Finite Group Databasebeta

← C2×#5668 #86060

Subgroups — C2×#5668 #86060

Order 384.

1728 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#239C211yesmaximal, characteristic, complemented
2C22×#237C211yesmaximal, characteristic, complemented
3C2×C3×#124C211yesmaximal, characteristic, complemented
4C2×#944C211yesmaximal, characteristic, complemented
5C2×#940C211yesmaximal, characteristic, complemented
6#5668C2116yesmaximal, direct product factor
7C2×#884C214yesmaximal, complemented
8C2×#67⋊C4C214yesmaximal, complemented
9C2×#875C211yesmaximal, characteristic, complemented
10C2×C42×D3C211yesmaximal, characteristic, complemented
11C2×#532—31nomaximal, Sylow
12C22×C3×Q8C2211yescharacteristic
13C22×C3×D4C2211yescharacteristic
14C22×#67C2212yes—
15C22×#67C2214yes—
16C23×Dic3C2212yes—
17C2×#239—28no—
18C2×#239C2218yescomplemented
19C2×#237—28no—
20C2×#237C2218yescomplemented
21C22×D12C2211yescharacteristic
22C22×D12C2212yes—
23C22×C4×D3C2212yes—
24C22×C4×D3C2212yes—
25C22×C4×D3C2211yescharacteristic
26C22×Dic6C2211yescharacteristic
27C3×#124C2218yescomplemented
28C2×C3×C22⋊C4C2214yes—
29C2×C3×C42C2211yescharacteristic
30#944C2218yescomplemented
31#940C2218yescomplemented
32C2×(C2×D3)⋊C4C2214yes—
33C2×Dic3⋊C4C2214yes—
34C2×C4×Dic3C2212yes—
35C2×C4×Dic3C2211yescharacteristic
36#884C22132yescomplemented
37#67⋊C4C22132yescomplemented
38#875C2218yescomplemented
39C42×D3C2218yescomplemented
40C22×#41—31no—
41C22×#41—31no—
42#532—316no—
43C2×C4⋊Q8—31no—
44C2×#127—31no—
45C2×#124—31no—
46C2×#124D311yescharacteristic, complemented
47C2×#121—34no—
48C2×C4×D4—34no—
49C22×C42—31no—
50C24×C3C2312yes—
51C23×D3C2312yes—
52C23×D3C2311yescharacteristic
53C2×C3×Q8—24no—
54C2×C3×Q8C2314yes—
55C2×C3×D4—24no—
56C2×C3×D4C2314yes—
57C22×C3×C4C2312yes—
58C22×C3×C4C2312yes—
59C22×C3×C4C2311yescharacteristic
60C2×#67—28no—
61C2×#67—216no—
62C2×#67C23116yes—
63C2×#67C2318yes—
64C22×Dic3—28no—
65C22×Dic3C2318yes—
66C22×Dic3C2312yes—
67C22×Dic3C2311yescharacteristic
68C22×Dic3C2312yes—
69#239—232no—
70#237—232no—
71C2×D12—28no—
72C2×D12—24no—
73C2×D12C2314yes—
74C2×D12C2318yes—
75C2×C4×D3—28no—
76C2×C4×D3—28no—
77C2×C4×D3D418yescomplemented
78C2×C4×D3C2318yes—
79C2×C4×D3C2318yes—
80C2×C4×D3C2314yes—
81C2×Dic6—24no—
82C2×Dic6C2314yes—
83C3×C22⋊C4C23116yes—
84C3×C42C2314yes—
85(C2×D3)⋊C4C23116yes—
86Dic3⋊C4C23116yes—
87C4×Dic3C2318yes—
88C4×Dic3C2314yes—
89C2×#41—38no—
90C2×#41—38no—
91C2×#41—68no—
92C2×#41—68no—
93C22×Q8—31no—
94C22×Q8C2×D311yescharacteristic
95C22×D4—34no—
96C22×D4—32no—
97C22×D4—32no—
98C22×D4—31no—
99C22×D4C2×D311yescharacteristic
100C23×C4—31no—
101C23×C4—32no—
102C23×C4—32no—
103C23×C4—32no—
104C4⋊Q8—38no—
105#127—38no—
106#124—38no—
107#124C2×D318yescomplemented
108#121—332no—
109C4×D4—332no—
110C2×C4⋊C4—34no—
111C2×C22⋊C4—34no—
112C2×C22⋊C4C2×D314yes—
113C2×C42—32no—
114C2×C42—31no—
115C2×C42—38no—
116C2×C42C2×D311yescharacteristic
117C23×C3—24no—
118C23×C3—28no—
119C23×C3C2414yes—
120C23×C3C2411yescharacteristic
121C22×D3—24no—
122C22×D3—24no—
123C22×D3—28no—
124C22×D3C2×D414yes—
125C22×D3C2414yes—
126C22×D3C2412yes—
127C3×Q8—28no—
128C3×D4—28no—
129C2×C3×C4—24no—
130C2×C3×C4—24no—
131C2×C3×C4C2×D414yes—
132C2×C3×C4C2414yes—
133C2×C3×C4C2414yes—
134C2×C3×C4C2412yes—
135#67—216no—
136#67—232no—
137C2×Dic3—24no—
138C2×Dic3—216no—
139C2×Dic3C2×D414yes—
140C2×Dic3#4118yescomplemented
141C2×Dic3C2414yes—
142C2×Dic3C2412yes—
143C2×Dic3C2414yes—
144D12—216no—
145D12—28no—
146C4×D3—216no—
147C4×D3—216no—
148C4×D3C2×D4116yescomplemented
149Dic6—28no—
150C24—31no—
151C24—32no—
152C24C22×D312yes—
153#41—632no—
154#41—632no—
155C2×Q8—24no—
156C2×Q8—34no—
157C2×Q8—64no—
158C2×Q8C22×D314yes—
159C2×D4—24no—
160C2×D4—316no—
161C2×D4—38no—
162C2×D4—38no—
163C2×D4—34no—
164C2×D4—68no—
165C2×D4—616no—
166C2×D4—68no—
167C2×D4—64no—
168C2×D4C22×D314yes—
169C22×C4—32no—
170C22×C4—31no—
171C22×C4—38no—
172C22×C4—34no—
173C22×C4—38no—
174C22×C4—32no—
175C22×C4—38no—
176C22×C4—38no—
177C22×C4—68no—
178C22×C4—68no—
179C22×C4—68no—
180C22×C4C22×D312yes—
181C22×C4C22×D312yes—
182C22×C4C22×D311yescharacteristic
183C4⋊C4—316no—
184C22⋊C4—316no—
185C22⋊C4C22×D3116yes—
186C42—38no—
187C42—34no—
188C42C22×D314yes—
189C22×C3—24no—
190C22×C3—28no—
191C22×C3—48no—
192C22×C3C22×D412yes—
193C22×C3C2×#4114yes—
194C22×C3C2511yescharacteristic
195C2×D3—28no—
196C2×D3—24no—
197C2×D3—24no—
198C2×D3—28no—
199C2×D3—48no—
200C2×D3C22×D414yes—
201C3×C4—24no—
202C3×C4—24no—
203C3×C4C22×D414yes—
204Dic3—24no—
205Dic3C22×D414yes—
206Dic3C2×#4118yescomplemented
207C23—24no—
208C23—28no—
209C23—34no—
210C23—32no—
211C23—34no—
212C23—64no—
213C23—64no—
214C23—68no—
215C23C23×D314yes—
216C23C23×D311yescharacteristic
217Q8—28no—
218Q8—68no—
219D4—28no—
220D4—616no—
221D4—632no—
222D4—616no—
223D4—68no—
224C2×C4—24no—
225C2×C4—24no—
226C2×C4—34no—
227C2×C4—38no—
228C2×C4—34no—
229C2×C4—32no—
230C2×C4—316no—
231C2×C4—34no—
232C2×C4—616no—
233C2×C4—616no—
234C2×C4—616no—
235C2×C4—64no—
236C2×C4D3×D414yes—
237C2×C4C23×D314yes—
238C2×C4C23×D314yes—
239C2×C4C23×D312yes—
240C2×C3—44no—
241C2×C3#53214yescomplemented
242C2×C3C23×D411yescharacteristic
243C2×C3C22×#4112yes—
244D3—24no—
245D3—44no—
246C22—24no—
247C22—28no—
248C22—34no—
249C22—48no—
250C22—68no—
251C22—64no—
252C22—64no—
253C22—68no—
254C22—128no—
255C22C2×D3×D412yes—
256C22D3×#4114yes—
257C22C24×D311yescharacteristic
258C4—24no—
259C4—24no—
260C4—34no—
261C4—38no—
262C4—64no—
263C4C2×D3×D414yes—
264C3C2×#53211yescharacteristic, complemented, Sylow
265C2—44no—
266C2—64no—
267C2—124no—
268C2#566814yesdirect product factor
269C2C22×D3×D411yescharacteristic
270C2C2×D3×#4112yes—
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×#124C2
socleC23×C3C24
hypercenterC2×#124D3
hypocenterC3C2×#532
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C22×#239C2semidirect
2. C22×#237C2semidirect
3. C2×C3×#124C2semidirect
4. C2×#944C2semidirect
5. C2×#940C2semidirect
6. #5668C2direct (N × H)
7. C2×#884C2semidirect
8. C2×#67⋊C4C2semidirect
9. C2×#875C2semidirect
10. C2×C42×D3C2semidirect
18. C2×#239C22semidirect
20. C2×#237C22semidirect
27. C3×#124C22semidirect
30. #944C22semidirect
31. #940C22semidirect
36. #884C22semidirect
37. #67⋊C4C22semidirect
38. #875C22semidirect
39. C42×D3C22semidirect
46. C2×#124D3semidirect
77. C2×C4×D3D4semidirect
107. #124C2×D3semidirect
140. C2×Dic3#41semidirect
148. C4×D3C2×D4semidirect
206. Dic3C2×#41semidirect
241. C2×C3#532semidirect
264. C3C2×#532semidirect
268. C2#5668direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#2392
C22×#2372
C2×C3×#1242
C2×#9442
C2×#9402
#56682
C2×#8842
C2×#67⋊C42
C2×#8752
C2×C42×D32
C2×#5323