Finite Group Databasebeta

← #10596078

Subgroups — #10596078

Order 576.

1002 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1002 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
2C2×#3828C211yesmaximal, characteristic, complemented
3C32×#121C211yesmaximal, characteristic, complemented
4#64149C212yesmaximal, complemented
5#64147C211yesmaximal, characteristic, complemented
6#64146C212yesmaximal, complemented
7#64139C211yesmaximal, characteristic, complemented
8#64104C211yesmaximal, characteristic, complemented
9#64101C211yesmaximal, characteristic, complemented
10#64094C212yesmaximal, complemented
11#64091C212yesmaximal, complemented
12#5607—34nomaximal
13C23×C3⋊D3C2212yes—
14C2×C32×D4C2211yescharacteristic
15C2×C32×D4C2212yescomplemented
16C22×C32×C4C2211yescharacteristic, complemented
17C2×#638C2212yes—
18C2×#638C2211yescharacteristic
19C2×#638C2211yescharacteristic, complemented
20C2×#638C2212yescomplemented
21C2×#638C2212yescomplemented
22D4×C3⋊D3—24no—
23#3828—24no—
24C2×C4⋊(C3⋊D3)C2211yescharacteristic
25C2×C4⋊(C3⋊D3)C2211yescharacteristic
26C2×C4⋊(C3⋊D3)C2212yescomplemented
27C2×C4×C3⋊D3C2211yescharacteristic
28C2×C4×C3⋊D3C2211yescharacteristic, complemented
29C2×#634C2211yescharacteristic, complemented
30C32×C4⋊C4C2211yescharacteristic, complemented
31C32×C22⋊C4C2212yescomplemented
32#3757C2212yescomplemented
33#3752C2212yes—
34#3752C2212yescomplemented
35#3752C2212yescomplemented
36C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
37C4×C3⋊Dic3C2212yescomplemented
38C2×D3×D4—34no—
39C2×#235—34no—
40C3×#121D314yescomplemented
41#940—38no—
42#938—34no—
43#937—38no—
44#930—34no—
45#895—34no—
46#892—34no—
47#885—38no—
48#882—38no—
49C23×C32C2312yes—
50C23×C32C2311yescharacteristic
51C22×C3⋊D3—24no—
52C22×C3⋊D3—42no—
53C22×C3⋊D3—42no—
54C22×C3⋊D3C2312yes—
55C22×C3⋊D3C2311yescharacteristic
56C22×C3⋊D3C2311yescharacteristic
57C32×D4—22no—
58C32×D4—42no—
59C2×C32×C4—22no—
60C2×C32×C4C2312yes—
61C2×C32×C4C2311yescharacteristic
62C2×C32×C4C2311yescharacteristic
63#638—22no—
64#638—22no—
65#638—24no—
66#638—42no—
67#638—42no—
68C2×C3⋊Dic3C2312yes—
69C2×C3⋊Dic3C2311yescharacteristic
70C2×C3⋊Dic3C2311yescharacteristic
71C4⋊(C3⋊D3)—22no—
72C4⋊(C3⋊D3)—22no—
73C4⋊(C3⋊D3)—42no—
74C4×C3⋊D3—22no—
75C4×C3⋊D3D414yescomplemented
76#634—22no—
77#535—91noSylow
78C23×D3—38no—
79C2×C3×D4C2×D314yes—
80C2×C3×D4C2×D318yescomplemented
81C22×C3×C4C2×D314yescomplemented
82C2×#67—38no—
83C2×#67—34no—
84C2×#67—34no—
85C2×#67—38no—
86C2×#67—38no—
87D3×D4—616no—
88#235—616no—
89C2×D12—34no—
90C2×D12—34no—
91C2×D12—38no—
92C2×C4×D3—34no—
93C2×C4×D3—34no—
94C2×Dic6—34no—
95C3×C4⋊C4C2×D314yescomplemented
96C3×C22⋊C4C2×D318yescomplemented
97#217—38no—
98(C2×D3)⋊C4—38no—
99(C2×D3)⋊C4—38no—
100(C2×D3)⋊C4—38no—
101C4⋊Dic3—34no—
102C4×Dic3—38no—
103C22×C32—21no—
104C22×C32—22no—
105C22×C32—21no—
106C22×C32—22no—
107C22×C32—21no—
108C22×C32—22no—
109C22×C32C2411yescharacteristic
110C2×C3⋊D3—21no—
111C2×C3⋊D3—21no—
112C2×C3⋊D3—21no—
113C2×C3⋊D3—22no—
114C2×C3⋊D3—21no—
115C2×C3⋊D3—22no—
116C2×C3⋊D3—21no—
117C2×C3⋊D3—22no—
118C2×C3⋊D3—48no—
119C2×C3⋊D3C2×D412yes—
120C32×C4—21no—
121C32×C4—22no—
122C32×C4C2×D412yes—
123C3⋊Dic3—21no—
124C3⋊Dic3—22no—
125C3⋊Dic3C2×D412yes—
126C2×#41—91no—
127C22×D4—91no—
128#127—92no—
129#124—92no—
130#121—91no—
131#121—91no—
132#121—91no—
133#121C3⋊D311yescharacteristic, complemented
134#120—92no—
135#120—92no—
136#117—91no—
137C23×C3C22×D318yes—
138C23×C3C22×D314yes—
139C22×D3—34no—
140C22×D3—38no—
141C22×D3—34no—
142C22×D3—616no—
143C22×D3—128no—
144C22×D3—128no—
145C3×D4—28no—
146C3×D4—48no—
147C2×C3×C4—28no—
148C2×C3×C4C22×D318yes—
149C2×C3×C4C22×D314yes—
150C2×C3×C4C22×D314yes—
151#67—68no—
152#67—616no—
153#67—68no—
154#67—128no—
155#67—128no—
156C2×Dic3—34no—
157C2×Dic3—38no—
158C2×Dic3—34no—
159D12—68no—
160D12—68no—
161D12—128no—
162C4×D3—316no—
163C4×D3—68no—
164Dic6—68no—
165C2×C32—42no—
166C2×C32—41no—
167C2×C32C22×D411yescharacteristic
168C2×C32H(5,𝔽2)11yescharacteristic
169C2×C32H(5,𝔽2)11yescharacteristic
170C3⋊D3—22no—
171C3⋊D3—41no—
172C3⋊D3—42no—
173C24—92no—
174#41—184no—
175C2×Q8—91no—
176C2×D4—91no—
177C2×D4—92no—
178C2×D4—91no—
179C2×D4—92no—
180C2×D4—92no—
181C2×D4—91no—
182C2×D4—91no—
183C2×D4—92no—
184C2×D4—184no—
185C2×D4C2×C3⋊D311yescharacteristic
186C2×D4C2×C3⋊D312yescomplemented
187C22×C4—91no—
188C22×C4—91no—
189C22×C4C2×C3⋊D311yescharacteristic, complemented
190C4⋊C4—91no—
191C4⋊C4C2×C3⋊D311yescharacteristic, complemented
192C22⋊C4—92no—
193C22⋊C4—92no—
194C22⋊C4—92no—
195C22⋊C4—92no—
196C22⋊C4C2×C3⋊D312yescomplemented
197C42—92no—
198C22×C3—24no—
199C22×C3—28no—
200C22×C3—24no—
201C22×C3—28no—
202C22×C3—24no—
203C22×C3—28no—
204C22×C3C23×D314yes—
205C2×D3—38no—
206C2×D3—64no—
207C2×D3—64no—
208C2×D3—64no—
209C2×D3—68no—
210C2×D3—64no—
211C2×D3—68no—
212C2×D3—64no—
213C2×D3—68no—
214C2×D3—1232no—
215C3×C4—24no—
216C3×C4—28no—
217C3×C4D3×D418yes—
218Dic3—38no—
219Dic3—64no—
220Dic3—68no—
221C32#53511yescharacteristic, complemented, Sylow
222C23—91no—
223C23—92no—
224C23—91no—
225C23—184no—
226C23—362no—
227C23—362no—
228C23C22×C3⋊D312yes—
229C23C22×C3⋊D311yescharacteristic
230Q8—182no—
231D4—22no—
232D4—42no—
233D4—182no—
234D4—184no—
235D4—182no—
236D4—182no—
237D4—182no—
238D4—362no—
239D4—362no—
240D4—362no—
241C2×C4—22no—
242C2×C4—91no—
243C2×C4—94no—
244C2×C4—92no—
245C2×C4—91no—
246C2×C4—182no—
247C2×C4C22×C3⋊D312yes—
248C2×C4C22×C3⋊D311yescharacteristic
249C2×C4C22×C3⋊D311yescharacteristic
250C2×C3—48no—
251C2×C3—44no—
252C2×C3C2×D3×D414yes—
253C2×C3#100414yes—
254C2×C3#100914yes—
255D3—68no—
256D3—128no—
257D3—124no—
258C22—21no—
259C22—22no—
260C22—21no—
261C22—22no—
262C22—21no—
263C22—22no—
264C22—92no—
265C22—181no—
266C22—181no—
267C22—181no—
268C22—182no—
269C22—181no—
270C22—182no—
271C22—181no—
272C22—182no—
273C22—368no—
274C22C23×C3⋊D311yescharacteristic
275C4—21no—
276C4—22no—
277C4—92no—
278C4—181no—
279C4—182no—
280C4D4×C3⋊D312yes—
281C3#560714yescomplemented
282C2—42no—
283C2—41no—
284C2—182no—
285C2—362no—
286C2—361no—
287C2C2×D4×C3⋊D311yescharacteristic
288C2#6436111yescharacteristic
289C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#121C2
hypercenter#121C3⋊D3
hypocenterC32#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×C3⋊D3C2semidirect
2. C2×#3828C2semidirect
3. C32×#121C2semidirect
4. #64149C2semidirect
5. #64147C2semidirect
6. #64146C2semidirect
7. #64139C2semidirect
8. #64104C2semidirect
9. #64101C2semidirect
10. #64094C2semidirect
11. #64091C2semidirect
15. C2×C32×D4C22semidirect
16. C22×C32×C4C22semidirect
19. C2×#638C22semidirect
20. C2×#638C22semidirect
21. C2×#638C22semidirect
26. C2×C4⋊(C3⋊D3)C22semidirect
28. C2×C4×C3⋊D3C22semidirect
29. C2×#634C22semidirect
30. C32×C4⋊C4C22semidirect
31. C32×C22⋊C4C22semidirect
32. #3757C22semidirect
34. #3752C22semidirect
35. #3752C22semidirect
36. C4⋊(C3⋊Dic3)C22semidirect
37. C4×C3⋊Dic3C22semidirect
40. C3×#121D3semidirect
75. C4×C3⋊D3D4semidirect
80. C2×C3×D4C2×D3semidirect
81. C22×C3×C4C2×D3semidirect
95. C3×C4⋊C4C2×D3semidirect
96. C3×C22⋊C4C2×D3semidirect
133. #121C3⋊D3semidirect
186. C2×D4C2×C3⋊D3semidirect
189. C22×C4C2×C3⋊D3semidirect
191. C4⋊C4C2×C3⋊D3semidirect
196. C22⋊C4C2×C3⋊D3semidirect
221. C32#535semidirect
281. C3#5607semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊D32
C2×#38282
C32×#1212
#641492
#641472
#641462
#641392
#641042
#641012
#640942
#640912
#56073