Finite Group Databasebeta

← C9×#914 #11721546

Subgroups — C9×#914 #11721546

Order 864.

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

normal subgroups only

This group has 224 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×#239C211yesmaximal, characteristic, complemented
2D3×D4×C9C211yesmaximal, characteristic, complemented
3C3×C9×SD16C211yesmaximal, characteristic, complemented
4C9×#215C211yesmaximal, characteristic, complemented
5C9×#213C211yesmaximal, characteristic, complemented
6C9×D24C211yesmaximal, characteristic, complemented
7C9×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
8C3×#914C311yesmaximal, characteristic
9C9×AGL(1,ℤ/8)—31nomaximal
10C22×D3×C9—21no—
11C3×Q8×C9C2211yescharacteristic, complemented
12C3×D4×C9C2211yescharacteristic
13C9×#67—21no—
14C9×D12—21no—
15C9×D12C2211yescharacteristic
16C9×D12C2211yescharacteristic, complemented
17C4×D3×C9—21no—
18C4×D3×C9C2211yescharacteristic
19C3×C8×C9C2211yescharacteristic, complemented
20C9×C3⋊C8C2211yescharacteristic, complemented
21C3×#239C2×C311yescharacteristic
22C3×D3×D4C2×C311yescharacteristic
23C32×SD16C2×C311yescharacteristic
24C3×#215C2×C311yescharacteristic
25C3×#213C2×C311yescharacteristic
26C3×D24C2×C311yescharacteristic
27C3×ΓL(1,𝔽25)C2×C311yescharacteristic
28C9×#41—31no—
29C2×D4×C9—31no—
30C9×SD16—21no—
31C9×SD16—31no—
32C9×SD16D311yescharacteristic, complemented
33C9×D8—31no—
34C9×D8—31no—
35C9×M16—31no—
36C22×C3×C9—21no—
37C2×D3×C9—21no—
38C2×D3×C9—21no—
39C2×D3×C9—42no—
40C2×D3×C9D411yescharacteristic
41C3×C4×C9—21no—
42C3×C4×C9C2311yescharacteristic
43C9×Dic3D411yescharacteristic
44C3×AGL(1,ℤ/8)—31no—
45#914C911yescharacteristic, direct product factor
46C22×C3×D3—21no—
47C32×Q8C22×C311yescharacteristic
48C32×D4C22×C311yescharacteristic
49C3×#67—21no—
50C3×D12—21no—
51C3×D12C22×C311yescharacteristic
52C3×D12C22×C311yescharacteristic
53C3×C4×D3—21no—
54C3×C4×D3C22×C311yescharacteristic
55C23×C9—61no—
56C32×C8C22×C311yescharacteristic
57C3×C3⋊C8C22×C311yescharacteristic
58Q8×C9—21no—
59Q8×C9C2×D311yescharacteristic, complemented
60D4×C9—21no—
61D4×C9—31no—
62D4×C9—31no—
63D4×C9—61no—
64D4×C9—61no—
65D4×C9C2×D311yescharacteristic
66C2×C4×C9—31no—
67C2×C4×C9—61no—
68C8×C9—21no—
69C8×C9—31no—
70C8×C9C2×D311yescharacteristic, complemented
71C2×C3×C9—41no—
72C2×C3×C9C2×D411yescharacteristic
73D3×C9—21no—
74D3×C9—41no—
75D3×C9—41no—
76C3×#41—31no—
77C2×C3×D4—31no—
78#239C2×C911yescharacteristic, complemented
79D3×D4C2×C911yescharacteristic, complemented
80C3×SD16—21no—
81C3×SD16—31no—
82C3×SD16C2×C911yescharacteristic, complemented
83C3×SD16C3×D311yescharacteristic
84C3×D8—31no—
85C3×D8—31no—
86C3×M16—31no—
87#215C2×C911yescharacteristic, complemented
88#213C2×C911yescharacteristic, complemented
89D24C2×C911yescharacteristic, complemented
90ΓL(1,𝔽25)C2×C911yescharacteristic, complemented
91C22×C32—21no—
92C2×C3×D3—21no—
93C2×C3×D3—21no—
94C2×C3×D3—42no—
95C2×C3×D3C3×D411yescharacteristic
96C32×C4—21no—
97C32×C4C23×C311yescharacteristic
98C3×Dic3C3×D411yescharacteristic
99C22×C9—21no—
100C22×C9—31no—
101C22×C9—41no—
102C22×C9—61no—
103C22×C9—61no—
104C22×C9—122no—
105C4×C9—21no—
106C4×C9—21no—
107C4×C9—31no—
108C4×C9—41no—
109C4×C9C22×D311yescharacteristic
110AGL(1,ℤ/8)—31noSylow
111C3×C9AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
112C23×C3—61no—
113C22×D3—21no—
114C3×Q8—21no—
115C3×Q8C22×C911yescharacteristic, complemented
116C3×Q8C2×C3×D311yescharacteristic
117C3×D4—21no—
118C3×D4—31no—
119C3×D4—31no—
120C3×D4—61no—
121C3×D4—61no—
122C3×D4C22×C911yescharacteristic
123C3×D4C2×C3×D311yescharacteristic
124C2×C3×C4—31no—
125C2×C3×C4—61no—
126#67—21no—
127D12—21no—
128D12C22×C911yescharacteristic
129D12C22×C911yescharacteristic, complemented
130C4×D3—21no—
131C4×D3C22×C911yescharacteristic
132C3×C8—21no—
133C3×C8—31no—
134C3×C8C22×C911yescharacteristic, complemented
135C3×C8C2×C3×D311yescharacteristic
136C3⋊C8C22×C911yescharacteristic, complemented
137C2×C32—41no—
138C2×C32C2×C3×D411yescharacteristic
139C3×D3—21no—
140C3×D3—41no—
141C3×D3—41no—
142C2×C9—21no—
143C2×C9—41no—
144C2×C9—61no—
145C2×C9—81no—
146C2×C9—121no—
147C2×C9—121no—
148C2×C9D3×D411yescharacteristic
149#41—31no—
150C2×D4—31no—
151SD16—31no—
152SD16D3×C911yescharacteristic, complemented
153D8—31no—
154D8—31no—
155M16—31no—
156C22×C3—21no—
157C22×C3—21no—
158C22×C3—31no—
159C22×C3—41no—
160C22×C3—61no—
161C22×C3—61no—
162C22×C3—122no—
163C2×D3—21no—
164C2×D3—21no—
165C2×D3—42no—
166C2×D3D4×C911yescharacteristic
167C3×C4—21no—
168C3×C4—21no—
169C3×C4—21no—
170C3×C4—31no—
171C3×C4—41no—
172C3×C4C23×C911yescharacteristic
173C3×C4C22×C3×D311yescharacteristic
174Dic3D4×C911yescharacteristic
175C32C3×AGL(1,ℤ/8)11yescharacteristic
176C9—21no—
177C9#91411yescharacteristic, direct product factor
178C23—61no—
179Q8C2×D3×C911yescharacteristic, complemented
180D4—31no—
181D4—31no—
182D4—61no—
183D4—61no—
184D4C2×D3×C911yescharacteristic
185C2×C4—31no—
186C2×C4—61no—
187C8—31no—
188C8C2×D3×C911yescharacteristic, complemented
189C2×C3—21no—
190C2×C3—41no—
191C2×C3—41no—
192C2×C3—61no—
193C2×C3—81no—
194C2×C3—121no—
195C2×C3—121no—
196C2×C3C2×D4×C911yescharacteristic
197C2×C3C3×D3×D411yescharacteristic
198D3—21no—
199D3—41no—
200D3—41no—
201C22—21no—
202C22—31no—
203C22—61no—
204C22—61no—
205C22—122no—
206C4—21no—
207C4—31no—
208C4C22×D3×C911yescharacteristic
209C3—21no—
210C3C9×AGL(1,ℤ/8)11yescharacteristic, complemented
211C3C3×#91411yescharacteristic
212C2—41no—
213C2—61no—
214C2—121no—
215C2—121no—
216C2D3×D4×C911yescharacteristic
subgroupG/H
centerC2×C9D3×D4
commutatorC3×C4C23×C9
FrattiniC3×C4C22×C3×D3
FittingC3×C9×SD16C2
socleC2×C32C2×C3×D4
prime radicalC3×#914C3
cosocleC32×C4C23×C3
hypercenterC9×SD16D3
hypocenterC3C9×AGL(1,ℤ/8)
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C4×C9 ◁ C9×SD16

Complements

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

NH ≅ G/Ntype
1. C9×#239C2semidirect
2. D3×D4×C9C2semidirect
3. C3×C9×SD16C2semidirect
4. C9×#215C2semidirect
5. C9×#213C2semidirect
6. C9×D24C2semidirect
7. C9×ΓL(1,𝔽25)C2semidirect
11. C3×Q8×C9C22semidirect
16. C9×D12C22semidirect
19. C3×C8×C9C22semidirect
20. C9×C3⋊C8C22semidirect
32. C9×SD16D3semidirect
45. #914C9direct (N × H)
59. Q8×C9C2×D3semidirect
70. C8×C9C2×D3semidirect
78. #239C2×C9semidirect
79. D3×D4C2×C9semidirect
82. C3×SD16C2×C9semidirect
87. #215C2×C9semidirect
88. #213C2×C9semidirect
89. D24C2×C9semidirect
90. ΓL(1,𝔽25)C2×C9semidirect
111. C3×C9AGL(1,ℤ/8)semidirect
115. C3×Q8C22×C9semidirect
129. D12C22×C9semidirect
134. C3×C8C22×C9semidirect
136. C3⋊C8C22×C9semidirect
152. SD16D3×C9semidirect
177. C9#914direct (N × H)
179. Q8C2×D3×C9semidirect
188. C8C2×D3×C9semidirect
210. 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×#2392
D3×D4×C92
C3×C9×SD162
C9×#2152
C9×#2132
C9×D242
C9×ΓL(1,𝔽25)2
C3×#9143
C9×AGL(1,ℤ/8)3