Finite Group Databasebeta

← #10589872

Subgroups — #10589872

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#63716C211yesmaximal, characteristic, complemented
2#63715C211yesmaximal, characteristic, complemented
3C9×#135C211yesmaximal, characteristic, complemented
4#63512C211yesmaximal, characteristic, complemented
5#63511C211yesmaximal, characteristic, complemented
6#63480C212yesmaximal, complemented
7#63478C212yesmaximal, complemented
8SD16×D9C212yesmaximal, complemented
9#63473C212yesmaximal, complemented
10C2×#3664C211yesmaximal, characteristic, complemented
11#63464C211yesmaximal, characteristic, complemented
12#5770—31nomaximal
13C9×#41C2211yescharacteristic
14C9×#41C2211yescharacteristic, complemented
15#3701C2212yes—
16Q8×D9C2212yescomplemented
17#3699—22no—
18#3699C2212yescomplemented
19D4×D9—22no—
20D4×D9C2212yes—
21#3697—21no—
22#3697—21no—
23#3697C2211yescharacteristic
24C2×D36—21no—
25C2×D36C2211yescharacteristic
26C2×Dic18—21no—
27C2×Dic18C2211yescharacteristic, complemented
28C9×Q16C2211yescharacteristic, complemented
29C9×SD16C2211yescharacteristic, complemented
30C9×SD16C2211yescharacteristic, complemented
31C9×D8C2211yescharacteristic, complemented
32C2×C8×C9C2211yescharacteristic, complemented
33#3675C2212yescomplemented
34#3674C2212yescomplemented
35#3673C2212yescomplemented
36#3672C2212yescomplemented
37#3667C2211yescharacteristic, complemented
38#3664C2212yescomplemented
39#3664C2212yescomplemented
40#3663C2212yescomplemented
41C8×D9C2212yescomplemented
42#1010—31no—
43#1009—31no—
44C3×#135D311yescharacteristic
45#951—31no—
46#949—31no—
47#918—32no—
48#916—32no—
49D3×SD16—32no—
50#911—32no—
51C2×#204—31no—
52#901—31no—
53C22×D9—21no—
54C22×D9—22no—
55Q8×C9C2311yescharacteristic
56Q8×C9C2311yescharacteristic
57D4×C9—21no—
58D4×C9—21no—
59D4×C9C2311yescharacteristic
60D4×C9C2311yescharacteristic
61C2×C4×C9—21no—
62C2×C4×C9—21no—
63C2×C4×C9C2311yescharacteristic
64#611—22no—
65#611—22no—
66#611D412yes—
67C2×Dic9—21no—
68C2×Dic9—22no—
69D36—22no—
70D36—21no—
71D36D411yescharacteristic
72D36C2312yes—
73C4×D9—22no—
74C4×D9—22no—
75C4×D9C2312yes—
76Dic18—21no—
77Dic18—22no—
78Dic18D411yescharacteristic
79Dic18C2312yes—
80C8×C9C2311yescharacteristic
81C8×C9C2311yescharacteristic
82C9⋊C8C2312yes—
83#577—91noSylow
84C3×#41C2×D311yescharacteristic
85C3×#41C2×D311yescharacteristic
86#239—32no—
87D3×Q8—32no—
88#237—32no—
89#237—62no—
90D3×D4—32no—
91D3×D4—62no—
92#235—31no—
93#235—61no—
94#235—61no—
95C2×D12—31no—
96C2×D12—61no—
97C2×Dic6—31no—
98C2×Dic6—61no—
99C3×Q16C2×D311yescharacteristic
100C3×SD16C2×D311yescharacteristic
101C3×SD16C2×D311yescharacteristic
102C3×D8C2×D311yescharacteristic
103C2×C3×C8C2×D311yescharacteristic
104#216—32no—
105#215—32no—
106#214—32no—
107#213—32no—
108#208—31no—
109#204—32no—
110#204—32no—
111ΓL(1,𝔽25)—32no—
112D3×C8—32no—
113C22×C9—21no—
114C22×C9—21no—
115C22×C9C2×D411yescharacteristic
116C2×D9—22no—
117C2×D9—82no—
118C2×D9—81no—
119C2×D9C2×D412yes—
120C4×C9—21no—
121C4×C9—21no—
122C4×C9C2×D411yescharacteristic
123C4×C9C2411yescharacteristic
124Dic9—22no—
125Dic9C2×D412yes—
1262−1+4—91no—
127H(5,𝔽2)—91no—
128#137—91no—
129#137—92no—
130AGL(1,ℤ/8)—91no—
131AGL(1,ℤ/8)—92no—
132#135—92no—
133#135D911yescharacteristic, complemented
134C2×SD16—92no—
135C2×SD16—91no—
136#131—91no—
137C22×D3—62no—
138C22×D3—61no—
139C3×Q8C22×D311yescharacteristic
140C3×Q8C22×D311yescharacteristic
141C3×D4—21no—
142C3×D4—21no—
143C3×D4C22×D311yescharacteristic
144C3×D4C22×D311yescharacteristic
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4C22×D311yescharacteristic
148#67—32no—
149#67—62no—
150#67—62no—
151C2×Dic3—62no—
152C2×Dic3—61no—
153D12—31no—
154D12—32no—
155D12—62no—
156D12—61no—
157C4×D3—32no—
158C4×D3—62no—
159C4×D3—62no—
160Dic6—31no—
161Dic6—32no—
162Dic6—62no—
163Dic6—61no—
164C3×C8C22×D311yescharacteristic
165C3×C8C22×D311yescharacteristic
166C3⋊C8—32no—
167C2×C9—21no—
168C2×C9—41no—
169C2×C9—41no—
170C2×C9C22×D411yescharacteristic
171D9—22no—
172D9—42no—
173#41—91no—
174#41—92no—
175#41—92no—
176#41—181no—
177#41—181no—
178#41—182no—
179#41C2×D911yescharacteristic
180#41C2×D911yescharacteristic, complemented
181C2×Q8—92no—
182C2×Q8—91no—
183C2×Q8—181no—
184C2×D4—92no—
185C2×D4—91no—
186C2×D4—181no—
187C2×D4—182no—
188Q16—92no—
189Q16C2×D911yescharacteristic, complemented
190SD16—92no—
191SD16—92no—
192SD16—92no—
193SD16—92no—
194SD16C2×D911yescharacteristic, complemented
195SD16C2×D911yescharacteristic, complemented
196D8—92no—
197D8C2×D911yescharacteristic, complemented
198M16—91no—
199M16—92no—
200C2×C8—92no—
201C2×C8C2×D911yescharacteristic, complemented
202C22×C3—21no—
203C22×C3—21no—
204C22×C3D3×D411yescharacteristic
205C2×D3—32no—
206C2×D3—62no—
207C2×D3—242no—
208C2×D3—241no—
209C3×C4—21no—
210C3×C4—21no—
211C3×C4D3×D411yescharacteristic
212C3×C4C23×D311yescharacteristic
213Dic3—32no—
214Dic3—62no—
215C9#57711yescharacteristic, complemented, Sylow
216C23—182no—
217C23—181no—
218Q8—91no—
219Q8—92no—
220Q8—182no—
221Q8—181no—
222Q8C22×D911yescharacteristic
223Q8C22×D911yescharacteristic
224D4—21no—
225D4—21no—
226D4—92no—
227D4—91no—
228D4—92no—
229D4—182no—
230D4—182no—
231D4—182no—
232D4—181no—
233D4C22×D911yescharacteristic
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—92no—
244C8C22×D911yescharacteristic
245C8C22×D911yescharacteristic
246C2×C3—21no—
247C2×C3—41no—
248C2×C3—41no—
249C2×C3C2×D3×D411yescharacteristic
250D3—62no—
251D3—122no—
252C22—21no—
253C22—21no—
254C22—92no—
255C22—182no—
256C22—721no—
257C22—722no—
258C22D4×D911yescharacteristic
259C4—21no—
260C4—21no—
261C4—92no—
262C4—182no—
263C4D4×D911yescharacteristic
264C4C23×D911yescharacteristic
265C3#577011yescharacteristic
266C2—21no—
267C2—41no—
268C2—41no—
269C2—182no—
270C2—362no—
271C2C2×D4×D911yescharacteristic
subgroupG/H
centerC2C2×D4×D9
commutator, cosocleC4×C9C24
FrattiniC3×C4C23×D3
FittingC9×#135C2
socleC2×C3C2×D3×D4
hypercenter#135D9
hypocenterC9#577
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #63716C2semidirect
2. #63715C2semidirect
3. C9×#135C2semidirect
4. #63512C2semidirect
5. #63511C2semidirect
6. #63480C2semidirect
7. #63478C2semidirect
8. SD16×D9C2semidirect
9. #63473C2semidirect
10. C2×#3664C2semidirect
11. #63464C2semidirect
14. C9×#41C22semidirect
16. Q8×D9C22semidirect
18. #3699C22semidirect
27. C2×Dic18C22semidirect
28. C9×Q16C22semidirect
29. C9×SD16C22semidirect
30. C9×SD16C22semidirect
31. C9×D8C22semidirect
32. C2×C8×C9C22semidirect
33. #3675C22semidirect
34. #3674C22semidirect
35. #3673C22semidirect
36. #3672C22semidirect
37. #3667C22semidirect
38. #3664C22semidirect
39. #3664C22semidirect
40. #3663C22semidirect
41. C8×D9C22semidirect
133. #135D9semidirect
180. #41C2×D9semidirect
189. Q16C2×D9semidirect
194. SD16C2×D9semidirect
195. SD16C2×D9semidirect
197. D8C2×D9semidirect
201. C2×C8C2×D9semidirect
215. C9#577semidirect

Maximal subgroups

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

subgroupindex
#637162
#637152
C9×#1352
#635122
#635112
#634802
#634782
SD16×D92
#634732
C2×#36642
#634642
#57703