescharacteristic
274C2C2×D3×#4112yes—
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#124C2
socleC23×C3C24
prime radicalC22×D3×D4C2
hypercenterC2×#124D3
hypocenterC3C22×#124
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×D3×Q8C2semidirect
3. C2×C3×#124C2semidirect
4. C2×#947C2semidirect
5. C2×#936C2semidirect
6. D3×#124C2direct (N × H)
7. C2×#885C2semidirect
8. C2×D3×C22⋊C4C2semidirect
9. C2×#875C2semidirect
10. C2×C42×D3C2semidirect
13. C22×C3×Q8C22semidirect
17. C2×D3×Q8C22semidirect
24. C22×Dic6C22semidirect
25. C22×Dic6C22semidirect
26. C3×#124C22semidirect
27. C2×C3×C22⋊C4C22semidirect
28. C2×C3×C42C22semidirect
29. C2×#217C22semidirect
30. #947C22semidirect
31. #936C22semidirect
32. C2×(C2×D3)⋊C4C22semidirect
33. C2×(C2×D3)⋊C4C22semidirect
34. C2×C4×Dic3C22semidirect
35. C2×C4×Dic3C22semidirect
36. #885C22semidirect
37. D3×C22⋊C4C22semidirect
38. #875C22semidirect
39. C42×D3C22semidirect
47. C2×#124D3direct (N × H)
57. C2×C3×Q8C23semidirect
80. C2×Dic6C23semidirect
81. C2×Dic6C23semidirect
82. C3×C22⋊C4C23semidirect
83. C3×C42C23semidirect
84. #217C23semidirect
85. (C2×D3)⋊C4C23semidirect
86. (C2×D3)⋊C4C23semidirect
87. C4×Dic3C23semidirect
88. C4×Dic3C23semidirect
94. C22×Q8C2×D3semidirect
107. #124C2×D3direct (N × H)
112. C2×C22⋊C4C2×D3semidirect
116. C2×C42C2×D3semidirect
163. C2×Q8C22×D3semidirect
186. C22⋊C4C22×D3semidirect
189. C42C22×D3semidirect
200. C2×D3#124direct (N × H)
244. C2×C3C2×#124semidirect
248. D3C2×#124direct (N × H)
268. C3C22×#124semidirect
272. C2D3×#124direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D3×Q82
C22×D3×D42
C2×C3×#1242
C2×#9472
C2×#9362
D3×#1242
C2×#8852
C2×D3×C22⋊C42
C2×#8752
C2×C42×D32
C22×#1243