Finite Group Databasebeta

← #10589732

Subgroups — #10589732

Order 576.

501 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 131 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×#3701C211yesmaximal, characteristic, complemented
2C22×D36C211yesmaximal, characteristic, complemented
3C9×C22⋊Q8C211yesmaximal, characteristic, complemented
4#63509C211yesmaximal, characteristic, complemented
5C4×#611C211yesmaximal, characteristic, complemented
6#63457C212yesmaximal, complemented
7#63457C211yesmaximal, characteristic, complemented
8#63456C212yesmaximal, complemented
9#63455C211yesmaximal, characteristic, complemented
10#63444C212yesmaximal, complemented
11C22⋊C4×D9C212yesmaximal, complemented
12#5630—31nomaximal
13C23×D9C2212yes—
14C2×Q8×C9C2211yescharacteristic, complemented
15C22×C4×C9C2211yescharacteristic
16C2×#611C2211yescharacteristic
17#3701—24no—
18C2×D36—24no—
19C2×D36C2211yescharacteristic
20C2×D36C2212yes—
21C2×D36C2211yescharacteristic
22C2×D36C2212yescomplemented
23C2×D36C2212yescomplemented
24C2×C4×D9C2211yescharacteristic, complemented
25C2×C4×D9C2212yescomplemented
26C2×C4×D9C2212yescomplemented
27C9×C4⋊C4C2211yescharacteristic, complemented
28C9×C4⋊C4C2212yescomplemented
29C9×C22⋊C4C2212yes—
30#3676C2211yescharacteristic
31(C2×D9)⋊C4C2211yescharacteristic, complemented
32(C2×D9)⋊C4C2212yescomplemented
33(C2×D9)⋊C4C2212yescomplemented
34(C2×D9)⋊C4C2212yescomplemented
35(C2×D9)⋊C4C2212yescomplemented
36Dic9⋊C4C2211yescharacteristic, complemented
37C4×Dic9C2211yescharacteristic, complemented
38C2×#239—31no—
39C22×D12—31no—
40C3×C22⋊Q8D311yescharacteristic
41#947—31no—
42C4×#67—31no—
43#895—31no—
44#895—32no—
45#894—32no—
46#893—31no—
47#882—32no—
48D3×C22⋊C4—32no—
49C23×C9C2311yescharacteristic
50C22×D9—24no—
51C22×D9—24no—
52C22×D9—42no—
53C22×D9C2312yes—
54C22×D9C2311yescharacteristic
55C22×D9C2312yes—
56Q8×C9—22no—
57C2×C4×C9—22no—
58C2×C4×C9C2311yescharacteristic
59C2×C4×C9C2312yes—
60C2×C4×C9C2312yes—
61C2×C4×C9C2311yescharacteristic
62#611—22no—
63C2×Dic9C2311yescharacteristic
64C2×Dic9C2311yescharacteristic
65C2×Dic9C2311yescharacteristic
66D36—24no—
67D36—24no—
68D36—22no—
69D36—42no—
70D36D414yescomplemented
71C4×D9—24no—
72C4×D9—22no—
73C4×D9—42no—
74#546—91noSylow
75C23×D3—32no—
76C2×C3×Q8C2×D311yescharacteristic
77C22×C3×C4C2×D311yescharacteristic
78C2×#67—31no—
79#239—64no—
80C2×D12—31no—
81C2×D12—32no—
82C2×D12—31no—
83C2×D12—32no—
84C2×D12—32no—
85C2×D12—64no—
86C2×C4×D3—32no—
87C2×C4×D3—31no—
88C2×C4×D3—32no—
89C3×C4⋊C4C2×D311yescharacteristic
90C3×C4⋊C4C2×D312yes—
91C3×C22⋊C4C2×D312yes—
92#217—31no—
93(C2×D3)⋊C4—32no—
94(C2×D3)⋊C4—32no—
95(C2×D3)⋊C4—32no—
96(C2×D3)⋊C4—32no—
97(C2×D3)⋊C4—31no—
98Dic3⋊C4—31no—
99C4×Dic3—31no—
100C22×C9—21no—
101C22×C9—21no—
102C22×C9#4112yescomplemented
103C22×C9C2411yescharacteristic
104C2×D9—22no—
105C2×D9—22no—
106C2×D9—21no—
107C2×D9—22no—
108C2×D9—21no—
109C2×D9—22no—
110C2×D9—21no—
111C2×D9—22no—
112C2×D9—48no—
113C2×D9C2×D414yes—
114C4×C9—22no—
115C4×C9—21no—
116C4×C9—22no—
117C4×C9C2×D412yes—
118Dic9—21no—
119Dic9—21no—
120Dic9—21no—
121C2×#41—91no—
122C22×D4—91no—
123#124—91no—
124#123—92no—
125C22⋊Q8D911yescharacteristic, complemented
126#121—91no—
127#121—92no—
128#120—92no—
129C4×D4—91no—
130C4×D4—91no—
131C2×C22⋊C4—92no—
132C23×C3C22×D311yescharacteristic
133C22×D3—32no—
134C22×D3—32no—
135C22×D3—31no—
136C22×D3—64no—
137C22×D3—64no—
138C22×D3—122no—
139C3×Q8—22no—
140C2×C3×C4—22no—
141C2×C3×C4C22×D311yescharacteristic
142C2×C3×C4C22×D312yes—
143C2×C3×C4C22×D312yes—
144C2×C3×C4C22×D311yescharacteristic
145#67—62no—
146C2×Dic3—31no—
147C2×Dic3—31no—
148C2×Dic3—31no—
149D12—34no—
150D12—64no—
151D12—64no—
152D12—62no—
153D12—122no—
154C4×D3—64no—
155C4×D3—62no—
156C4×D3—122no—
157C2×C9—22no—
158C2×C9C22×D411yescharacteristic
159C2×C9C2×#4111yescharacteristic
160C2×C9H(5,𝔽2)11yescharacteristic
161D9—24no—
162D9—41no—
163D9—42no—
164C24—92no—
165#41—184no—
166C2×Q8C2×D911yescharacteristic, complemented
167C2×D4—91no—
168C2×D4—91no—
169C2×D4—92no—
170C2×D4—92no—
171C2×D4—91no—
172C2×D4—92no—
173C2×D4—184no—
174C22×C4—92no—
175C22×C4—91no—
176C22×C4—92no—
177C22×C4C2×D911yescharacteristic
178C4⋊C4—91no—
179C4⋊C4C2×D911yescharacteristic, complemented
180C4⋊C4C2×D912yescomplemented
181C22⋊C4—91no—
182C22⋊C4—92no—
183C22⋊C4—92no—
184C22⋊C4—92no—
185C22⋊C4—91no—
186C22⋊C4—92no—
187C22⋊C4C2×D912yes—
188C42—91no—
189C22×C3—21no—
190C22×C3—21no—
191C22×C3#23912yes—
192C22×C3C23×D311yescharacteristic
193C2×D3—34no—
194C2×D3—62no—
195C2×D3—62no—
196C2×D3—61no—
197C2×D3—62no—
198C2×D3—61no—
199C2×D3—62no—
200C2×D3—61no—
201C2×D3—62no—
202C2×D3—128no—
203C3×C4—22no—
204C3×C4—21no—
205C3×C4—22no—
206C3×C4D3×D412yes—
207Dic3—61no—
208Dic3—61no—
209Dic3—61no—
210C9#54611yescharacteristic, complemented, Sylow
211C23—92no—
212C23—92no—
213C23—91no—
214C23—184no—
215C23—184no—
216C23—362no—
217C23C22×D911yescharacteristic
218Q8—22no—
219D4—94no—
220D4—184no—
221D4—184no—
222D4—182no—
223D4—182no—
224D4—362no—
225C2×C4—22no—
226C2×C4—91no—
227C2×C4—91no—
228C2×C4—91no—
229C2×C4—184no—
230C2×C4—182no—
231C2×C4—362no—
232C2×C4C22×D912yes—
233C2×C4C22×D912yes—
234C2×C4C22×D911yescharacteristic
235C2×C4C22×D911yescharacteristic
236C2×C3—22no—
237C2×C3C2×D3×D411yescharacteristic
238C2×C3C2×#23911yescharacteristic
239C2×C3#100911yescharacteristic
240D3—64no—
241D3—122no—
242D3—121no—
243C22—21no—
244C22—21no—
245C22—94no—
246C22—182no—
247C22—182no—
248C22—181no—
249C22—182no—
250C22—181no—
251C22—182no—
252C22—181no—
253C22—182no—
254C22—368no—
255C22#370112yescomplemented
256C22C23×D911yescharacteristic
257C4—21no—
258C4—22no—
259C4—22no—
260C4—181no—
261C4—181no—
262C4—181no—
263C4D4×D912yes—
264C3#563011yescharacteristic
265C2—22no—
266C2—184no—
267C2—362no—
268C2—361no—
269C2C2×D4×D911yescharacteristic
270C2C2×#370111yescharacteristic
271C2#6371511yescharacteristic
subgroupG/H
centerC22C23×D9
commutator, cosocleC22×C9C24
Frattini, socleC22×C3C23×D3
FittingC9×C22⋊Q8C2
hypercenterC22⋊Q8D9
hypocenterC9#546
LCSG ▷ C22×C9 ▷ C9
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#3701C2semidirect
2. C22×D36C2semidirect
3. C9×C22⋊Q8C2semidirect
4. #63509C2semidirect
5. C4×#611C2semidirect
6. #63457C2semidirect
7. #63457C2semidirect
8. #63456C2semidirect
9. #63455C2semidirect
10. #63444C2semidirect
11. C22⋊C4×D9C2semidirect
14. C2×Q8×C9C22semidirect
22. C2×D36C22semidirect
23. C2×D36C22semidirect
24. C2×C4×D9C22semidirect
25. C2×C4×D9C22semidirect
26. C2×C4×D9C22semidirect
27. C9×C4⋊C4C22semidirect
28. C9×C4⋊C4C22semidirect
31. (C2×D9)⋊C4C22semidirect
32. (C2×D9)⋊C4C22semidirect
33. (C2×D9)⋊C4C22semidirect
34. (C2×D9)⋊C4C22semidirect
35. (C2×D9)⋊C4C22semidirect
36. Dic9⋊C4C22semidirect
37. C4×Dic9C22semidirect
70. D36D4semidirect
102. C22×C9#41semidirect
125. C22⋊Q8D9semidirect
166. C2×Q8C2×D9semidirect
179. C4⋊C4C2×D9semidirect
180. C4⋊C4C2×D9semidirect
210. C9#546semidirect
255. C22#3701semidirect

Maximal subgroups

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

subgroupindex
C2×#37012
C22×D362
C9×C22⋊Q82
#635092
C4×#6112
#634572
#634562
#634552
#634442
C22⋊C4×D92
#56303