Finite Group Databasebeta

← C2×#5580 #85807

Subgroups — C2×#5580 #85807

Order 384.

1232 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#947C211yesmaximal, characteristic, complemented
3C2×#946C211yesmaximal, characteristic, complemented
4#5580C2116yesmaximal, direct product factor
5C2×#898C212yesmaximal, complemented
6C2×#895C211yesmaximal, characteristic
7C2×#894C212yesmaximal, complemented
8C2×#890C211yesmaximal, characteristic, complemented
9C2×C4×D12C212yesmaximal, complemented
10C2×C4×D12C211yesmaximal, characteristic, complemented
11C2×#872C212yesmaximal, complemented
12C2×C42×D3C211yesmaximal, characteristic, complemented
13C2×#529—31nomaximal, Sylow
14C22×C3×Q8C2211yescharacteristic
15C22×D12C2212yes—
16C22×D12C2211yescharacteristic
17C22×C4×D3C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2212yes—
21C3×C4×Q8C2218yescomplemented
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2212yes—
24C2×C3×C42C2211yescharacteristic
25C2×C3×C42C2212yes—
26#947C2218yescomplemented
27#946C2218yescomplemented
28C2×(C2×D3)⋊C4C2212yes—
29C2×(C2×D3)⋊C4C2212yes—
30C2×(C2×D3)⋊C4C2212yes—
31C2×(C2×D3)⋊C4C2212yes—
32C2×(C2×D3)⋊C4C2212yes—
33C2×C4⋊Dic3C2212yes—
34C2×C4⋊Dic3C2211yescharacteristic
35C2×Dic3⋊C4C2212yes—
36C2×Dic3⋊C4C2212yes—
37C2×C4×Dic3C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40#898C22116yescomplemented
41#895C2218yes—
42#894C22116yescomplemented
43#890C2218yescomplemented
44C4×D12C22116yescomplemented
45C4×D12C2218yescomplemented
46#872C22116yescomplemented
47C42×D3C2218yescomplemented
48#529—316no—
49C2×#126—32no—
50C2×#125—31no—
51C2×#124—31no—
52C2×#123—32no—
53C2×C22⋊Q8—31no—
54C2×#121—31no—
55C2×C4×Q8D311yescharacteristic, complemented
56C2×C4×D4—32no—
57C2×C4×D4—31no—
58C2×#117—32no—
59C22×C42—31no—
60C23×D3C2312yes—
61C23×D3C2311yescharacteristic
62C2×C3×Q8—24no—
63C2×C3×Q8C2314yes—
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2312yes—
66C22×C3×C4C2312yes—
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×Dic3C2311yescharacteristic
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2311yescharacteristic
72C22×Dic3C2312yes—
73C2×D12—28no—
74C2×D12—24no—
75C2×D12C2318yes—
76C2×D12C2314yes—
77C2×C4×D3—24no—
78C2×C4×D3—24no—
79C2×C4×D3—24no—
80C2×C4×D3—28no—
81C2×C4×D3C2314yes—
82C2×C4×D3C2314yes—
83C2×C4×D3C2314yes—
84C2×C4×D3C2318yes—
85C3×C4⋊C4C2314yes—
86C3×C4⋊C4C2318yes—
87C3×C42C2314yes—
88C3×C42C2318yes—
89(C2×D3)⋊C4C2318yes—
90(C2×D3)⋊C4C2318yes—
91(C2×D3)⋊C4C2318yes—
92(C2×D3)⋊C4C2318yes—
93(C2×D3)⋊C4C2318yes—
94C4⋊Dic3C2318yes—
95C4⋊Dic3C2314yes—
96Dic3⋊C4C2318yes—
97Dic3⋊C4C2318yes—
98C4×Dic3C2314yes—
99C4×Dic3C2314yes—
100C4×Dic3C2314yes—
101C22×Q8C2×D311yescharacteristic
102C22×D4—32no—
103C22×D4—31no—
104C23×C4—31no—
105C23×C4—31no—
106C23×C4—32no—
107C23×C4—31no—
108#126—316no—
109#125—38no—
110#124—38no—
111#123—316no—
112C22⋊Q8—38no—
113#121—38no—
114C4×Q8C2×D318yescomplemented
115C4×D4—38no—
116C4×D4—316no—
117#117—316no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—32no—
122C2×C4⋊C4C2×D311yescharacteristic
123C2×C4⋊C4C2×D312yes—
124C2×C22⋊C4—32no—
125C2×C22⋊C4—32no—
126C2×C22⋊C4—32no—
127C2×C22⋊C4—32no—
128C2×C22⋊C4—32no—
129C2×C42—31no—
130C2×C42—31no—
131C2×C42—31no—
132C2×C42—38no—
133C2×C42C2×D312yes—
134C2×C42C2×D311yescharacteristic
135C23×C3C2411yescharacteristic
136C22×D3—22no—
137C22×D3—22no—
138C22×D3—24no—
139C22×D3—24no—
140C22×D3—24no—
141C22×D3#4114yes—
142C22×D3C2414yes—
143C22×D3C2412yes—
144C3×Q8—28no—
145C2×C3×C4—22no—
146C2×C3×C4—24no—
147C2×C3×C4—24no—
148C2×C3×C4#4114yes—
149C2×C3×C4#4114yes—
150C2×C3×C4C2412yes—
151C2×C3×C4C2414yes—
152C2×C3×C4C2414yes—
153C2×C3×C4C2412yes—
154C2×C3×C4C2412yes—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—24no—
158C2×Dic3—22no—
159C2×Dic3C2412yes—
160C2×Dic3C2412yes—
161C2×Dic3C2412yes—
162C2×Dic3C2414yes—
163D12—216no—
164D12—28no—
165C4×D3—28no—
166C4×D3—28no—
167C4×D3—28no—
168C4×D3—216no—
169C24—31no—
170C24—32no—
171C2×Q8—24no—
172C2×Q8C22×D314yes—
173C2×D4—38no—
174C2×D4—34no—
175C2×D4—68no—
176C2×D4—64no—
177C22×C4—31no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—32no—
181C22×C4—34no—
182C22×C4—34no—
183C22×C4—34no—
184C22×C4—38no—
185C22×C4—64no—
186C22×C4—64no—
187C22×C4—64no—
188C22×C4—68no—
189C22×C4C22×D311yescharacteristic
190C22×C4C22×D312yes—
191C22×C4C22×D312yes—
192C22×C4C22×D311yescharacteristic
193C22×C4C22×D311yescharacteristic
194C4⋊C4—38no—
195C4⋊C4—38no—
196C4⋊C4—34no—
197C4⋊C4—38no—
198C4⋊C4C22×D314yes—
199C4⋊C4C22×D318yes—
200C22⋊C4—38no—
201C22⋊C4—38no—
202C22⋊C4—38no—
203C22⋊C4—38no—
204C22⋊C4—38no—
205C42—34no—
206C42—34no—
207C42—34no—
208C42C22×D314yes—
209C42C22×D318yes—
210C22×C3C2×#4112yes—
211C22×C3C2×#4112yes—
212C22×C3C2×#4112yes—
213C22×C3C2511yescharacteristic
214C2×D3—28no—
215C2×D3—22no—
216C2×D3—22no—
217C2×D3—24no—
218C2×D3—24no—
219C2×D3—24no—
220C2×D3—48no—
221C2×D3C2×#4114yes—
222C3×C4—22no—
223C3×C4—24no—
224C3×C4—24no—
225C3×C4C2×#4114yes—
226C3×C4C2×#4114yes—
227Dic3—22no—
228Dic3—22no—
229Dic3—24no—
230Dic3—22no—
231C23—34no—
232C23—32no—
233C23—34no—
234C23—62no—
235C23—62no—
236C23—64no—
237C23—64no—
238C23—64no—
239C23C23×D311yescharacteristic
240Q8—28no—
241D4—616no—
242D4—68no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—32no—
249C2×C4—34no—
250C2×C4—68no—
251C2×C4—62no—
252C2×C4—62no—
253C2×C4—64no—
254C2×C4—62no—
255C2×C4—68no—
256C2×C4—68no—
257C2×C4—616no—
258C2×C4#23514yes—
259C2×C4#23914yes—
260C2×C4C23×D312yes—
261C2×C4C23×D314yes—
262C2×C4C23×D314yes—
263C2×C4C23×D312yes—
264C2×C4C23×D312yes—
265C2×C3#52914yescomplemented
266C2×C3C22×#4111yescharacteristic
267C2×C3C22×#4111yescharacteristic
268C2×C3C22×#4111yescharacteristic
269D3—24no—
270D3—44no—
271C22—34no—
272C22—62no—
273C22—62no—
274C22—68no—
275C22—64no—
276C22—64no—
277C22—64no—
278C22—128no—
279C22C2×#23512yes—
280C22C2×#23912yes—
281C22D3×#4112yes—
282C22C24×D311yescharacteristic
283C4—24no—
284C4—22no—
285C4—24no—
286C4—62no—
287C4—62no—
288C4—64no—
289C4—62no—
290C4C2×#23514yes—
291C4C2×#23914yes—
292C3C2×#52911yescharacteristic, complemented, Sylow
293C2—64no—
294C2—124no—
295C2#558014yesdirect product factor
296C2C22×#23511yescharacteristic
297C2C22×#23911yescharacteristic
298C2C2×D3×#4111yescharacteristic
subgroupG/H
centerC22×C4C22×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×Q8C2
socleC23×C3C24
prime radicalC2×#895C2
hypercenterC2×C4×Q8D3
hypocenterC3C2×#529
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q8C2semidirect
2. C2×#947C2semidirect
3. C2×#946C2semidirect
4. #5580C2direct (N × H)
5. C2×#898C2semidirect
7. C2×#894C2semidirect
8. C2×#890C2semidirect
9. C2×C4×D12C2semidirect
10. C2×C4×D12C2semidirect
11. C2×#872C2semidirect
12. C2×C42×D3C2semidirect
21. C3×C4×Q8C22semidirect
26. #947C22semidirect
27. #946C22semidirect
40. #898C22semidirect
42. #894C22semidirect
43. #890C22semidirect
44. C4×D12C22semidirect
45. C4×D12C22semidirect
46. #872C22semidirect
47. C42×D3C22semidirect
55. C2×C4×Q8D3semidirect
114. C4×Q8C2×D3semidirect
265. C2×C3#529semidirect
292. C3C2×#529semidirect
295. C2#5580direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q82
C2×#9472
C2×#9462
#55802
C2×#8982
C2×#8952
C2×#8942
C2×#8902
C2×C4×D122
C2×#8722
C2×C42×D32
C2×#5293