Finite Group Databasebeta

← #10589880

Subgroups — #10589880

Order 576.

387 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 119 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#63715C211yesmaximal, characteristic, complemented
2#63713C211yesmaximal, characteristic, complemented
3C9×#137C211yesmaximal, characteristic, complemented
4#63513C211yesmaximal, characteristic, complemented
5C2×#3675C211yesmaximal, characteristic, complemented
6#63481C212yesmaximal, complemented
7#63480C212yesmaximal, complemented
8#63476C212yesmaximal, complemented
9SD16×D9C212yesmaximal, complemented
10#63470C211yesmaximal, characteristic, complemented
11#63469C211yesmaximal, characteristic, complemented
12#5778—31nomaximal
13C9×#41C2211yescharacteristic
14C2×Q8×C9C2211yescharacteristic, complemented
15#3701—21no—
16#3701C2212yes—
17#3701C2212yescomplemented
18Q8×D9—21no—
19Q8×D9C2212yescomplemented
20D4×D9—22no—
21D4×D9C2212yes—
22#3697—22no—
23#3697—21no—
24#3697C2211yescharacteristic
25#3697C2211yescharacteristic, complemented
26C2×D36—21no—
27C2×D36C2211yescharacteristic
28C9×Q16C2212yescomplemented
29C9×SD16C2212yescomplemented
30C9×M16C2211yescharacteristic, complemented
31#3675C2212yescomplemented
32#3675C2212yescomplemented
33#3675C2211yescharacteristic, complemented
34#3674C2211yescharacteristic, complemented
35#3673C2211yescharacteristic, complemented
36#3672C2211yescharacteristic, complemented
37C2×C9⋊C8C2211yescharacteristic, complemented
38D72C2212yescomplemented
39#3664C2212yescomplemented
40#3663C2212yescomplemented
41C8×D9C2212yescomplemented
42#1009—31no—
43#1007—31no—
44C3×#137D311yescharacteristic
45#950—31no—
46C2×#215—31no—
47#919—32no—
48#918—32no—
49#914—32no—
50D3×SD16—32no—
51#908—31no—
52#907—31no—
53C22×D9—21no—
54C22×D9—22no—
55Q8×C9—21no—
56Q8×C9C2311yescharacteristic
57Q8×C9C2312yes—
58D4×C9—21no—
59D4×C9C2311yescharacteristic
60C2×C4×C9—21no—
61C2×C4×C9—21no—
62C2×C4×C9C2311yescharacteristic
63#611—21no—
64#611—22no—
65#611D412yes—
66D36—22no—
67D36—22no—
68D36—21no—
69D36D411yescharacteristic
70D36C2312yes—
71D36C2311yescharacteristic
72C4×D9—22no—
73C4×D9—21no—
74C4×D9—22no—
75C4×D9—22no—
76C4×D9C2312yes—
77Dic18—22no—
78Dic18D411yescharacteristic
79Dic18C2311yescharacteristic
80C8×C9C2312yes—
81C9⋊C8C2311yescharacteristic
82C9⋊C8C2311yescharacteristic
83#577—91noSylow
84C3×#41C2×D311yescharacteristic
85C2×C3×Q8C2×D311yescharacteristic
86#239—32no—
87#239—32no—
88#239—61no—
89D3×Q8—32no—
90D3×Q8—61no—
91D3×D4—32no—
92D3×D4—62no—
93#235—31no—
94#235—31no—
95#235—61no—
96#235—62no—
97C2×D12—31no—
98C2×D12—61no—
99C3×Q16C2×D312yes—
100C3×SD16C2×D312yes—
101C3×M16C2×D311yescharacteristic
102#216—31no—
103#215—32no—
104#215—32no—
105#215—31no—
106#214—31no—
107#213—31no—
108C2×C3⋊C8—31no—
109D24—32no—
110#204—32no—
111ΓL(1,𝔽25)—32no—
112D3×C8—32no—
113C22×C9—21no—
114C22×C9C2×D411yescharacteristic
115C2×D9—21no—
116C2×D9—22no—
117C2×D9—82no—
118C2×D9—81no—
119C2×D9C2×D412yes—
120C4×C9—22no—
121C4×C9—21no—
122C4×C9C2×D411yescharacteristic
123C4×C9C2411yescharacteristic
124Dic9—21no—
125Dic9C2×D412yes—
1262−1+4—91no—
127H(5,𝔽2)—91no—
128#137—92no—
129#137D911yescharacteristic, 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—62no—
138C22×D3—61no—
139C3×Q8—21no—
140C3×Q8C22×D312yes—
141C3×Q8C22×D311yescharacteristic
142C3×D4—21no—
143C3×D4C22×D311yescharacteristic
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4C22×D311yescharacteristic
147#67—32no—
148#67—62no—
149#67—61no—
150D12—31no—
151D12—32no—
152D12—31no—
153D12—62no—
154D12—62no—
155D12—61no—
156C4×D3—32no—
157C4×D3—62no—
158C4×D3—62no—
159C4×D3—62no—
160C4×D3—61no—
161Dic6—31no—
162Dic6—31no—
163Dic6—62no—
164C3×C8C22×D312yes—
165C3⋊C8—31no—
166C3⋊C8—31no—
167C2×C9—21no—
168C2×C9—41no—
169C2×C9C22×D411yescharacteristic
170D9—22no—
171D9—41no—
172D9—42no—
173#41—91no—
174#41—92no—
175#41—92no—
176#41—91no—
177#41—182no—
178#41—181no—
179#41—181no—
180#41C2×D911yescharacteristic
181C2×Q8—92no—
182C2×Q8—181no—
183C2×Q8C2×D911yescharacteristic, complemented
184C2×D4—92no—
185C2×D4—91no—
186C2×D4—181no—
187C2×D4—182no—
188Q16—91no—
189Q16C2×D912yescomplemented
190SD16—92no—
191SD16—92no—
192SD16—91no—
193SD16—91no—
194SD16—92no—
195SD16C2×D912yescomplemented
196D8—91no—
197D8—92no—
198M16—92no—
199M16C2×D911yescharacteristic, complemented
200C2×C8—91no—
201C2×C8—92no—
202C22×C3—21no—
203C22×C3D3×D411yescharacteristic
204C2×D3—32no—
205C2×D3—61no—
206C2×D3—62no—
207C2×D3—242no—
208C2×D3—241no—
209C3×C4—22no—
210C3×C4—21no—
211C3×C4D3×D411yescharacteristic
212C3×C4C23×D311yescharacteristic
213Dic3—32no—
214Dic3—61no—
215C9#57711yescharacteristic, complemented, Sylow
216C23—182no—
217C23—181no—
218Q8—21no—
219Q8—91no—
220Q8—91no—
221Q8—182no—
222Q8C22×D912yes—
223Q8C22×D911yescharacteristic
224D4—21no—
225D4—92no—
226D4—91no—
227D4—91no—
228D4—92no—
229D4—182no—
230D4—182no—
231D4—182no—
232D4—181no—
233D4—181no—
234D4C22×D911yescharacteristic
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—92no—
238C2×C4—182no—
239C2×C4—182no—
240C2×C4—182no—
241C2×C4—181no—
242C2×C4C22×D911yescharacteristic
243C8—91no—
244C8—91no—
245C8C22×D912yes—
246C2×C3—21no—
247C2×C3—41no—
248C2×C3C2×D3×D411yescharacteristic
249D3—62no—
250D3—122no—
251D3—121no—
252C22—21no—
253C22—92no—
254C22—181no—
255C22—182no—
256C22—721no—
257C22—722no—
258C22D4×D911yescharacteristic
259C4—21no—
260C4—22no—
261C4—92no—
262C4—181no—
263C4D4×D911yescharacteristic
264C4C23×D911yescharacteristic
265C3#577811yescharacteristic
266C2—21no—
267C2—41no—
268C2—182no—
269C2—362no—
270C2—361no—
271C2C2×D4×D911yescharacteristic
subgroupG/H
centerC2C2×D4×D9
commutator, cosocleC4×C9C24
FrattiniC3×C4C23×D3
FittingC9×#137C2
socleC2×C3C2×D3×D4
hypercenter#137D9
hypocenterC9#577
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. #63715C2semidirect
2. #63713C2semidirect
3. C9×#137C2semidirect
4. #63513C2semidirect
5. C2×#3675C2semidirect
6. #63481C2semidirect
7. #63480C2semidirect
8. #63476C2semidirect
9. SD16×D9C2semidirect
10. #63470C2semidirect
11. #63469C2semidirect
14. C2×Q8×C9C22semidirect
17. #3701C22semidirect
19. Q8×D9C22semidirect
25. #3697C22semidirect
28. C9×Q16C22semidirect
29. C9×SD16C22semidirect
30. C9×M16C22semidirect
31. #3675C22semidirect
32. #3675C22semidirect
33. #3675C22semidirect
34. #3674C22semidirect
35. #3673C22semidirect
36. #3672C22semidirect
37. C2×C9⋊C8C22semidirect
38. D72C22semidirect
39. #3664C22semidirect
40. #3663C22semidirect
41. C8×D9C22semidirect
129. #137D9semidirect
183. C2×Q8C2×D9semidirect
189. Q16C2×D9semidirect
195. SD16C2×D9semidirect
199. M16C2×D9semidirect
215. C9#577semidirect

Maximal subgroups

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

subgroupindex
#637152
#637132
C9×#1372
#635132
C2×#36752
#634812
#634802
#634762
SD16×D92
#634702
#634692
#57783