Finite Group Databasebeta

← #10596083

Subgroups — #10596083

Order 576.

1002 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 193 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#64147C212yesmaximal, complemented
5#64146C212yesmaximal, complemented
6#64145C211yesmaximal, characteristic, complemented
7C4×#638C211yesmaximal, characteristic, complemented
8#64106C211yesmaximal, characteristic, complemented
9#64102C211yesmaximal, characteristic, complemented
10#64092C212yesmaximal, complemented
11C22⋊C4×C3⋊D3C212yesmaximal, complemented
12#5612—34nomaximal
13C23×C3⋊D3C2212yes—
14C2×C32×D4C2211yescharacteristic
15C2×C32×D4C2212yescomplemented
16C22×C32×C4C2211yescharacteristic, complemented
17C2×#638C2212yes—
18C2×#638C2211yescharacteristic
19C2×#638C2212yescomplemented
20D4×C3⋊D3—24no—
21#3828—24no—
22C2×C4⋊(C3⋊D3)C2211yescharacteristic
23C2×C4×C3⋊D3C2211yescharacteristic
24C2×C4×C3⋊D3C2212yescomplemented
25C2×C4×C3⋊D3C2212yescomplemented
26C2×#634C2211yescharacteristic, complemented
27C32×C4⋊C4C2211yescharacteristic, complemented
28C32×C22⋊C4C2212yescomplemented
29#3757C2212yescomplemented
30#3757C2212yescomplemented
31#3757C2211yescharacteristic, complemented
32#3752C2211yescharacteristic
33#3752C2212yes—
34#3752C2212yescomplemented
35C4⋊(C3⋊Dic3)C2212yescomplemented
36#3750C2211yescharacteristic, complemented
37C4×C3⋊Dic3C2211yescharacteristic, complemented
38C2×D3×D4—34no—
39C2×#235—34no—
40C3×#121D314yescomplemented
41#938—38no—
42#937—38no—
43#936—34no—
44C4×#67—34no—
45#897—34no—
46#893—34no—
47#883—38no—
48D3×C22⋊C4—38no—
49C23×C32C2312yes—
50C23×C32C2311yescharacteristic
51C22×C3⋊D3—24no—
52C22×C3⋊D3—24no—
53C22×C3⋊D3—42no—
54C22×C3⋊D3C2312yes—
55C22×C3⋊D3C2311yescharacteristic
56C32×D4—22no—
57C32×D4—42no—
58C2×C32×C4—22no—
59C2×C32×C4C2311yescharacteristic
60C2×C32×C4C2312yes—
61C2×C32×C4C2311yescharacteristic
62#638—24no—
63#638—24no—
64#638—22no—
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2312yes—
69C4⋊(C3⋊D3)D414yescomplemented
70C4×C3⋊D3—24no—
71C4×C3⋊D3—22no—
72C4×C3⋊D3—42no—
73#634—22no—
74#546—91noSylow
75C23×D3—38no—
76C2×C3×D4C2×D314yes—
77C2×C3×D4C2×D318yescomplemented
78C22×C3×C4C2×D314yescomplemented
79C2×#67—38no—
80C2×#67—34no—
81C2×#67—38no—
82D3×D4—616no—
83#235—616no—
84C2×D12—34no—
85C2×C4×D3—38no—
86C2×C4×D3—34no—
87C2×C4×D3—38no—
88C2×Dic6—34no—
89C3×C4⋊C4C2×D314yescomplemented
90C3×C22⋊C4C2×D318yescomplemented
91#217—38no—
92#217—38no—
93#217—34no—
94(C2×D3)⋊C4—34no—
95(C2×D3)⋊C4—38no—
96(C2×D3)⋊C4—38no—
97C4⋊Dic3—38no—
98Dic3⋊C4—34no—
99C4×Dic3—34no—
100C22×C32—21no—
101C22×C32—22no—
102C22×C32—21no—
103C22×C32—22no—
104C22×C32—21no—
105C22×C32—22no—
106C22×C32C2411yescharacteristic
107C2×C3⋊D3—21no—
108C2×C3⋊D3—22no—
109C2×C3⋊D3—22no—
110C2×C3⋊D3—21no—
111C2×C3⋊D3—48no—
112C2×C3⋊D3C2×D414yes—
113C2×C3⋊D3#4112yescomplemented
114C32×C4—21no—
115C32×C4—22no—
116C32×C4C2×D412yes—
117C3⋊Dic3—22no—
118C3⋊Dic3—21no—
119C3⋊Dic3—21no—
120C3⋊Dic3—21no—
121C2×#41—91no—
122C22×D4—91no—
123#124—91no—
124#123—92no—
125C22⋊Q8—91no—
126#121—92no—
127#121C3⋊D311yescharacteristic, complemented
128#120—92no—
129C4×D4—91no—
130C4×D4—91no—
131C2×C22⋊C4—92no—
132C23×C3C22×D318yes—
133C23×C3C22×D314yes—
134C22×D3—34no—
135C22×D3—38no—
136C22×D3—616no—
137C22×D3—616no—
138C22×D3—128no—
139C3×D4—28no—
140C3×D4—48no—
141C2×C3×C4—28no—
142C2×C3×C4C22×D314yes—
143C2×C3×C4C22×D318yes—
144C2×C3×C4C22×D314yes—
145#67—616no—
146#67—616no—
147#67—68no—
148C2×Dic3—34no—
149C2×Dic3—34no—
150C2×Dic3—34no—
151C2×Dic3—38no—
152D12—316no—
153C4×D3—616no—
154C4×D3—68no—
155C4×D3—128no—
156Dic6—68no—
157C2×C32—42no—
158C2×C32—41no—
159C2×C32C22×D411yescharacteristic
160C2×C32C2×#4111yescharacteristic
161C2×C32H(5,𝔽2)11yescharacteristic
162C3⋊D3—22no—
163C3⋊D3—24no—
164C24—92no—
165#41—184no—
166C2×Q8—91no—
167C2×D4—91no—
168C2×D4—92no—
169C2×D4—92no—
170C2×D4—91no—
171C2×D4—184no—
172C2×D4C2×C3⋊D311yescharacteristic
173C2×D4C2×C3⋊D312yescomplemented
174C22×C4—92no—
175C22×C4—91no—
176C22×C4—92no—
177C22×C4C2×C3⋊D311yescharacteristic, complemented
178C4⋊C4—91no—
179C4⋊C4—92no—
180C4⋊C4C2×C3⋊D311yescharacteristic, complemented
181C22⋊C4—92no—
182C22⋊C4—91no—
183C22⋊C4—92no—
184C22⋊C4—91no—
185C22⋊C4—92no—
186C22⋊C4—92no—
187C22⋊C4C2×C3⋊D312yescomplemented
188C42—91no—
189C22×C3—24no—
190C22×C3—28no—
191C22×C3—24no—
192C22×C3—28no—
193C22×C3—24no—
194C22×C3—28no—
195C22×C3C23×D314yes—
196C2×D3—316no—
197C2×D3—38no—
198C2×D3—64no—
199C2×D3—68no—
200C2×D3—64no—
201C2×D3—68no—
202C2×D3—1232no—
203C3×C4—24no—
204C3×C4—28no—
205C3×C4D3×D418yes—
206Dic3—68no—
207Dic3—64no—
208Dic3—64no—
209Dic3—64no—
210C32#54611yescharacteristic, complemented, Sylow
211C23—91no—
212C23—92no—
213C23—184no—
214C23—184no—
215C23—362no—
216C23C22×C3⋊D312yes—
217C23C22×C3⋊D311yescharacteristic
218Q8—182no—
219D4—22no—
220D4—42no—
221D4—94no—
222D4—184no—
223D4—184no—
224D4—182no—
225C2×C4—22no—
226C2×C4—91no—
227C2×C4—91no—
228C2×C4—92no—
229C2×C4—91no—
230C2×C4—184no—
231C2×C4—182no—
232C2×C4—362no—
233C2×C4C22×C3⋊D312yes—
234C2×C4C22×C3⋊D311yescharacteristic
235C2×C4C22×C3⋊D311yescharacteristic
236C2×C3—48no—
237C2×C3—44no—
238C2×C3C2×D3×D414yes—
239C2×C3#100414yes—
240C2×C3D3×#4114yes—
241D3—68no—
242D3—616no—
243C22—21no—
244C22—22no—
245C22—21no—
246C22—22no—
247C22—21no—
248C22—22no—
249C22—94no—
250C22—92no—
251C22—181no—
252C22—182no—
253C22—181no—
254C22—182no—
255C22—368no—
256C22C23×C3⋊D311yescharacteristic
257C4—21no—
258C4—22no—
259C4—181no—
260C4—182no—
261C4—181no—
262C4—181no—
263C4D4×C3⋊D312yes—
264C3#561214yescomplemented
265C2—42no—
266C2—41no—
267C2—182no—
268C2—184no—
269C2C2×D4×C3⋊D311yescharacteristic
270C2#6436111yescharacteristic
271C2#41×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#121C2
hypercenter#121C3⋊D3
hypocenterC32#546
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. #64147C2semidirect
5. #64146C2semidirect
6. #64145C2semidirect
7. C4×#638C2semidirect
8. #64106C2semidirect
9. #64102C2semidirect
10. #64092C2semidirect
11. C22⋊C4×C3⋊D3C2semidirect
15. C2×C32×D4C22semidirect
16. C22×C32×C4C22semidirect
19. C2×#638C22semidirect
24. C2×C4×C3⋊D3C22semidirect
25. C2×C4×C3⋊D3C22semidirect
26. C2×#634C22semidirect
27. C32×C4⋊C4C22semidirect
28. C32×C22⋊C4C22semidirect
29. #3757C22semidirect
30. #3757C22semidirect
31. #3757C22semidirect
34. #3752C22semidirect
35. C4⋊(C3⋊Dic3)C22semidirect
36. #3750C22semidirect
37. C4×C3⋊Dic3C22semidirect
40. C3×#121D3semidirect
69. C4⋊(C3⋊D3)D4semidirect
77. C2×C3×D4C2×D3semidirect
78. C22×C3×C4C2×D3semidirect
89. C3×C4⋊C4C2×D3semidirect
90. C3×C22⋊C4C2×D3semidirect
113. C2×C3⋊D3#41semidirect
127. #121C3⋊D3semidirect
173. C2×D4C2×C3⋊D3semidirect
177. C22×C4C2×C3⋊D3semidirect
180. C4⋊C4C2×C3⋊D3semidirect
187. C22⋊C4C2×C3⋊D3semidirect
210. C32#546semidirect
264. C3#5612semidirect

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
#641472
#641462
#641452
C4×#6382
#641062
#641022
#640922
C22⋊C4×C3⋊D32
#56123