Finite Group Databasebeta

← #11722641

Subgroups — #11722641

Order 864.

518 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 187 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#41×3−1+2C211yesmaximal, characteristic, complemented
2#88724C212yesmaximal, complemented
3D4×AGL(1,ℤ/9)C216yesmaximal, complemented
4#3697⋊C3C213yesmaximal, complemented
5C2×C4⋊AGL(1,ℤ/9)C213yesmaximal, complemented
6C3×#1009—31nomaximal
7#63715C311yesmaximal, characteristic, complemented
8C22×AGL(1,ℤ/9)C2216yescomplemented
9Q8×3−1+2C2211yescharacteristic, complemented
10D4×3−1+2C2213yescomplemented
11C2×C4×3−1+2C2213yescomplemented
12#611⋊C3C2216yescomplemented
13C4⋊AGL(1,ℤ/9)C2213yescomplemented
14C4⋊AGL(1,ℤ/9)C2216yescomplemented
15C4×AGL(1,ℤ/9)C2216yescomplemented
163−1+2⋊Q8C2211yescharacteristic, complemented
17C32×#41D311yescharacteristic
18C3×#239—32no—
19C3×D3×D4—36no—
20C3×#235—33no—
21C2×C3×D12—33no—
22C9×#41—21no—
23C9×#41C2×C311yescharacteristic, complemented
24#3701C2×C312yescomplemented
25D4×D9C2×C316yescomplemented
26#3697C2×C313yescomplemented
27C2×D36C2×C313yescomplemented
28C22×3−1+2C2313yes—
29C2×AGL(1,ℤ/9)—46no—
30C2×AGL(1,ℤ/9)C2316yes—
31C4×3−1+2C2311yescharacteristic
32C4×3−1+2C2313yes—
333−1+2⋊C4C2312yes—
34C3×H(5,𝔽2)—91no—
35#1009—31no—
36C22×C3×D3—36no—
37C32×Q8C2×D311yescharacteristic
38C32×D4C2×D313yes—
39C2×C32×C4C2×D313yes—
40C3×#67—36no—
41C3×D12—33no—
42C3×D12—36no—
43C3×C4×D3—36no—
44C3×Dic6—31no—
45C22×D9C22×C316yescomplemented
46Q8×C9—21no—
47Q8×C9C22×C311yescharacteristic, complemented
48D4×C9—23no—
49D4×C9C22×C313yescomplemented
50C2×C4×C9—23no—
51C2×C4×C9C22×C313yescomplemented
52#611C22×C316yescomplemented
53D36C22×C313yescomplemented
54D36C22×C316yescomplemented
55C4×D9C22×C316yescomplemented
56Dic18C22×C311yescharacteristic, complemented
57C2×3−1+2—23no—
58C2×3−1+2C2411yescharacteristic
59AGL(1,ℤ/9)—26no—
60C3×#41—31no—
61C3×#41—93no—
62C3×#41—92no—
63C3×#41C3×D311yescharacteristic
64C2×C3×D4—93no—
65C2×C3×D4—96no—
66#239—32no—
67D3×D4—36no—
68#235—33no—
69C2×D12—33no—
70C22×C32C22×D313yes—
71C2×C3×D3—36no—
72C2×C3×D3—126no—
73C32×C4C22×D311yescharacteristic
74C32×C4C22×D313yes—
75C3×Dic3—32no—
76C22×C9—23no—
77C22×C9C23×C313yes—
78C2×D9—46no—
79C2×D9C23×C316yes—
80C4×C9—21no—
81C4×C9—23no—
82C4×C9C23×C311yescharacteristic
83C4×C9C23×C313yes—
84Dic9C23×C312yes—
85H(5,𝔽2)—91noSylow
863−1+2H(5,𝔽2)11yescharacteristic, complemented, Sylow
87C23×C3—96no—
88C22×D3—36no—
89C3×Q8—31no—
90C3×Q8—91no—
91C3×Q8C2×C3×D311yescharacteristic
92C3×D4—33no—
93C3×D4—93no—
94C3×D4—96no—
95C3×D4—96no—
96C3×D4C2×C3×D313yes—
97C2×C3×C4—33no—
98C2×C3×C4—96no—
99C2×C3×C4C2×C3×D313yes—
100#67—36no—
101D12—33no—
102D12—36no—
103C4×D3—36no—
104Dic6—31no—
105C2×C32—23no—
106C2×C32C23×D311yescharacteristic
107C3×D3—66no—
108C2×C9—23no—
109C2×C9—21no—
110C2×C9—43no—
111C2×C9C24×C311yescharacteristic
112D9—26no—
113#41—92no—
114#41—93no—
115#41AGL(1,ℤ/9)11yescharacteristic, complemented
116C2×D4—96no—
117C2×D4—93no—
118C22×C3—33no—
119C22×C3—96no—
120C22×C3—366no—
121C22×C3C22×C3×D313yes—
122C2×D3—36no—
123C2×D3—126no—
124C3×C4—31no—
125C3×C4—33no—
126C3×C4—92no—
127C3×C4C22×C3×D311yescharacteristic
128C3×C4C22×C3×D313yes—
129Dic3—32no—
130C32#100911yescharacteristic
131C9—21no—
132C9C3×H(5,𝔽2)11yescharacteristic, complemented
133C23—96no—
134Q8—91no—
135Q8C2×AGL(1,ℤ/9)11yescharacteristic, complemented
136D4—96no—
137D4—96no—
138D4—93no—
139D4C2×AGL(1,ℤ/9)13yescomplemented
140C2×C4—96no—
141C2×C4C2×AGL(1,ℤ/9)13yescomplemented
142C2×C3—23no—
143C2×C3—31no—
144C2×C3—63no—
145C2×C3—186no—
146C2×C3C23×C3×D311yescharacteristic
147D3—66no—
148C22—96no—
149C22—366no—
150C22C22×AGL(1,ℤ/9)13yes—
151C4—92no—
152C4C22×AGL(1,ℤ/9)13yes—
153C4C22×AGL(1,ℤ/9)11yescharacteristic
154C3—31no—
155C3C3×#100911yescharacteristic
156C2—23no—
157C2—186no—
158C2C23×AGL(1,ℤ/9)11yescharacteristic
subgroupG/H
centerC2C23×AGL(1,ℤ/9)
commutator, cosocleC2×C9C24×C3
Frattini, socleC2×C3C23×C3×D3
Fitting#41×3−1+2C2
hypercenter#41AGL(1,ℤ/9)
hypocenterC9C3×H(5,𝔽2)
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #41×3−1+2C2semidirect
2. #88724C2semidirect
3. D4×AGL(1,ℤ/9)C2semidirect
4. #3697⋊C3C2semidirect
5. C2×C4⋊AGL(1,ℤ/9)C2semidirect
7. #63715C3semidirect
8. C22×AGL(1,ℤ/9)C22semidirect
9. Q8×3−1+2C22semidirect
10. D4×3−1+2C22semidirect
11. C2×C4×3−1+2C22semidirect
12. #611⋊C3C22semidirect
13. C4⋊AGL(1,ℤ/9)C22semidirect
14. C4⋊AGL(1,ℤ/9)C22semidirect
15. C4×AGL(1,ℤ/9)C22semidirect
16. 3−1+2⋊Q8C22semidirect
23. C9×#41C2×C3semidirect
24. #3701C2×C3semidirect
25. D4×D9C2×C3semidirect
26. #3697C2×C3semidirect
27. C2×D36C2×C3semidirect
45. C22×D9C22×C3semidirect
47. Q8×C9C22×C3semidirect
49. D4×C9C22×C3semidirect
51. C2×C4×C9C22×C3semidirect
52. #611C22×C3semidirect
53. D36C22×C3semidirect
54. D36C22×C3semidirect
55. C4×D9C22×C3semidirect
56. Dic18C22×C3semidirect
86. 3−1+2H(5,𝔽2)semidirect
115. #41AGL(1,ℤ/9)semidirect
132. C9C3×H(5,𝔽2)semidirect
135. Q8C2×AGL(1,ℤ/9)semidirect
139. D4C2×AGL(1,ℤ/9)semidirect
141. C2×C4C2×AGL(1,ℤ/9)semidirect

Maximal subgroups

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

subgroupindex
#41×3−1+22
#887242
D4×AGL(1,ℤ/9)2
#3697⋊C32
C2×C4⋊AGL(1,ℤ/9)2
C3×#10093
#637153