Finite Group Databasebeta

← #10589871

Subgroups — #10589871

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#63715C212yesmaximal, complemented
2C9×#135C211yesmaximal, characteristic, complemented
3#63512C212yesmaximal, complemented
4#63481C212yesmaximal, complemented
5#63476C214yesmaximal, complemented
6D8×D9C212yesmaximal, complemented
7C2×D72C211yesmaximal, characteristic, complemented
8#63464C211yesmaximal, characteristic, complemented
9#5769—31nomaximal
10C9×#41C2212yescomplemented
11#3701C2214yescomplemented
12D4×D9—24no—
13D4×D9C2214yescomplemented
14#3697—22no—
15#3697C2211yescharacteristic
16C2×D36—22no—
17C2×D36C2212yescomplemented
18C9×Q16C2211yescharacteristic, complemented
19C9×SD16C2212yescomplemented
20C9×D8C2211yescharacteristic, complemented
21C2×C8×C9C2211yescharacteristic, complemented
22#3675C2214yescomplemented
23#3673C2214yescomplemented
24#3667C2211yescharacteristic, complemented
25D72C2212yescomplemented
26D72C2212yescomplemented
27#3663C2212yescomplemented
28C8×D9C2212yescomplemented
29#1009—32no—
30C3×#135D311yescharacteristic
31#949—32no—
32#919—32no—
33#914—34no—
34D3×D8—32no—
35C2×D24—31no—
36#901—31no—
37C22×D9—22no—
38C22×D9—24no—
39Q8×C9C2312yes—
40D4×C9—22no—
41D4×C9C2312yes—
42C2×C4×C9—22no—
43C2×C4×C9C2311yescharacteristic
44#611—24no—
45#611D412yes—
46D36—24no—
47D36—22no—
48D36D411yescharacteristic
49D36C2314yes—
50C4×D9—24no—
51C4×D9C2312yes—
52Dic18D411yescharacteristic
53C8×C9C2311yescharacteristic
54C8×C9C2311yescharacteristic
55C9⋊C8C2312yes—
56#576—91noSylow
57C3×#41C2×D312yes—
58#239—34no—
59D3×D4—34no—
60D3×D4—64no—
61#235—31no—
62#235—62no—
63C2×D12—32no—
64C2×D12—62no—
65C3×Q16C2×D311yescharacteristic
66C3×SD16C2×D312yes—
67C3×D8C2×D311yescharacteristic
68C2×C3×C8C2×D311yescharacteristic
69#215—34no—
70#213—34no—
71#208—31no—
72D24—32no—
73D24—32no—
74ΓL(1,𝔽25)—32no—
75D3×C8—32no—
76C22×C9—22no—
77C22×C9C2×D411yescharacteristic
78C2×D9—24no—
79C2×D9—84no—
80C2×D9—82no—
81C2×D9C2×D412yes—
82C4×C9—22no—
83C4×C9C2×D411yescharacteristic
84C4×C9C2411yescharacteristic
85Dic9C2×D412yes—
86H(5,𝔽2)—92no—
87AGL(1,ℤ/8)—92no—
88AGL(1,ℤ/8)—94no—
89#135—92no—
90#135D911yescharacteristic, complemented
91C2×D8—92no—
92C2×D8—91no—
93#131—91no—
94C22×D3—64no—
95C22×D3—62no—
96C3×Q8C22×D312yes—
97C3×D4—22no—
98C3×D4C22×D312yes—
99C2×C3×C4—22no—
100C2×C3×C4C22×D311yescharacteristic
101#67—32no—
102#67—64no—
103D12—31no—
104D12—34no—
105D12—64no—
106D12—62no—
107C4×D3—32no—
108C4×D3—64no—
109Dic6—31no—
110C3×C8C22×D311yescharacteristic
111C3×C8C22×D311yescharacteristic
112C3⋊C8—32no—
113C2×C9—21no—
114C2×C9—42no—
115C2×C9C22×D411yescharacteristic
116D9—22no—
117D9—44no—
118#41—91no—
119#41—94no—
120#41—182no—
121#41C2×D912yescomplemented
122C2×D4—94no—
123C2×D4—92no—
124C2×D4—182no—
125C2×D4—184no—
126Q16C2×D911yescharacteristic, complemented
127SD16—94no—
128SD16C2×D912yescomplemented
129D8—94no—
130D8—92no—
131D8—92no—
132D8C2×D911yescharacteristic, complemented
133M16—91no—
134M16—92no—
135C2×C8—92no—
136C2×C8C2×D911yescharacteristic, complemented
137C22×C3—22no—
138C22×C3D3×D411yescharacteristic
139C2×D3—32no—
140C2×D3—64no—
141C2×D3—244no—
142C2×D3—242no—
143C3×C4—22no—
144C3×C4D3×D411yescharacteristic
145C3×C4C23×D311yescharacteristic
146Dic3—32no—
147C9#57611yescharacteristic, complemented, Sylow
148C23—184no—
149C23—182no—
150Q8—91no—
151Q8C22×D912yes—
152D4—22no—
153D4—92no—
154D4—91no—
155D4—94no—
156D4—184no—
157D4—184no—
158D4—182no—
159D4C22×D912yes—
160C2×C4—22no—
161C2×C4—92no—
162C2×C4—184no—
163C2×C4C22×D911yescharacteristic
164C8—92no—
165C8C22×D911yescharacteristic
166C8C22×D911yescharacteristic
167C2×C3—21no—
168C2×C3—42no—
169C2×C3C2×D3×D411yescharacteristic
170D3—62no—
171D3—124no—
172C22—22no—
173C22—92no—
174C22—184no—
175C22—722no—
176C22—724no—
177C22D4×D911yescharacteristic
178C4—22no—
179C4—92no—
180C4D4×D911yescharacteristic
181C4C23×D911yescharacteristic
182C3#576911yescharacteristic
183C2—21no—
184C2—42no—
185C2—182no—
186C2—364no—
187C2C2×D4×D911yescharacteristic
subgroupG/H
centerC2C2×D4×D9
commutator, cosocleC4×C9C24
FrattiniC3×C4C23×D3
FittingC9×#135C2
socleC2×C3C2×D3×D4
hypercenter#135D9
hypocenterC9#576
LCSG ▷ C4×C9 ▷ C2×C9 ▷ C9
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. #63715C2semidirect
2. C9×#135C2semidirect
3. #63512C2semidirect
4. #63481C2semidirect
5. #63476C2semidirect
6. D8×D9C2semidirect
7. C2×D72C2semidirect
8. #63464C2semidirect
10. C9×#41C22semidirect
11. #3701C22semidirect
13. D4×D9C22semidirect
17. C2×D36C22semidirect
18. C9×Q16C22semidirect
19. C9×SD16C22semidirect
20. C9×D8C22semidirect
21. C2×C8×C9C22semidirect
22. #3675C22semidirect
23. #3673C22semidirect
24. #3667C22semidirect
25. D72C22semidirect
26. D72C22semidirect
27. #3663C22semidirect
28. C8×D9C22semidirect
90. #135D9semidirect
121. #41C2×D9semidirect
126. Q16C2×D9semidirect
128. SD16C2×D9semidirect
132. D8C2×D9semidirect
136. C2×C8C2×D9semidirect
147. C9#576semidirect

Maximal subgroups

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

subgroupindex
#637152
C9×#1352
#635122
#634812
#634762
D8×D92
C2×D722
#634642
#57693