Finite Group Databasebeta

← #10596249

Subgroups — #10596249

Order 576.

774 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 179 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
1#64366C211yesmaximal, characteristic, complemented
2#64364C211yesmaximal, characteristic, complemented
3C32×#137C211yesmaximal, characteristic, complemented
4#64159C211yesmaximal, characteristic, complemented
5C2×#3755C211yesmaximal, characteristic, complemented
6#64128C212yesmaximal, complemented
7#64127C212yesmaximal, complemented
8#64123C212yesmaximal, complemented
9SD16×C3⋊D3C212yesmaximal, complemented
10#64117C211yesmaximal, characteristic, complemented
11#64116C211yesmaximal, characteristic, complemented
12#5778—34nomaximal
13C32×#41C2211yescharacteristic
14C2×C32×Q8C2211yescharacteristic, complemented
15#3832—21no—
16#3832C2212yes—
17#3832C2212yescomplemented
18Q8×C3⋊D3—21no—
19Q8×C3⋊D3C2212yescomplemented
20D4×C3⋊D3—22no—
21D4×C3⋊D3C2212yes—
22#3828—22no—
23#3828—21no—
24#3828C2211yescharacteristic
25#3828C2211yescharacteristic, complemented
26C2×C4⋊(C3⋊D3)—21no—
27C2×C4⋊(C3⋊D3)C2211yescharacteristic
28C32×Q16C2212yescomplemented
29C32×SD16C2212yescomplemented
30C32×M16C2211yescharacteristic, complemented
31#3756C2211yescharacteristic, complemented
32#3755C2212yescomplemented
33#3755C2212yescomplemented
34#3755C2211yescharacteristic, complemented
35#3754C2211yescharacteristic, complemented
36#3753C2211yescharacteristic, complemented
37C2×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
38#3745C2212yescomplemented
39#3744C2212yescomplemented
40#3743C2212yescomplemented
41C8×C3⋊D3C2212yescomplemented
42#1009—34no—
43#1007—34no—
44C3×#137D314yescomplemented
45#950—34no—
46C2×#215—34no—
47#919—38no—
48#918—38no—
49#914—38no—
50D3×SD16—38no—
51#908—34no—
52#907—34no—
53C22×C3⋊D3—21no—
54C22×C3⋊D3—22no—
55C32×Q8—21no—
56C32×Q8C2312yes—
57C32×Q8C2311yescharacteristic
58C32×D4—21no—
59C32×D4C2311yescharacteristic
60C2×C32×C4—21no—
61C2×C32×C4—21no—
62C2×C32×C4C2311yescharacteristic
63#638—21no—
64#638—22no—
65#638D412yes—
66C4⋊(C3⋊D3)—22no—
67C4⋊(C3⋊D3)—22no—
68C4⋊(C3⋊D3)—21no—
69C4⋊(C3⋊D3)D411yescharacteristic
70C4⋊(C3⋊D3)C2312yes—
71C4⋊(C3⋊D3)C2311yescharacteristic
72C4×C3⋊D3—22no—
73C4×C3⋊D3—21no—
74C4×C3⋊D3—22no—
75C4×C3⋊D3—22no—
76C4×C3⋊D3C2312yes—
77#634—22no—
78#634D411yescharacteristic
79#634C2311yescharacteristic
80C32×C8C2312yes—
81C3⋊(C3⋊C8)C2311yescharacteristic
82C3⋊(C3⋊C8)C2311yescharacteristic
83#577—91noSylow
84C3×#41C2×D314yes—
85C2×C3×Q8C2×D314yescomplemented
86#239—38no—
87#239—38no—
88#239—64no—
89D3×Q8—38no—
90D3×Q8—64no—
91D3×D4—38no—
92D3×D4—68no—
93#235—34no—
94#235—34no—
95#235—64no—
96#235—68no—
97C2×D12—34no—
98C2×D12—64no—
99C3×Q16C2×D318yescomplemented
100C3×SD16C2×D318yescomplemented
101C3×M16C2×D314yescomplemented
102#216—34no—
103#215—38no—
104#215—38no—
105#215—34no—
106#214—34no—
107#213—34no—
108C2×C3⋊C8—34no—
109D24—38no—
110#204—38no—
111ΓL(1,𝔽25)—38no—
112D3×C8—38no—
113C22×C32—21no—
114C22×C32C2×D411yescharacteristic
115C2×C3⋊D3—21no—
116C2×C3⋊D3—22no—
117C2×C3⋊D3—82no—
118C2×C3⋊D3—81no—
119C2×C3⋊D3C2×D412yes—
120C32×C4—22no—
121C32×C4—21no—
122C32×C4C2×D411yescharacteristic
123C32×C4C2411yescharacteristic
124C3⋊Dic3—21no—
125C3⋊Dic3C2×D412yes—
1262−1+4—91no—
127H(5,𝔽2)—91no—
128#137—92no—
129#137C3⋊D311yescharacteristic, complemented
130AGL(1,ℤ/8)—92no—
131AGL(1,ℤ/8)—91no—
132#135—91no—
133#135—92no—
134C2×SD16—91no—
135C2×SD16—92no—
136#131—91no—
137C22×D3—68no—
138C22×D3—64no—
139C3×Q8—24no—
140C3×Q8C22×D318yes—
141C3×Q8C22×D314yes—
142C3×D4—24no—
143C3×D4C22×D314yes—
144C2×C3×C4—24no—
145C2×C3×C4—24no—
146C2×C3×C4C22×D314yes—
147#67—38no—
148#67—68no—
149#67—64no—
150D12—34no—
151D12—38no—
152D12—34no—
153D12—68no—
154D12—68no—
155D12—64no—
156C4×D3—38no—
157C4×D3—68no—
158C4×D3—68no—
159C4×D3—68no—
160C4×D3—64no—
161Dic6—34no—
162Dic6—34no—
163Dic6—68no—
164C3×C8C22×D318yes—
165C3⋊C8—34no—
166C3⋊C8—34no—
167C2×C32—21no—
168C2×C32—41no—
169C2×C32C22×D411yescharacteristic
170C3⋊D3—22no—
171C3⋊D3—41no—
172C3⋊D3—42no—
173#41—91no—
174#41—92no—
175#41—92no—
176#41—91no—
177#41—182no—
178#41—181no—
179#41—181no—
180#41C2×C3⋊D311yescharacteristic
181C2×Q8—92no—
182C2×Q8—181no—
183C2×Q8C2×C3⋊D311yescharacteristic, complemented
184C2×D4—92no—
185C2×D4—91no—
186C2×D4—181no—
187C2×D4—182no—
188Q16—91no—
189Q16C2×C3⋊D312yescomplemented
190SD16—92no—
191SD16—92no—
192SD16—91no—
193SD16—91no—
194SD16—92no—
195SD16C2×C3⋊D312yescomplemented
196D8—91no—
197D8—92no—
198M16—92no—
199M16C2×C3⋊D311yescharacteristic, complemented
200C2×C8—91no—
201C2×C8—92no—
202C22×C3—24no—
203C22×C3D3×D414yes—
204C2×D3—38no—
205C2×D3—64no—
206C2×D3—68no—
207C2×D3—248no—
208C2×D3—244no—
209C3×C4—28no—
210C3×C4—24no—
211C3×C4D3×D414yes—
212C3×C4C23×D314yes—
213Dic3—38no—
214Dic3—64no—
215C32#57711yescharacteristic, complemented, Sylow
216C23—182no—
217C23—181no—
218Q8—21no—
219Q8—91no—
220Q8—91no—
221Q8—182no—
222Q8C22×C3⋊D312yes—
223Q8C22×C3⋊D311yescharacteristic
224D4—21no—
225D4—92no—
226D4—91no—
227D4—92no—
228D4—91no—
229D4—182no—
230D4—182no—
231D4—182no—
232D4—181no—
233D4—181no—
234D4C22×C3⋊D311yescharacteristic
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—92no—
238C2×C4—182no—
239C2×C4—182no—
240C2×C4—182no—
241C2×C4—181no—
242C2×C4C22×C3⋊D311yescharacteristic
243C8—91no—
244C8—91no—
245C8C22×C3⋊D312yes—
246C2×C3—24no—
247C2×C3—44no—
248C2×C3C2×D3×D414yes—
249D3—68no—
250D3—128no—
251D3—124no—
252C22—21no—
253C22—92no—
254C22—181no—
255C22—182no—
256C22—721no—
257C22—722no—
258C22D4×C3⋊D311yescharacteristic
259C4—21no—
260C4—22no—
261C4—92no—
262C4—181no—
263C4D4×C3⋊D311yescharacteristic
264C4C23×C3⋊D311yescharacteristic
265C3#577814yescomplemented
266C2—21no—
267C2—41no—
268C2—182no—
269C2—362no—
270C2—361no—
271C2C2×D4×C3⋊D311yescharacteristic
subgroupG/H
centerC2C2×D4×C3⋊D3
commutator, cosocleC32×C4C24
FrattiniC4C23×C3⋊D3
FittingC32×#137C2
socleC2×C32C22×D4
hypercenter#137C3⋊D3
hypocenterC32#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #64366C2semidirect
2. #64364C2semidirect
3. C32×#137C2semidirect
4. #64159C2semidirect
5. C2×#3755C2semidirect
6. #64128C2semidirect
7. #64127C2semidirect
8. #64123C2semidirect
9. SD16×C3⋊D3C2semidirect
10. #64117C2semidirect
11. #64116C2semidirect
14. C2×C32×Q8C22semidirect
17. #3832C22semidirect
19. Q8×C3⋊D3C22semidirect
25. #3828C22semidirect
28. C32×Q16C22semidirect
29. C32×SD16C22semidirect
30. C32×M16C22semidirect
31. #3756C22semidirect
32. #3755C22semidirect
33. #3755C22semidirect
34. #3755C22semidirect
35. #3754C22semidirect
36. #3753C22semidirect
37. C2×C3⋊(C3⋊C8)C22semidirect
38. #3745C22semidirect
39. #3744C22semidirect
40. #3743C22semidirect
41. C8×C3⋊D3C22semidirect
44. C3×#137D3semidirect
85. C2×C3×Q8C2×D3semidirect
99. C3×Q16C2×D3semidirect
100. C3×SD16C2×D3semidirect
101. C3×M16C2×D3semidirect
129. #137C3⋊D3semidirect
183. C2×Q8C2×C3⋊D3semidirect
189. Q16C2×C3⋊D3semidirect
195. SD16C2×C3⋊D3semidirect
199. M16C2×C3⋊D3semidirect
215. C32#577semidirect
265. C3#5778semidirect

Maximal subgroups

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

subgroupindex
#643662
#643642
C32×#1372
#641592
C2×#37552
#641282
#641272
#641232
SD16×C3⋊D32
#641172
#641162
#57783