Finite Group Databasebeta

← C2×#5729 #86210

Subgroups — C2×#5729 #86210

Order 384.

1080 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
1C2×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#946C212yesmaximal
3C2×#944C212yesmaximal, complemented
4#5729C2116yesmaximal, direct product factor
5C2×#896C212yesmaximal
6C2×#892C211yesmaximal, characteristic
7C2×D3×C4⋊C4C211yesmaximal, characteristic
8C2×#890C211yesmaximal, characteristic, complemented
9C2×#889C212yesmaximal, complemented
10C2×#888C211yesmaximal, characteristic, complemented
11C2×#872C211yesmaximal, characteristic
12C2×#870C211yesmaximal, characteristic, complemented
13C2×#544—31nomaximal, Sylow
14C22×C3×Q8C2212yes—
15C22×C4×D3C2212yes—
16C22×C4×D3C2211yescharacteristic
17C22×Dic6C2212yes—
18C3×C4⋊Q8C2218yescomplemented
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C4⋊C4C2212yes—
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C42C2211yescharacteristic
23#946C22116yes—
24#944C22116yescomplemented
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2212yes—
28C2×C4⋊Dic3C2211yescharacteristic
29C2×C4⋊Dic3C2211yescharacteristic
30C2×C4⋊Dic3C2212yes—
31C2×Dic3⋊C4C2212yes—
32C2×Dic3⋊C4C2212yes—
33C2×Dic3⋊C4C2211yescharacteristic
34C2×Dic3⋊C4C2211yescharacteristic
35C2×Dic3⋊C4C2212yes—
36C2×Dic3⋊C4C2212yes—
37C2×Dic3⋊C4C2212yes—
38C2×C4×Dic3C2211yescharacteristic
39C2×C4×Dic3C2212yes—
40#896C22116yes—
41#892C2218yes—
42D3×C4⋊C4C2218yes—
43#890C2218yescomplemented
44#889C22116yescomplemented
45#888C2218yescomplemented
46#872C2218yes—
47#870C2218yescomplemented
48#544—316no—
49C2×C4⋊Q8—32no—
50C2×C4⋊Q8—31no—
51C2×C4⋊Q8D311yescharacteristic, complemented
52C2×#125—32no—
53C2×#125—31no—
54C2×#125—31no—
55C2×C22⋊Q8—32no—
56C2×C22⋊Q8—32no—
57C2×#117—31no—
58C2×#117—31no—
59C22×C4⋊C4—31no—
60C23×D3C2311yescharacteristic
61C2×C3×Q8—44no—
62C2×C3×Q8C2318yes—
63C22×C3×C4C2312yes—
64C22×C3×C4C2311yescharacteristic
65C22×C3×C4C2311yescharacteristic
66C22×C3×C4C2312yes—
67C22×C3×C4C2311yescharacteristic
68C22×Dic3C2311yescharacteristic
69C22×Dic3C2312yes—
70C22×Dic3C2311yescharacteristic
71C22×Dic3C2312yes—
72C22×Dic3C2311yescharacteristic
73C2×C4×D3—44no—
74C2×C4×D3Q818yescomplemented
75C2×C4×D3C2318yes—
76C2×C4×D3C2314yes—
77C2×Dic6—44no—
78C2×Dic6C2318yes—
79C3×C4⋊C4C2314yes—
80C3×C4⋊C4C2318yes—
81C3×C4⋊C4C2314yes—
82C3×C42C2314yes—
83(C2×D3)⋊C4C2314yes—
84(C2×D3)⋊C4C2314yes—
85(C2×D3)⋊C4C2318yes—
86C4⋊Dic3C2314yes—
87C4⋊Dic3C2314yes—
88C4⋊Dic3C2318yes—
89Dic3⋊C4C2318yes—
90Dic3⋊C4C2318yes—
91Dic3⋊C4C2314yes—
92Dic3⋊C4C2314yes—
93Dic3⋊C4C2318yes—
94Dic3⋊C4C2318yes—
95Dic3⋊C4C2318yes—
96C4×Dic3C2314yes—
97C4×Dic3C2318yes—
98C22×Q8—32no—
99C22×Q8C2×D312yes—
100C23×C4—31no—
101C23×C4—32no—
102C4⋊Q8—316no—
103C4⋊Q8—38no—
104C4⋊Q8C2×D318yescomplemented
105#125—316no—
106#125—38no—
107#125—38no—
108C22⋊Q8—316no—
109C22⋊Q8—316no—
110#117—38no—
111#117—38no—
112C2×C4⋊C4—32no—
113C2×C4⋊C4—31no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—32no—
121C2×C4⋊C4—32no—
122C2×C4⋊C4—38no—
123C2×C4⋊C4C2×D311yescharacteristic
124C2×C4⋊C4C2×D311yescharacteristic
125C2×C4⋊C4C2×D312yes—
126C2×C22⋊C4—32no—
127C2×C22⋊C4—31no—
128C2×C22⋊C4—31no—
129C2×C42—32no—
130C2×C42—31no—
131C2×C42C2×D311yescharacteristic
132C23×C3C2411yescharacteristic
133C22×D3—22no—
134C22×D3—22no—
135C22×D3C2×Q814yes—
136C22×D3C2412yes—
137C3×Q8—48no—
138C2×C3×C4—24no—
139C2×C3×C4—24no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4C2×Q814yes—
143C2×C3×C4C2414yes—
144C2×C3×C4C2412yes—
145C2×C3×C4C2412yes—
146C2×C3×C4C2414yes—
147C2×C3×C4C2412yes—
148C2×Dic3—22no—
149C2×Dic3—24no—
150C2×Dic3—22no—
151C2×Dic3—24no—
152C2×Dic3C2×Q814yes—
153C2×Dic3C2412yes—
154C2×Dic3C2414yes—
155C2×Dic3C2412yes—
156C2×Dic3C2414yes—
157C2×Dic3C2412yes—
158C4×D3—48no—
159C4×D3C2×Q8116yescomplemented
160Dic6—48no—
161C24—31no—
162C2×Q8—38no—
163C2×Q8—44no—
164C2×Q8—124no—
165C2×Q8C22×D318yes—
166C22×C4—31no—
167C22×C4—38no—
168C22×C4—34no—
169C22×C4—31no—
170C22×C4—32no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—38no—
174C22×C4—124no—
175C22×C4C22×D312yes—
176C22×C4C22×D311yescharacteristic
177C22×C4C22×D311yescharacteristic
178C22×C4C22×D312yes—
179C22×C4C22×D311yescharacteristic
180C4⋊C4—38no—
181C4⋊C4—34no—
182C4⋊C4—34no—
183C4⋊C4—34no—
184C4⋊C4—34no—
185C4⋊C4—38no—
186C4⋊C4—38no—
187C4⋊C4—38no—
188C4⋊C4—38no—
189C4⋊C4—38no—
190C4⋊C4C22×D314yes—
191C4⋊C4C22×D314yes—
192C4⋊C4C22×D318yes—
193C22⋊C4—38no—
194C22⋊C4—34no—
195C22⋊C4—34no—
196C42—38no—
197C42—34no—
198C42C22×D314yes—
199C22×C3C22×Q812yes—
200C22×C3H(5,𝔽2)12yes—
201C22×C32−1+412yes—
202C22×C3C2511yescharacteristic
203C2×D3—28no—
204C2×D3—22no—
205C2×D3—22no—
206C2×D3C22×Q814yes—
207C3×C4—24no—
208C3×C4—24no—
209C3×C4—22no—
210C3×C4—22no—
211C3×C4C22×Q814yes—
212Dic3—22no—
213Dic3—24no—
214Dic3—22no—
215Dic3—24no—
216Dic3C22×Q814yes—
217C23—34no—
218C23—32no—
219C23—62no—
220C23—62no—
221C23C23×D311yescharacteristic
222Q8—48no—
223Q8—128no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—34no—
229C2×C4—32no—
230C2×C4—316no—
231C2×C4—32no—
232C2×C4—34no—
233C2×C4—32no—
234C2×C4—34no—
235C2×C4—64no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—64no—
239C2×C4—128no—
240C2×C4D3×Q814yes—
241C2×C4C23×D314yes—
242C2×C4C23×D312yes—
243C2×C4C23×D312yes—
244C2×C4C23×D314yes—
245C2×C4C23×D312yes—
246C2×C3#54414yescomplemented
247C2×C3C23×Q811yescharacteristic
248C2×C3C2×H(5,𝔽2)11yescharacteristic
249C2×C3C2×2−1+411yescharacteristic
250D3—24no—
251C22—34no—
252C22—62no—
253C22—62no—
254C22—68no—
255C22#100412yes—
256C22C2×D3×Q812yes—
257C22#100712yes—
258C22C24×D311yescharacteristic
259C4—24no—
260C4—22no—
261C4—24no—
262C4—22no—
263C4—34no—
264C4—64no—
265C4—62no—
266C4—62no—
267C4—64no—
268C4C2×D3×Q814yes—
269C3C2×#54411yescharacteristic, complemented, Sylow
270C2—64no—
271C2#572914yesdirect product factor
272C2C2×#100411yescharacteristic
273C2C22×D3×Q811yescharacteristic
274C2C2×#100711yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4⋊Q8C2
socleC23×C3C24
prime radicalC23×D3C23
hypercenterC2×C4⋊Q8D3
hypocenterC3C2×#544
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4⋊Q8C2semidirect
3. C2×#944C2semidirect
4. #5729C2direct (N × H)
8. C2×#890C2semidirect
9. C2×#889C2semidirect
10. C2×#888C2semidirect
12. C2×#870C2semidirect
18. C3×C4⋊Q8C22semidirect
24. #944C22semidirect
43. #890C22semidirect
44. #889C22semidirect
45. #888C22semidirect
47. #870C22semidirect
51. C2×C4⋊Q8D3semidirect
74. C2×C4×D3Q8semidirect
104. C4⋊Q8C2×D3semidirect
159. C4×D3C2×Q8semidirect
246. C2×C3#544semidirect
269. C3C2×#544semidirect
271. C2#5729direct (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×#9462
C2×#9442
#57292
C2×#8962
C2×#8922
C2×D3×C4⋊C42
C2×#8902
C2×#8892
C2×#8882
C2×#8722
C2×#8702
C2×#5443