Finite Group Databasebeta

← #10596124

Subgroups — #10596124

Order 576.

1038 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1038 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×#123C211yesmaximal, characteristic, complemented
4#64146C211yesmaximal, characteristic, complemented
5#64138C211yesmaximal, characteristic, complemented
6#64104C212yesmaximal, complemented
7#64103C212yesmaximal, complemented
8#64093C212yesmaximal, complemented
9#64091C212yesmaximal, complemented
10#64091C211yesmaximal, characteristic, complemented
11C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
12#5653—34nomaximal
13C23×C3⋊D3C2211yescharacteristic
14C23×C3⋊D3C2211yescharacteristic, complemented
15C23×C3⋊D3C2211yescharacteristic, complemented
16C2×C32×D4C2211yescharacteristic
17C22×C32×C4C2211yescharacteristic, complemented
18C2×#638C2211yescharacteristic
19C2×#638C2211yescharacteristic, complemented
20C2×#638C2212yescomplemented
21D4×C3⋊D3—24no—
22C2×C4⋊(C3⋊D3)—24no—
23C2×C4⋊(C3⋊D3)C2212yes—
24C2×C4⋊(C3⋊D3)C2212yescomplemented
25C2×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
26C2×C4⋊(C3⋊D3)C2212yescomplemented
27C2×C4×C3⋊D3C2211yescharacteristic, complemented
28C2×C4×C3⋊D3C2212yescomplemented
29C32×C4⋊C4C2212yescomplemented
30C32×C22⋊C4C2211yescharacteristic, complemented
31C32×C22⋊C4C2212yescomplemented
32#3757C2211yescharacteristic, complemented
33#3752C2212yescomplemented
34#3752C2212yescomplemented
35#3752C2212yescomplemented
36#3752C2212yescomplemented
37#3750C2212yescomplemented
38C2×D3×D4—34no—
39C22×D12—34no—
40C3×#123D314yescomplemented
41#937—34no—
42#929—34no—
43#895—38no—
44#894—38no—
45#884—38no—
46#882—38no—
47#882—34no—
48D3×C22⋊C4—34no—
49C23×C32C2311yescharacteristic
50C23×C32C2311yescharacteristic
51C22×C3⋊D3—22no—
52C22×C3⋊D3—22no—
53C22×C3⋊D3—42no—
54C22×C3⋊D3—42no—
55C22×C3⋊D3—42no—
56C22×C3⋊D3D414yescomplemented
57C22×C3⋊D3C2312yes—
58C22×C3⋊D3C2311yescharacteristic
59C22×C3⋊D3C2311yescharacteristic
60C22×C3⋊D3C2311yescharacteristic
61C32×D4—22no—
62C2×C32×C4—22no—
63C2×C32×C4C2312yes—
64C2×C32×C4C2312yes—
65C2×C32×C4C2311yescharacteristic
66#638—22no—
67#638—22no—
68#638—42no—
69C2×C3⋊Dic3C2311yescharacteristic
70C2×C3⋊Dic3C2312yes—
71C4⋊(C3⋊D3)—24no—
72C4⋊(C3⋊D3)—24no—
73C4⋊(C3⋊D3)—22no—
74C4⋊(C3⋊D3)—42no—
75C4×C3⋊D3—22no—
76C4×C3⋊D3—42no—
77#534—91noSylow
78C23×D3—34no—
79C23×D3—34no—
80C23×D3—34no—
81C2×C3×D4C2×D314yes—
82C22×C3×C4C2×D314yescomplemented
83C2×#67—34no—
84C2×#67—34no—
85C2×#67—38no—
86D3×D4—616no—
87C2×D12—38no—
88C2×D12—38no—
89C2×D12—34no—
90C2×D12—38no—
91C2×D12—616no—
92C2×C4×D3—34no—
93C2×C4×D3—38no—
94C3×C4⋊C4C2×D318yescomplemented
95C3×C22⋊C4C2×D314yescomplemented
96C3×C22⋊C4C2×D318yescomplemented
97#217—34no—
98(C2×D3)⋊C4—38no—
99(C2×D3)⋊C4—38no—
100(C2×D3)⋊C4—38no—
101(C2×D3)⋊C4—38no—
102Dic3⋊C4—38no—
103C22×C32—22no—
104C22×C32—21no—
105C22×C32—22no—
106C22×C32C2×D412yes—
107C22×C32C2411yescharacteristic
108C2×C3⋊D3—22no—
109C2×C3⋊D3—22no—
110C2×C3⋊D3—22no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3—22no—
113C2×C3⋊D3—22no—
114C2×C3⋊D3—22no—
115C2×C3⋊D3—44no—
116C2×C3⋊D3—44no—
117C2×C3⋊D3—44no—
118C2×C3⋊D3C2×D412yes—
119C2×C3⋊D3C2×D412yes—
120C32×C4—21no—
121C32×C4—22no—
122C32×C4—22no—
123C3⋊Dic3—21no—
124C3⋊Dic3—22no—
125C22×D4—91no—
126C22×D4—91no—
127#123—91no—
128#123—92no—
129#123C3⋊D311yescharacteristic, complemented
130#121—92no—
131#121—92no—
132#120—91no—
133#120—92no—
134#120—91no—
135C2×C22⋊C4—91no—
136C23×C3C22×D314yes—
137C23×C3C22×D314yes—
138C22×D3—34no—
139C22×D3—34no—
140C22×D3—316no—
141C22×D3—38no—
142C22×D3—34no—
143C22×D3—68no—
144C22×D3—68no—
145C22×D3—128no—
146C22×D3—128no—
147C22×D3—128no—
148C3×D4—28no—
149C2×C3×C4—28no—
150C2×C3×C4C22×D318yes—
151C2×C3×C4C22×D318yes—
152C2×C3×C4C22×D314yes—
153#67—68no—
154#67—68no—
155#67—128no—
156C2×Dic3—34no—
157C2×Dic3—38no—
158D12—616no—
159D12—616no—
160D12—68no—
161D12—128no—
162C4×D3—68no—
163C4×D3—128no—
164C2×C32—22no—
165C2×C32—41no—
166C2×C32C22×D411yescharacteristic
167C2×C32H(5,𝔽2)12yes—
168C3⋊D3—22no—
169C3⋊D3—22no—
170C3⋊D3—42no—
171C3⋊D3—41no—
172C24—91no—
173C24—91no—
174C24—91no—
175C2×D4—91no—
176C2×D4—91no—
177C2×D4—92no—
178C2×D4—92no—
179C2×D4—91no—
180C2×D4—92no—
181C2×D4—92no—
182C2×D4—184no—
183C2×D4—184no—
184C2×D4C2×C3⋊D311yescharacteristic
185C22×C4—91no—
186C22×C4—92no—
187C22×C4C2×C3⋊D311yescharacteristic, complemented
188C4⋊C4—92no—
189C4⋊C4C2×C3⋊D312yescomplemented
190C22⋊C4—91no—
191C22⋊C4—92no—
192C22⋊C4—92no—
193C22⋊C4—92no—
194C22⋊C4—92no—
195C22⋊C4C2×C3⋊D311yescharacteristic, complemented
196C22⋊C4C2×C3⋊D312yescomplemented
197C22×C3—28no—
198C22×C3—24no—
199C22×C3—28no—
200C22×C3D3×D418yes—
201C22×C3C23×D314yes—
202C2×D3—38no—
203C2×D3—38no—
204C2×D3—68no—
205C2×D3—68no—
206C2×D3—68no—
207C2×D3—64no—
208C2×D3—68no—
209C2×D3—68no—
210C2×D3—68no—
211C2×D3—1216no—
212C2×D3—1216no—
213C2×D3—1216no—
214C3×C4—24no—
215C3×C4—28no—
216C3×C4—28no—
217Dic3—64no—
218Dic3—68no—
219C32#53411yescharacteristic, complemented, Sylow
220C23—91no—
221C23—91no—
222C23—94no—
223C23—91no—
224C23—92no—
225C23—182no—
226C23—182no—
227C23—362no—
228C23—362no—
229C23—362no—
230C23C22×C3⋊D311yescharacteristic
231C23C22×C3⋊D311yescharacteristic
232D4—22no—
233D4—184no—
234D4—184no—
235D4—182no—
236D4—182no—
237D4—182no—
238D4—362no—
239D4—362no—
240C2×C4—22no—
241C2×C4—92no—
242C2×C4—91no—
243C2×C4—182no—
244C2×C4—362no—
245C2×C4C22×C3⋊D312yes—
246C2×C4C22×C3⋊D311yescharacteristic
247C2×C4C22×C3⋊D312yes—
248C2×C3—28no—
249C2×C3—44no—
250C2×C3C2×D3×D414yes—
251C2×C3#100918yes—
252D3—68no—
253D3—68no—
254D3—128no—
255D3—124no—
256C22—22no—
257C22—21no—
258C22—22no—
259C22—92no—
260C22—92no—
261C22—182no—
262C22—182no—
263C22—182no—
264C22—181no—
265C22—182no—
266C22—182no—
267C22—182no—
268C22—364no—
269C22—364no—
270C22—364no—
271C22D4×C3⋊D312yes—
272C22C23×C3⋊D311yescharacteristic
273C4—22no—
274C4—21no—
275C4—22no—
276C4—181no—
277C4—182no—
278C3#565314yescomplemented
279C2—22no—
280C2—41no—
281C2—182no—
282C2—182no—
283C2—362no—
284C2—361no—
285C2C2×D4×C3⋊D311yescharacteristic
286C2#6436612yes—
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#123C2
hypercenter#123C3⋊D3
hypocenterC32#534
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C2×D4×C3⋊D3C2semidirect
2. C22×C4⋊(C3⋊D3)C2semidirect
3. C32×#123C2semidirect
4. #64146C2semidirect
5. #64138C2semidirect
6. #64104C2semidirect
7. #64103C2semidirect
8. #64093C2semidirect
9. #64091C2semidirect
10. #64091C2semidirect
11. C22⋊C4×C3⋊D3C2semidirect
14. C23×C3⋊D3C22semidirect
15. C23×C3⋊D3C22semidirect
17. C22×C32×C4C22semidirect
19. C2×#638C22semidirect
20. C2×#638C22semidirect
24. C2×C4⋊(C3⋊D3)C22semidirect
25. C2×C4⋊(C3⋊D3)C22semidirect
26. C2×C4⋊(C3⋊D3)C22semidirect
27. C2×C4×C3⋊D3C22semidirect
28. C2×C4×C3⋊D3C22semidirect
29. C32×C4⋊C4C22semidirect
30. C32×C22⋊C4C22semidirect
31. C32×C22⋊C4C22semidirect
32. #3757C22semidirect
33. #3752C22semidirect
34. #3752C22semidirect
35. #3752C22semidirect
36. #3752C22semidirect
37. #3750C22semidirect
40. C3×#123D3semidirect
56. C22×C3⋊D3D4semidirect
82. C22×C3×C4C2×D3semidirect
94. C3×C4⋊C4C2×D3semidirect
95. C3×C22⋊C4C2×D3semidirect
96. C3×C22⋊C4C2×D3semidirect
129. #123C3⋊D3semidirect
187. C22×C4C2×C3⋊D3semidirect
189. C4⋊C4C2×C3⋊D3semidirect
195. C22⋊C4C2×C3⋊D3semidirect
196. C22⋊C4C2×C3⋊D3semidirect
219. C32#534semidirect
278. C3#5653semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D32
C22×C4⋊(C3⋊D3)2
C32×#1232
#641462
#641382
#641042
#641032
#640932
#640912
C22⋊C4×C3⋊D32
#56533