Finite Group Databasebeta

← #11722515

Subgroups — #11722515

Order 864.

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

normal subgroups only

This group has 538 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
1#88746C211yesmaximal, characteristic, complemented
2C9×#239C211yesmaximal, characteristic, complemented
3C3×#3701C211yesmaximal, characteristic, complemented
4#88646C213yesmaximal, complemented
5D9×D12C213yesmaximal, complemented
6D3×D36C213yesmaximal, complemented
7#88640C213yesmaximal, complemented
8#64320—31nomaximal
9#63715—31nomaximal
10C2×D3×D9C2216yescomplemented
11C3×Q8×C9C2211yescharacteristic, complemented
12C4⋊(C3⋊D9)C2213yescomplemented
13C4×C3⋊D9C2213yescomplemented
14C9×D12C2213yescomplemented
15C4×D3×C9C2213yescomplemented
16C3×D36C2213yescomplemented
17C3×C4×D9C2213yescomplemented
18#6195C2213yescomplemented
19#6194C2213yescomplemented
20#6192C2213yescomplemented
21#6189C2211yescharacteristic, complemented
22#3832—31no—
23C3×#239—31no—
24C3×#239D311yescharacteristic
25#3802—33no—
26D3×D12—33no—
27D3×D12—33no—
28#3798—33no—
29C9×#41—31no—
30#3701—31no—
31#3701—61no—
32#3701D311yescharacteristic, complemented
33D4×D9—33no—
34D4×D9—33no—
35#3697—33no—
36C2×D36—33no—
37C2×C3⋊D9C2313yes—
38C2×D3×C9C2313yes—
39C2×C3×D9C2313yes—
40D3×D9—46no—
41C3×C4×C9C2313yes—
42C3⋊Dic9C2311yescharacteristic
43C9×Dic3C2311yescharacteristic
44C3×Dic9C2311yescharacteristic
45#1009—91no—
46#1009—91no—
47C2×D32—36no—
48C32×Q8C2×D311yescharacteristic
49C4⋊(C3⋊D3)—33no—
50C4×C3⋊D3—33no—
51C3×D12—33no—
52C3×D12C2×D313yes—
53C3×C4×D3—33no—
54C3×C4×D3C2×D313yes—
55#627—31no—
56#626—33no—
57#626—33no—
58#625—33no—
59C22×D9—36no—
60Q8×C9—21no—
61Q8×C9C2×D311yescharacteristic, complemented
62D4×C9—33no—
63C2×C4×C9—33no—
64#611—33no—
65#611—33no—
66D36—33no—
67D36—33no—
68D36—63no—
69D36C2×D313yescomplemented
70C4×D9—33no—
71C4×D9—63no—
72C4×D9C2×D313yescomplemented
73Dic18—31no—
74C2×C3×C9C2411yescharacteristic
75C3⋊D9—23no—
76D3×C9—23no—
77C3×D9—23no—
78C3×#41—31no—
79C3×#41—91no—
80#239—31no—
81#239—91no—
82#239—91no—
83#239—181no—
84#239D911yescharacteristic, complemented
85D3×D4—93no—
86D3×D4—93no—
87D3×D4—93no—
88D3×D4—93no—
89#235—93no—
90#235—93no—
91C2×D12—93no—
92C2×D12—93no—
93C2×C3⋊D3—33no—
94C2×C3×D3—33no—
95C2×C3×D3C22×D313yes—
96D32—126no—
97C32×C4C22×D313yes—
98C3⋊Dic3—31no—
99C3×Dic3—31no—
100C3×Dic3C22×D311yescharacteristic
101C22×C9—33no—
102C2×D9—33no—
103C2×D9—63no—
104C2×D9—126no—
105C2×D9C22×D313yes—
106C4×C9—23no—
107C4×C9—31no—
108C4×C9C22×D313yes—
109Dic9—31no—
110Dic9—61no—
111Dic9C22×D311yescharacteristic
112H(5,𝔽2)—271noSylow
113C3×C9H(5,𝔽2)11yescharacteristic, complemented, Sylow
114C22×D3—96no—
115C22×D3—96no—
116C3×Q8—21no—
117C3×Q8C2×D911yescharacteristic, complemented
118C3×Q8D3211yescharacteristic
119C3×D4—33no—
120C3×D4—93no—
121C2×C3×C4—33no—
122C2×C3×C4—93no—
123#67—93no—
124#67—93no—
125#67—93no—
126#67—93no—
127D12—33no—
128D12—93no—
129D12—93no—
130D12—93no—
131D12—93no—
132D12—183no—
133D12C2×D913yescomplemented
134C4×D3—33no—
135C4×D3—93no—
136C4×D3—93no—
137C4×D3—183no—
138C4×D3C2×D913yescomplemented
139Dic6—91no—
140Dic6—91no—
141C2×C32C23×D311yescharacteristic
142C3⋊D3—63no—
143C3×D3—23no—
144C3×D3—63no—
145C2×C9—21no—
146C2×C9—63no—
147C2×C9C23×D311yescharacteristic
148D9—23no—
149D9—63no—
150D9—123no—
151#41—31no—
152#41—91no—
153#41—273no—
154#41—271no—
155C2×D4—273no—
156C2×D4—273no—
157C2×D4—273no—
158C22×C3—33no—
159C22×C3—93no—
160C2×D3—33no—
161C2×D3—93no—
162C2×D3—93no—
163C2×D3—183no—
164C2×D3—366no—
165C2×D3—366no—
166C2×D3C22×D913yes—
167C3×C4—23no—
168C3×C4—31no—
169C3×C4—91no—
170C3×C4C22×D913yes—
171C3×C4C2×D3213yes—
172Dic3—31no—
173Dic3—91no—
174Dic3—91no—
175Dic3—181no—
176Dic3C22×D911yescharacteristic
177C32#100911yescharacteristic
178C9—21no—
179C9#100911yescharacteristic, complemented
180C23—276no—
181Q8—271no—
182Q8D3×D911yescharacteristic, complemented
183D4—33no—
184D4—93no—
185D4—273no—
186D4—273no—
187D4—273no—
188D4—273no—
189C2×C4—33no—
190C2×C4—93no—
191C2×C4—273no—
192C2×C3—21no—
193C2×C3—63no—
194C2×C3—183no—
195C2×C3C23×D911yescharacteristic
196C2×C3C22×D3211yescharacteristic
197D3—23no—
198D3—63no—
199D3—183no—
200D3—183no—
201D3—363no—
202C22—33no—
203C22—93no—
204C22—273no—
205C22—1086no—
206C4—31no—
207C4—91no—
208C4—271no—
209C4C2×D3×D913yes—
210C3—21no—
211C3#6371511yescharacteristic, complemented
212C3#6432011yescharacteristic
213C2—63no—
214C2—183no—
215C2—543no—
216C2C22×D3×D911yescharacteristic
subgroupG/H
centerC2C22×D3×D9
commutator, cosocleC2×C3×C9C24
FrattiniC2×C3C22×D32
FittingC3×Q8×C9C22
socleC2×C32C23×D3
hypercenterQ8D3×D9
hypocenterC3×C9H(5,𝔽2)
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #88746C2semidirect
2. C9×#239C2semidirect
3. C3×#3701C2semidirect
4. #88646C2semidirect
5. D9×D12C2semidirect
6. D3×D36C2semidirect
7. #88640C2semidirect
10. C2×D3×D9C22semidirect
11. C3×Q8×C9C22semidirect
12. C4⋊(C3⋊D9)C22semidirect
13. C4×C3⋊D9C22semidirect
14. C9×D12C22semidirect
15. C4×D3×C9C22semidirect
16. C3×D36C22semidirect
17. C3×C4×D9C22semidirect
18. #6195C22semidirect
19. #6194C22semidirect
20. #6192C22semidirect
21. #6189C22semidirect
32. #3701D3semidirect
61. Q8×C9C2×D3semidirect
69. D36C2×D3semidirect
72. C4×D9C2×D3semidirect
84. #239D9semidirect
113. C3×C9H(5,𝔽2)semidirect
117. C3×Q8C2×D9semidirect
133. D12C2×D9semidirect
138. C4×D3C2×D9semidirect
179. C9#1009semidirect
182. Q8D3×D9semidirect
211. C3#63715semidirect

Maximal subgroups

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

subgroupindex
#887462
C9×#2392
C3×#37012
#886462
D9×D122
D3×D362
#886402
#643203
#637153