Finite Group Databasebeta

← #10589876

Subgroups — #10589876

Order 576.

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

normal subgroups only

This group has 402 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#63710C211yesmaximal, characteristic, complemented
3C9×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
4#63513C211yesmaximal, characteristic, complemented
5C2×#3673C211yesmaximal, characteristic, complemented
6#63478C212yesmaximal, complemented
7#63476C212yesmaximal, complemented
8#63473C212yesmaximal, complemented
9D8×D9C212yesmaximal, complemented
10#63470C211yesmaximal, characteristic, complemented
11#63469C211yesmaximal, characteristic, complemented
12#5774—31nomaximal
13C9×#41C2211yescharacteristic, complemented
14C2×D4×C9C2211yescharacteristic, complemented
15C2×#611—22no—
16#3701C2212yescomplemented
17#3699—21no—
18#3699C2212yescomplemented
19D4×D9—22no—
20D4×D9—21no—
21D4×D9C2212yescomplemented
22D4×D9C2212yescomplemented
23#3697—21no—
24#3697C2211yescharacteristic
25#3697C2211yescharacteristic, complemented
26C2×D36—21no—
27C2×D36C2211yescharacteristic, complemented
28C9×SD16C2212yescomplemented
29C9×D8C2212yescomplemented
30C9×M16C2211yescharacteristic, complemented
31#3675C2211yescharacteristic, complemented
32#3674C2211yescharacteristic, complemented
33#3673C2212yescomplemented
34#3673C2212yescomplemented
35#3673C2211yescharacteristic, complemented
36#3672C2211yescharacteristic, complemented
37C2×C9⋊C8C2211yescharacteristic, complemented
38D72C2212yescomplemented
39#3664C2212yescomplemented
40#3663C2212yescomplemented
41C8×D9C2212yescomplemented
42#1009—31no—
43#1004—31no—
44C3×AGL(1,ℤ/8)D311yescharacteristic
45#950—31no—
46C2×#213—31no—
47#916—32no—
48#914—32no—
49#911—32no—
50D3×D8—32no—
51#908—31no—
52#907—31no—
53C23×C9—21no—
54C22×D9—21no—
55C22×D9—22no—
56C22×D9—22no—
57Q8×C9C2311yescharacteristic
58D4×C9—21no—
59D4×C9—21no—
60D4×C9C2311yescharacteristic
61D4×C9C2312yes—
62C2×C4×C9—21no—
63C2×C4×C9C2311yescharacteristic
64#611—21no—
65#611—22no—
66#611—22no—
67#611—22no—
68#611D412yes—
69C2×Dic9—22no—
70D36—22no—
71D36—21no—
72D36D411yescharacteristic
73D36C2312yes—
74D36C2311yescharacteristic
75C4×D9—21no—
76C4×D9—22no—
77C4×D9C2312yes—
78Dic18D411yescharacteristic
79Dic18C2311yescharacteristic
80C8×C9C2312yes—
81C9⋊C8C2311yescharacteristic
82C9⋊C8C2311yescharacteristic
83#576—91noSylow
84C3×#41C2×D311yescharacteristic
85C2×C3×D4C2×D311yescharacteristic
86C2×#67—62no—
87#239—32no—
88#237—32no—
89#237—61no—
90D3×D4—32no—
91D3×D4—32no—
92D3×D4—61no—
93D3×D4—62no—
94#235—31no—
95#235—31no—
96#235—61no—
97C2×D12—31no—
98C2×D12—61no—
99C3×SD16C2×D312yes—
100C3×D8C2×D312yes—
101C3×M16C2×D311yescharacteristic
102#216—31no—
103#215—31no—
104#214—31no—
105#213—31no—
106#213—32no—
107#213—32no—
108C2×C3⋊C8—31no—
109D24—32no—
110#204—32no—
111ΓL(1,𝔽25)—32no—
112D3×C8—32no—
113C22×C9—22no—
114C22×C9—21no—
115C22×C9—81no—
116C22×C9C2×D411yescharacteristic
117C2×D9—22no—
118C2×D9—21no—
119C2×D9—82no—
120C2×D9—82no—
121C2×D9—81no—
122C2×D9C2×D412yes—
123C4×C9—21no—
124C4×C9C2×D411yescharacteristic
125C4×C9C2411yescharacteristic
126Dic9—21no—
127Dic9C2×D412yes—
128H(5,𝔽2)—91no—
129H(5,𝔽2)—91no—
130AGL(1,ℤ/8)—92no—
131AGL(1,ℤ/8)—92no—
132AGL(1,ℤ/8)—91no—
133AGL(1,ℤ/8)D911yescharacteristic, complemented
134#135—91no—
135#135—92no—
136C2×D8—91no—
137C2×D8—92no—
138#131—91no—
139C23×C3—21no—
140C22×D3—62no—
141C22×D3—62no—
142C22×D3—61no—
143C3×Q8C22×D311yescharacteristic
144C3×D4—21no—
145C3×D4—21no—
146C3×D4C22×D312yes—
147C3×D4C22×D311yescharacteristic
148C2×C3×C4—21no—
149C2×C3×C4C22×D311yescharacteristic
150#67—32no—
151#67—62no—
152#67—62no—
153#67—62no—
154#67—61no—
155C2×Dic3—62no—
156D12—31no—
157D12—32no—
158D12—31no—
159D12—62no—
160D12—61no—
161C4×D3—32no—
162C4×D3—62no—
163C4×D3—61no—
164Dic6—31no—
165Dic6—31no—
166C3×C8C22×D312yes—
167C3⋊C8—31no—
168C3⋊C8—31no—
169C2×C9—21no—
170C2×C9—41no—
171C2×C9—42no—
172C2×C9C22×D411yescharacteristic
173D9—22no—
174D9—42no—
175D9—41no—
176#41—91no—
177#41—92no—
178#41—92no—
179#41—91no—
180#41—181no—
181#41—181no—
182#41C2×D911yescharacteristic, complemented
183C2×D4—92no—
184C2×D4—92no—
185C2×D4—91no—
186C2×D4—181no—
187C2×D4—182no—
188C2×D4—181no—
189C2×D4—182no—
190C2×D4C2×D911yescharacteristic, complemented
191Q16—91no—
192SD16—91no—
193SD16—91no—
194SD16—92no—
195SD16C2×D912yescomplemented
196D8—92no—
197D8—91no—
198D8—92no—
199D8—92no—
200D8C2×D912yescomplemented
201M16—92no—
202M16C2×D911yescharacteristic, complemented
203C2×C8—91no—
204C2×C8—92no—
205C22×C3—22no—
206C22×C3—21no—
207C22×C3—81no—
208C22×C3D3×D411yescharacteristic
209C2×D3—32no—
210C2×D3—62no—
211C2×D3—61no—
212C2×D3—242no—
213C2×D3—242no—
214C2×D3—241no—
215C3×C4—21no—
216C3×C4D3×D411yescharacteristic
217C3×C4C23×D311yescharacteristic
218Dic3—32no—
219Dic3—61no—
220C9#57611yescharacteristic, complemented, Sylow
221C23—21no—
222C23—182no—
223C23—182no—
224C23—181no—
225Q8—91no—
226Q8—91no—
227Q8C22×D911yescharacteristic
228D4—21no—
229D4—21no—
230D4—92no—
231D4—91no—
232D4—92no—
233D4—91no—
234D4—182no—
235D4—182no—
236D4—182no—
237D4—182no—
238D4—181no—
239D4—181no—
240D4C22×D912yes—
241D4C22×D911yescharacteristic
242C2×C4—21no—
243C2×C4—92no—
244C2×C4—182no—
245C2×C4—182no—
246C2×C4—181no—
247C2×C4C22×D911yescharacteristic
248C8—91no—
249C8—91no—
250C8C22×D912yes—
251C2×C3—21no—
252C2×C3—41no—
253C2×C3—42no—
254C2×C3C2×D3×D411yescharacteristic
255D3—62no—
256D3—122no—
257D3—121no—
258C22—21no—
259C22—22no—
260C22—81no—
261C22—92no—
262C22—182no—
263C22—181no—
264C22—721no—
265C22—722no—
266C22—722no—
267C22D4×D911yescharacteristic
268C4—21no—
269C4—92no—
270C4—181no—
271C4D4×D911yescharacteristic
272C4C23×D911yescharacteristic
273C3#577411yescharacteristic
274C2—21no—
275C2—42no—
276C2—41no—
277C2—182no—
278C2—362no—
279C2—361no—
280C2C2×D4×D911yescharacteristic
subgroupG/H
centerC2C2×D4×D9
commutator, cosocleC4×C9C24
FrattiniC3×C4C23×D3
FittingC9×AGL(1,ℤ/8)C2
socleC2×C3C2×D3×D4
hypercenterAGL(1,ℤ/8)D9
hypocenterC9#576
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #63715C2semidirect
2. #63710C2semidirect
3. C9×AGL(1,ℤ/8)C2semidirect
4. #63513C2semidirect
5. C2×#3673C2semidirect
6. #63478C2semidirect
7. #63476C2semidirect
8. #63473C2semidirect
9. D8×D9C2semidirect
10. #63470C2semidirect
11. #63469C2semidirect
13. C9×#41C22semidirect
14. C2×D4×C9C22semidirect
16. #3701C22semidirect
18. #3699C22semidirect
21. D4×D9C22semidirect
22. D4×D9C22semidirect
25. #3697C22semidirect
27. C2×D36C22semidirect
28. C9×SD16C22semidirect
29. C9×D8C22semidirect
30. C9×M16C22semidirect
31. #3675C22semidirect
32. #3674C22semidirect
33. #3673C22semidirect
34. #3673C22semidirect
35. #3673C22semidirect
36. #3672C22semidirect
37. C2×C9⋊C8C22semidirect
38. D72C22semidirect
39. #3664C22semidirect
40. #3663C22semidirect
41. C8×D9C22semidirect
133. AGL(1,ℤ/8)D9semidirect
182. #41C2×D9semidirect
190. C2×D4C2×D9semidirect
195. SD16C2×D9semidirect
200. D8C2×D9semidirect
202. M16C2×D9semidirect
220. C9#576semidirect

Maximal subgroups

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

subgroupindex
#637152
#637102
C9×AGL(1,ℤ/8)2
#635132
C2×#36732
#634782
#634762
#634732
D8×D92
#634702
#634692
#57743