Finite Group Databasebeta

← #10589714

Subgroups — #10589714

Order 576.

501 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 127 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×D9C211yesmaximal, characteristic, complemented
2C2×#3697C211yesmaximal, characteristic, complemented
3C9×#121C211yesmaximal, characteristic, complemented
4#63500C212yesmaximal, complemented
5#63499C212yesmaximal, complemented
6#63498C211yesmaximal, characteristic, complemented
7C4×#611C211yesmaximal, characteristic, complemented
8#63459C211yesmaximal, characteristic, complemented
9#63455C211yesmaximal, characteristic, complemented
10#63445C212yesmaximal, complemented
11C22⋊C4×D9C212yesmaximal, complemented
12#5612—31nomaximal
13C23×D9C2212yes—
14C2×D4×C9C2211yescharacteristic
15C2×D4×C9C2212yescomplemented
16C22×C4×C9C2211yescharacteristic, complemented
17C2×#611C2211yescharacteristic
18C2×#611C2212yes—
19C2×#611C2212yescomplemented
20D4×D9—24no—
21#3697—24no—
22C2×D36C2211yescharacteristic
23C2×C4×D9C2211yescharacteristic
24C2×C4×D9C2212yescomplemented
25C2×C4×D9C2212yescomplemented
26C2×Dic18C2211yescharacteristic, complemented
27C9×C4⋊C4C2211yescharacteristic, complemented
28C9×C22⋊C4C2212yescomplemented
29#3676C2211yescharacteristic, complemented
30#3676C2212yescomplemented
31#3676C2212yescomplemented
32(C2×D9)⋊C4C2211yescharacteristic
33(C2×D9)⋊C4C2212yes—
34(C2×D9)⋊C4C2212yescomplemented
35C4⋊Dic9C2212yescomplemented
36Dic9⋊C4C2211yescharacteristic, complemented
37C4×Dic9C2211yescharacteristic, complemented
38C2×D3×D4—31no—
39C2×#235—31no—
40C3×#121D311yescharacteristic
41#938—32no—
42#937—32no—
43#936—31no—
44C4×#67—31no—
45#897—31no—
46#893—31no—
47#883—32no—
48D3×C22⋊C4—32no—
49C23×C9C2312yes—
50C23×C9C2311yescharacteristic
51C22×D9—24no—
52C22×D9—24no—
53C22×D9—42no—
54C22×D9C2312yes—
55C22×D9C2311yescharacteristic
56D4×C9—22no—
57D4×C9—42no—
58C2×C4×C9—22no—
59C2×C4×C9C2311yescharacteristic
60C2×C4×C9C2312yes—
61C2×C4×C9C2311yescharacteristic
62#611—24no—
63#611—24no—
64#611—22no—
65C2×Dic9C2311yescharacteristic
66C2×Dic9C2311yescharacteristic
67C2×Dic9C2311yescharacteristic
68C2×Dic9C2312yes—
69D36D414yescomplemented
70C4×D9—24no—
71C4×D9—22no—
72C4×D9—42no—
73Dic18—22no—
74#546—91noSylow
75C23×D3—32no—
76C2×C3×D4C2×D311yescharacteristic
77C2×C3×D4C2×D312yes—
78C22×C3×C4C2×D311yescharacteristic
79C2×#67—32no—
80C2×#67—31no—
81C2×#67—32no—
82D3×D4—64no—
83#235—64no—
84C2×D12—31no—
85C2×C4×D3—31no—
86C2×C4×D3—32no—
87C2×C4×D3—32no—
88C2×Dic6—31no—
89C3×C4⋊C4C2×D311yescharacteristic
90C3×C22⋊C4C2×D312yes—
91#217—32no—
92#217—32no—
93#217—31no—
94(C2×D3)⋊C4—31no—
95(C2×D3)⋊C4—32no—
96(C2×D3)⋊C4—32no—
97C4⋊Dic3—32no—
98Dic3⋊C4—31no—
99C4×Dic3—31no—
100C22×C9—21no—
101C22×C9—22no—
102C22×C9—21no—
103C22×C9—22no—
104C22×C9—21no—
105C22×C9—22no—
106C22×C9C2411yescharacteristic
107C2×D9—21no—
108C2×D9—22no—
109C2×D9—22no—
110C2×D9—21no—
111C2×D9—48no—
112C2×D9C2×D414yes—
113C2×D9#4112yescomplemented
114C4×C9—21no—
115C4×C9—22no—
116C4×C9C2×D412yes—
117Dic9—22no—
118Dic9—21no—
119Dic9—21no—
120Dic9—21no—
121C2×#41—91no—
122C22×D4—91no—
123#124—91no—
124#123—92no—
125C22⋊Q8—91no—
126#121—92no—
127#121D911yescharacteristic, complemented
128#120—92no—
129C4×D4—91no—
130C4×D4—91no—
131C2×C22⋊C4—92no—
132C23×C3C22×D312yes—
133C23×C3C22×D311yescharacteristic
134C22×D3—31no—
135C22×D3—32no—
136C22×D3—64no—
137C22×D3—64no—
138C22×D3—122no—
139C3×D4—22no—
140C3×D4—42no—
141C2×C3×C4—22no—
142C2×C3×C4C22×D311yescharacteristic
143C2×C3×C4C22×D312yes—
144C2×C3×C4C22×D311yescharacteristic
145#67—64no—
146#67—64no—
147#67—62no—
148C2×Dic3—31no—
149C2×Dic3—31no—
150C2×Dic3—31no—
151C2×Dic3—32no—
152D12—34no—
153C4×D3—64no—
154C4×D3—62no—
155C4×D3—122no—
156Dic6—62no—
157C2×C9—42no—
158C2×C9—41no—
159C2×C9C22×D411yescharacteristic
160C2×C9C2×#4111yescharacteristic
161C2×C9H(5,𝔽2)11yescharacteristic
162D9—22no—
163D9—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×D911yescharacteristic
173C2×D4C2×D912yescomplemented
174C22×C4—92no—
175C22×C4—91no—
176C22×C4—92no—
177C22×C4C2×D911yescharacteristic, complemented
178C4⋊C4—91no—
179C4⋊C4—92no—
180C4⋊C4C2×D911yescharacteristic, complemented
181C22⋊C4—92no—
182C22⋊C4—91no—
183C22⋊C4—92no—
184C22⋊C4—91no—
185C22⋊C4—92no—
186C22⋊C4—92no—
187C22⋊C4C2×D912yescomplemented
188C42—91no—
189C22×C3—21no—
190C22×C3—22no—
191C22×C3—21no—
192C22×C3—22no—
193C22×C3—21no—
194C22×C3—22no—
195C22×C3C23×D311yescharacteristic
196C2×D3—34no—
197C2×D3—32no—
198C2×D3—61no—
199C2×D3—62no—
200C2×D3—61no—
201C2×D3—62no—
202C2×D3—128no—
203C3×C4—21no—
204C3×C4—22no—
205C3×C4D3×D412yes—
206Dic3—62no—
207Dic3—61no—
208Dic3—61no—
209Dic3—61no—
210C9#54611yescharacteristic, complemented, Sylow
211C23—91no—
212C23—92no—
213C23—184no—
214C23—184no—
215C23—362no—
216C23C22×D912yes—
217C23C22×D911yescharacteristic
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×D912yes—
234C2×C4C22×D911yescharacteristic
235C2×C4C22×D911yescharacteristic
236C2×C3—42no—
237C2×C3—41no—
238C2×C3C2×D3×D411yescharacteristic
239C2×C3#100411yescharacteristic
240C2×C3D3×#4111yescharacteristic
241D3—62no—
242D3—64no—
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×D911yescharacteristic
257C4—21no—
258C4—22no—
259C4—181no—
260C4—182no—
261C4—181no—
262C4—181no—
263C4D4×D912yes—
264C3#561211yescharacteristic
265C2—42no—
266C2—41no—
267C2—182no—
268C2—184no—
269C2C2×D4×D911yescharacteristic
270C2#6371011yescharacteristic
271C2#41×D911yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×#121C2
hypercenter#121D9
hypocenterC9#546
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×D9C2semidirect
2. C2×#3697C2semidirect
3. C9×#121C2semidirect
4. #63500C2semidirect
5. #63499C2semidirect
6. #63498C2semidirect
7. C4×#611C2semidirect
8. #63459C2semidirect
9. #63455C2semidirect
10. #63445C2semidirect
11. C22⋊C4×D9C2semidirect
15. C2×D4×C9C22semidirect
16. C22×C4×C9C22semidirect
19. C2×#611C22semidirect
24. C2×C4×D9C22semidirect
25. C2×C4×D9C22semidirect
26. C2×Dic18C22semidirect
27. C9×C4⋊C4C22semidirect
28. C9×C22⋊C4C22semidirect
29. #3676C22semidirect
30. #3676C22semidirect
31. #3676C22semidirect
34. (C2×D9)⋊C4C22semidirect
35. C4⋊Dic9C22semidirect
36. Dic9⋊C4C22semidirect
37. C4×Dic9C22semidirect
69. D36D4semidirect
113. C2×D9#41semidirect
127. #121D9semidirect
173. C2×D4C2×D9semidirect
177. C22×C4C2×D9semidirect
180. C4⋊C4C2×D9semidirect
187. C22⋊C4C2×D9semidirect
210. C9#546semidirect

Maximal subgroups

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

subgroupindex
C2×D4×D92
C2×#36972
C9×#1212
#635002
#634992
#634982
C4×#6112
#634592
#634552
#634452
C22⋊C4×D92
#56123