Finite Group Databasebeta

← C9×#911 #11721533

Subgroups — C9×#911 #11721533

Order 864.

226 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C9×#237C211yesmaximal, characteristic, complemented
2D3×D4×C9C211yesmaximal, characteristic, complemented
3C3×C9×D8C211yesmaximal, characteristic, complemented
4C9×#214C211yesmaximal, characteristic, complemented
5C9×#213C211yesmaximal, characteristic, complemented
6C9×#204C211yesmaximal, characteristic, complemented
7C9×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
8C3×#911C311yesmaximal, characteristic
9C9×AGL(1,ℤ/8)—31nomaximal
10C22×D3×C9—21no—
11C3×D4×C9C2211yescharacteristic
12C3×D4×C9C2211yescharacteristic, complemented
13C9×#67—21no—
14C9×#67—21no—
15C2×C9×Dic3—21no—
16C9×D12C2211yescharacteristic
17C4×D3×C9C2211yescharacteristic
18C9×Dic6C2211yescharacteristic, complemented
19C3×C8×C9C2211yescharacteristic, complemented
20C9×C3⋊C8C2211yescharacteristic, complemented
21C3×#237C2×C311yescharacteristic
22C3×D3×D4C2×C311yescharacteristic
23C32×D8C2×C311yescharacteristic
24C3×#214C2×C311yescharacteristic
25C3×#213C2×C311yescharacteristic
26C3×#204C2×C311yescharacteristic
27C3×ΓL(1,𝔽25)C2×C311yescharacteristic
28C9×#41—31no—
29C2×D4×C9—31no—
30C9×SD16—31no—
31C9×SD16—31no—
32C9×D8—21no—
33C9×D8—31no—
34C9×D8D311yescharacteristic, complemented
35C9×M16—31no—
36C22×C3×C9—21no—
37C22×C3×C9—21no—
38C2×D3×C9—21no—
39C2×D3×C9—42no—
40C2×D3×C9D411yescharacteristic
41C3×C4×C9C2311yescharacteristic
42C9×Dic3—21no—
43C9×Dic3D411yescharacteristic
44C3×AGL(1,ℤ/8)—31no—
45#911C911yescharacteristic, direct product factor
46C22×C3×D3—21no—
47C32×D4C22×C311yescharacteristic
48C32×D4C22×C311yescharacteristic
49C3×#67—21no—
50C3×#67—21no—
51C2×C3×Dic3—21no—
52C3×D12C22×C311yescharacteristic
53C3×C4×D3C22×C311yescharacteristic
54C3×Dic6C22×C311yescharacteristic
55C23×C9—61no—
56C32×C8C22×C311yescharacteristic
57C3×C3⋊C8C22×C311yescharacteristic
58Q8×C9—31no—
59D4×C9—21no—
60D4×C9—21no—
61D4×C9—31no—
62D4×C9—61no—
63D4×C9—61no—
64D4×C9C2×D311yescharacteristic
65D4×C9C2×D311yescharacteristic, complemented
66C2×C4×C9—31no—
67C2×C4×C9—61no—
68C8×C9—21no—
69C8×C9—31no—
70C8×C9C2×D311yescharacteristic, complemented
71C2×C3×C9—41no—
72C2×C3×C9—41no—
73C2×C3×C9C2×D411yescharacteristic
74D3×C9—21no—
75D3×C9—41no—
76C3×#41—31no—
77C2×C3×D4—31no—
78#237C2×C911yescharacteristic, complemented
79D3×D4C2×C911yescharacteristic, complemented
80C3×SD16—31no—
81C3×SD16—31no—
82C3×D8—21no—
83C3×D8—31no—
84C3×D8C2×C911yescharacteristic, complemented
85C3×D8C3×D311yescharacteristic
86C3×M16—31no—
87#214C2×C911yescharacteristic, complemented
88#213C2×C911yescharacteristic, complemented
89#204C2×C911yescharacteristic, complemented
90ΓL(1,𝔽25)C2×C911yescharacteristic, complemented
91C22×C32—21no—
92C22×C32—21no—
93C2×C3×D3—21no—
94C2×C3×D3—42no—
95C2×C3×D3C3×D411yescharacteristic
96C32×C4C23×C311yescharacteristic
97C3×Dic3—21no—
98C3×Dic3C3×D411yescharacteristic
99C22×C9—21no—
100C22×C9—21no—
101C22×C9—31no—
102C22×C9—41no—
103C22×C9—41no—
104C22×C9—61no—
105C22×C9—122no—
106C4×C9—21no—
107C4×C9—31no—
108C4×C9—61no—
109C4×C9C22×D311yescharacteristic
110AGL(1,ℤ/8)—31noSylow
111C3×C9AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
112C23×C3—61no—
113C22×D3—21no—
114C3×Q8—31no—
115C3×D4—21no—
116C3×D4—21no—
117C3×D4—31no—
118C3×D4—61no—
119C3×D4—61no—
120C3×D4C22×C911yescharacteristic
121C3×D4C22×C911yescharacteristic, complemented
122C3×D4C2×C3×D311yescharacteristic
123C3×D4C2×C3×D311yescharacteristic
124C2×C3×C4—31no—
125C2×C3×C4—61no—
126#67—21no—
127#67—21no—
128C2×Dic3—21no—
129D12C22×C911yescharacteristic
130C4×D3C22×C911yescharacteristic
131Dic6C22×C911yescharacteristic, complemented
132C3×C8—21no—
133C3×C8—31no—
134C3×C8C22×C911yescharacteristic, complemented
135C3×C8C2×C3×D311yescharacteristic
136C3⋊C8C22×C911yescharacteristic, complemented
137C2×C32—41no—
138C2×C32—41no—
139C2×C32C2×C3×D411yescharacteristic
140C3×D3—21no—
141C3×D3—41no—
142C2×C9—21no—
143C2×C9—41no—
144C2×C9—41no—
145C2×C9—61no—
146C2×C9—81no—
147C2×C9—81no—
148C2×C9—121no—
149C2×C9D3×D411yescharacteristic
150#41—31no—
151C2×D4—31no—
152SD16—31no—
153SD16—31no—
154D8—31no—
155D8D3×C911yescharacteristic, complemented
156M16—31no—
157C22×C3—21no—
158C22×C3—21no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—31no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3—61no—
165C22×C3—122no—
166C2×D3—21no—
167C2×D3—42no—
168C2×D3D4×C911yescharacteristic
169C3×C4—21no—
170C3×C4—31no—
171C3×C4—61no—
172C3×C4C23×C911yescharacteristic
173C3×C4C22×C3×D311yescharacteristic
174Dic3—21no—
175Dic3D4×C911yescharacteristic
176C32C3×AGL(1,ℤ/8)11yescharacteristic
177C9—21no—
178C9#91111yescharacteristic, direct product factor
179C23—61no—
180Q8—31no—
181D4—31no—
182D4—61no—
183D4—61no—
184D4C2×D3×C911yescharacteristic
185D4C2×D3×C911yescharacteristic, complemented
186C2×C4—31no—
187C2×C4—61no—
188C8—31no—
189C8C2×D3×C911yescharacteristic, complemented
190C2×C3—21no—
191C2×C3—41no—
192C2×C3—41no—
193C2×C3—41no—
194C2×C3—41no—
195C2×C3—61no—
196C2×C3—81no—
197C2×C3—81no—
198C2×C3—121no—
199C2×C3C2×D4×C911yescharacteristic
200C2×C3C3×D3×D411yescharacteristic
201D3—21no—
202D3—41no—
203C22—21no—
204C22—21no—
205C22—31no—
206C22—61no—
207C22—122no—
208C4—31no—
209C4—61no—
210C4C22×D3×C911yescharacteristic
211C3—21no—
212C3C9×AGL(1,ℤ/8)11yescharacteristic, complemented
213C3C3×#91111yescharacteristic
214C2—41no—
215C2—41no—
216C2—61no—
217C2—121no—
218C2D3×D4×C911yescharacteristic
subgroupG/H
centerC2×C9D3×D4
commutatorC3×C4C23×C9
FrattiniC3×C4C22×C3×D3
FittingC3×C9×D8C2
socleC2×C32C2×C3×D4
prime radicalC3×#911C3
cosocleC32×C4C23×C3
hypercenterC9×D8D3
hypocenterC3C9×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C4×C9 ◁ C9×D8

Complements

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

NH ≅ G/Ntype
1. C9×#237C2semidirect
2. D3×D4×C9C2semidirect
3. C3×C9×D8C2semidirect
4. C9×#214C2semidirect
5. C9×#213C2semidirect
6. C9×#204C2semidirect
7. C9×ΓL(1,𝔽25)C2semidirect
12. C3×D4×C9C22semidirect
18. C9×Dic6C22semidirect
19. C3×C8×C9C22semidirect
20. C9×C3⋊C8C22semidirect
34. C9×D8D3semidirect
45. #911C9direct (N × H)
65. D4×C9C2×D3semidirect
70. C8×C9C2×D3semidirect
78. #237C2×C9semidirect
79. D3×D4C2×C9semidirect
84. C3×D8C2×C9semidirect
87. #214C2×C9semidirect
88. #213C2×C9semidirect
89. #204C2×C9semidirect
90. ΓL(1,𝔽25)C2×C9semidirect
111. C3×C9AGL(1,ℤ/8)semidirect
121. C3×D4C22×C9semidirect
131. Dic6C22×C9semidirect
134. C3×C8C22×C9semidirect
136. C3⋊C8C22×C9semidirect
155. D8D3×C9semidirect
178. C9#911direct (N × H)
185. D4C2×D3×C9semidirect
189. C8C2×D3×C9semidirect
212. C3C9×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
C9×#2372
D3×D4×C92
C3×C9×D82
C9×#2142
C9×#2132
C9×#2042
C9×ΓL(1,𝔽25)2
C3×#9113
C9×AGL(1,ℤ/8)3