Finite Group Databasebeta

← C2×#5565 #85758

Subgroups — C2×#5565 #85758

Order 384.

1252 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D4C211yesmaximal, characteristic, complemented
2C2×#939C212yesmaximal, complemented
3C2×#930C211yesmaximal, characteristic, complemented
4C2×#930C211yesmaximal, characteristic, complemented
5C2×#926C211yesmaximal, characteristic, complemented
6C22×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#5565C2116yesmaximal, direct product factor
8C2×#898C212yesmaximal, complemented
9C2×#883C212yesmaximal, complemented
10C2×#67⋊C4C212yesmaximal, complemented
11C2×C4×D12C211yesmaximal, characteristic, complemented
12C2×#870C211yesmaximal, characteristic, complemented
13C2×#553—31nomaximal, Sylow
14C22×C3×D4C2211yescharacteristic
15C23×C3×C4C2211yescharacteristic
16C23×C3×C4C2211yescharacteristic
17C22×#67C2212yes—
18C22×#67C2212yes—
19C23×Dic3C2212yes—
20C22×D12C2211yescharacteristic
21C22×C4×D3C2212yes—
22C3×C4×D4C2218yescomplemented
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C42C2211yescharacteristic
27C2×#217C2212yes—
28#939C22116yescomplemented
29#930C2218yescomplemented
30#930C2218yescomplemented
31C2×(C2×D3)⋊C4C2212yes—
32C2×(C2×D3)⋊C4C2212yes—
33C2×(C2×D3)⋊C4C2212yes—
34#926C2218yescomplemented
35C2×C4⋊Dic3C2211yescharacteristic
36C2×C4⋊Dic3C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic
38C2×C4⋊Dic3C2212yes—
39C2×C4⋊Dic3C2218yescomplemented
40C2×Dic3⋊C4C2212yes—
41C2×Dic3⋊C4C2212yes—
42C2×C4×Dic3C2212yes—
43#898C22116yescomplemented
44#883C22116yescomplemented
45#67⋊C4C22116yescomplemented
46C4×D12C2218yescomplemented
47#870C2218yescomplemented
48#553—316no—
49C2×#126—32no—
50C2×#125—31no—
51C2×#123—32no—
52C2×#121—32no—
53C2×#121—31no—
54C2×#121—31no—
55C2×C4×D4—32no—
56C2×C4×D4—31no—
57C2×C4×D4D311yescharacteristic, complemented
58C2×#117—31no—
59C22×C4⋊C4—31no—
60C24×C3C2311yescharacteristic
61C24×C3C2311yescharacteristic
62C23×D3C2312yes—
63C2×C3×D4—24no—
64C2×C3×D4C2314yes—
65C22×C3×C4—24no—
66C22×C3×C4—24no—
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2311yescharacteristic
69C22×C3×C4C2311yescharacteristic
70C22×C3×C4C2311yescharacteristic
71C22×C3×C4C2314yes—
72C22×C3×C4C2314yes—
73C22×C3×C4C2311yescharacteristic
74C2×#67—28no—
75C2×#67—44no—
76C2×#67C2318yes—
77C2×#67C2318yes—
78C22×Dic3—44no—
79C22×Dic3C2312yes—
80C22×Dic3C2312yes—
81C22×Dic3C2312yes—
82C22×Dic3C2318yes—
83C2×D12—24no—
84C2×D12C2314yes—
85C2×C4×D3—44no—
86C2×C4×D3C2318yes—
87C3×C4⋊C4C2314yes—
88C3×C22⋊C4C2314yes—
89C3×C22⋊C4C2314yes—
90C3×C42C2314yes—
91#217C2318yes—
92(C2×D3)⋊C4C2318yes—
93(C2×D3)⋊C4C2318yes—
94(C2×D3)⋊C4C2318yes—
95C4⋊Dic3C2314yes—
96C4⋊Dic3C2314yes—
97C4⋊Dic3C2314yes—
98C4⋊Dic3C2318yes—
99Dic3⋊C4C2318yes—
100Dic3⋊C4C2318yes—
101C4×Dic3C2318yes—
102C22×D4—32no—
103C22×D4—32no—
104C22×D4—31no—
105C22×D4C2×D311yescharacteristic
106C23×C4—32no—
107C23×C4—32no—
108C23×C4C2×D311yescharacteristic
109C23×C4C2×D311yescharacteristic
110#126—316no—
111#125—38no—
112#123—316no—
113#121—316no—
114#121—38no—
115#121—38no—
116C4×D4—316no—
117C4×D4—38no—
118C4×D4C2×D318yescomplemented
119#117—38no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—32no—
123C2×C4⋊C4—38no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—32no—
126C2×C4⋊C4—32no—
127C2×C4⋊C4C2×D311yescharacteristic
128C2×C22⋊C4—32no—
129C2×C22⋊C4—32no—
130C2×C22⋊C4—32no—
131C2×C22⋊C4—32no—
132C2×C22⋊C4C2×D311yescharacteristic
133C2×C22⋊C4C2×D311yescharacteristic
134C2×C42—32no—
135C2×C42C2×D311yescharacteristic
136C23×C3—22no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3#4114yes—
142C23×C3C2412yes—
143C23×C3C2412yes—
144C23×C3C2411yescharacteristic
145C22×D3—24no—
146C22×D3—24no—
147C22×D3—24no—
148C22×D3C2414yes—
149C3×D4—28no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—28no—
155C2×C3×C4—28no—
156C2×C3×C4#4114yes—
157C2×C3×C4C2412yes—
158C2×C3×C4C2412yes—
159C2×C3×C4C2412yes—
160C2×C3×C4C2412yes—
161C2×C3×C4C2412yes—
162#67—216no—
163#67—48no—
164C2×Dic3—24no—
165C2×Dic3—24no—
166C2×Dic3—24no—
167C2×Dic3—48no—
168C2×Dic3C2414yes—
169C2×Dic3C2414yes—
170C2×Dic3C2414yes—
171D12—28no—
172C4×D3—48no—
173C24—32no—
174C24C22×D311yescharacteristic
175C24C22×D311yescharacteristic
176C2×D4—24no—
177C2×D4—38no—
178C2×D4—38no—
179C2×D4—34no—
180C2×D4—68no—
181C2×D4—64no—
182C2×D4—124no—
183C2×D4C22×D314yes—
184C22×C4—24no—
185C22×C4—24no—
186C22×C4—32no—
187C22×C4—32no—
188C22×C4—32no—
189C22×C4—38no—
190C22×C4—38no—
191C22×C4—124no—
192C22×C4—124no—
193C22×C4C22×D314yes—
194C22×C4C22×D314yes—
195C22×C4C22×D311yescharacteristic
196C22×C4C22×D311yescharacteristic
197C22×C4C22×D311yescharacteristic
198C22×C4C22×D311yescharacteristic
199C22×C4C22×D311yescharacteristic
200C4⋊C4—34no—
201C4⋊C4—34no—
202C4⋊C4—38no—
203C4⋊C4—34no—
204C4⋊C4—38no—
205C4⋊C4—38no—
206C4⋊C4C22×D314yes—
207C22⋊C4—38no—
208C22⋊C4—38no—
209C22⋊C4—38no—
210C22⋊C4—38no—
211C22⋊C4C22×D314yes—
212C22⋊C4C22×D314yes—
213C42—38no—
214C42C22×D314yes—
215C22×C3—28no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—22no—
219C22×C3—22no—
220C22×C3—22no—
221C22×C3—44no—
222C22×C3C2×#4114yes—
223C22×C3C2×#4112yes—
224C22×C3C2×#4112yes—
225C22×C3H(5,𝔽2)12yes—
226C22×C3C2511yescharacteristic
227C2×D3—24no—
228C2×D3—24no—
229C2×D3—24no—
230C2×D3—48no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4C2×#4114yes—
236Dic3—24no—
237Dic3—24no—
238Dic3—24no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—34no—
245C23—64no—
246C23—64no—
247C23—64no—
248C23#23514yes—
249C23C23×D312yes—
250C23C23×D312yes—
251C23C23×D311yescharacteristic
252D4—28no—
253D4—616no—
254D4—68no—
255D4—128no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—28no—
261C2×C4—28no—
262C2×C4—34no—
263C2×C4—34no—
264C2×C4—34no—
265C2×C4—64no—
266C2×C4—64no—
267C2×C4—64no—
268C2×C4—128no—
269C2×C4—128no—
270C2×C4#23714yes—
271C2×C4C23×D312yes—
272C2×C4C23×D312yes—
273C2×C4C23×D312yes—
274C2×C4C23×D312yes—
275C2×C4C23×D312yes—
276C2×C3—24no—
277C2×C3—42no—
278C2×C3#55314yescomplemented
279C2×C3C22×#4111yescharacteristic
280C2×C3C22×#4111yescharacteristic
281C2×C3C2×H(5,𝔽2)11yescharacteristic
282D3—44no—
283C22—22no—
284C22—22no—
285C22—28no—
286C22—22no—
287C22—22no—
288C22—22no—
289C22—44no—
290C22—64no—
291C22—64no—
292C22—64no—
293C22—128no—
294C22C2×#23514yes—
295C22C2×#23512yes—
296C22C2×#23712yes—
297C22#100912yes—
298C22C24×D311yescharacteristic
299C4—22no—
300C4—22no—
301C4—22no—
302C4—22no—
303C4—64no—
304C4—64no—
305C4—64no—
306C4C2×#23714yes—
307C3C2×#55311yescharacteristic, complemented, Sylow
308C2—24no—
309C2—42no—
310C2—124no—
311C2#556514yesdirect product factor
312C2C22×#23511yescharacteristic
313C2C22×#23711yescharacteristic
314C2C2×#100911yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC22×C3C25
FrattiniC22C24×D3
FittingC2×C3×C4×D4C2
socleC23×C3C24
hypercenterC2×C4×D4D3
hypocenterC3C2×#553
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×D4C2semidirect
2. C2×#939C2semidirect
3. C2×#930C2semidirect
4. C2×#930C2semidirect
5. C2×#926C2semidirect
6. C22×C4⋊Dic3C2semidirect
7. #5565C2direct (N × H)
8. C2×#898C2semidirect
9. C2×#883C2semidirect
10. C2×#67⋊C4C2semidirect
11. C2×C4×D12C2semidirect
12. C2×#870C2semidirect
22. C3×C4×D4C22semidirect
28. #939C22semidirect
29. #930C22semidirect
30. #930C22semidirect
34. #926C22semidirect
39. C2×C4⋊Dic3C22semidirect
43. #898C22semidirect
44. #883C22semidirect
45. #67⋊C4C22semidirect
46. C4×D12C22semidirect
47. #870C22semidirect
57. C2×C4×D4D3semidirect
118. C4×D4C2×D3semidirect
278. C2×C3#553semidirect
307. C3C2×#553semidirect
311. C2#5565direct (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×D42
C2×#9392
C2×#9302
C2×#9262
C22×C4⋊Dic32
#55652
C2×#8982
C2×#8832
C2×#67⋊C42
C2×C4×D122
C2×#8702
C2×#5533