Finite Group Databasebeta

← C9×#916 #11721548

Subgroups — C9×#916 #11721548

Order 864.

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

normal subgroups only

This group has 206 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
2C9×#237C211yesmaximal, characteristic, complemented
3C3×C9×SD16C211yesmaximal, characteristic, complemented
4C9×#216C211yesmaximal, characteristic, complemented
5C9×#213C211yesmaximal, characteristic, complemented
6C9×#204C211yesmaximal, characteristic, complemented
7D3×C8×C9C211yesmaximal, characteristic, complemented
8C3×#916C311yesmaximal, characteristic
9C9×#135—31nomaximal
10C3×Q8×C9C2211yescharacteristic
11C3×D4×C9C2211yescharacteristic
12C9×#67—21no—
13C2×C9×Dic3—21no—
14C9×D12—21no—
15C9×D12C2211yescharacteristic
16C4×D3×C9—21no—
17C4×D3×C9C2211yescharacteristic
18C9×Dic6C2211yescharacteristic
19C3×C8×C9C2211yescharacteristic
20C9×C3⋊C8C2211yescharacteristic
21C3×#239C2×C311yescharacteristic
22C3×#237C2×C311yescharacteristic
23C32×SD16C2×C311yescharacteristic
24C3×#216C2×C311yescharacteristic
25C3×#213C2×C311yescharacteristic
26C3×#204C2×C311yescharacteristic
27C3×D3×C8C2×C311yescharacteristic
28C9×#41—31no—
29C9×#41—31no—
30C9×Q16—31no—
31C9×SD16—21no—
32C9×SD16—31no—
33C9×SD16D311yescharacteristic, complemented
34C9×D8—31no—
35C2×C8×C9—31no—
36C22×C3×C9—21no—
37C2×D3×C9—21no—
38C2×D3×C9D411yescharacteristic
39C3×C4×C9—21no—
40C3×C4×C9C2311yescharacteristic
41C9×Dic3—21no—
42C9×Dic3D411yescharacteristic
43C3×#135—31no—
44#916C911yescharacteristic, direct product factor
45C32×Q8C22×C311yescharacteristic
46C32×D4C22×C311yescharacteristic
47C3×#67—21no—
48C2×C3×Dic3—21no—
49C3×D12—21no—
50C3×D12C22×C311yescharacteristic
51C3×C4×D3—21no—
52C3×C4×D3C22×C311yescharacteristic
53C3×Dic6C22×C311yescharacteristic
54C32×C8C22×C311yescharacteristic
55C3×C3⋊C8C22×C311yescharacteristic
56Q8×C9—21no—
57Q8×C9—31no—
58Q8×C9C2×D311yescharacteristic
59D4×C9—21no—
60D4×C9—31no—
61D4×C9—61no—
62D4×C9—61no—
63D4×C9C2×D311yescharacteristic
64C2×C4×C9—31no—
65C2×C4×C9—61no—
66C2×C4×C9—61no—
67C8×C9—21no—
68C8×C9—31no—
69C8×C9C2×D311yescharacteristic
70C2×C3×C9—41no—
71C2×C3×C9C2×D411yescharacteristic
72D3×C9—21no—
73D3×C9—41no—
74C3×#41—31no—
75C3×#41—31no—
76#239C2×C911yescharacteristic, complemented
77#237C2×C911yescharacteristic, complemented
78C3×Q16—31no—
79C3×SD16—21no—
80C3×SD16—31no—
81C3×SD16C2×C911yescharacteristic, complemented
82C3×SD16C3×D311yescharacteristic
83C3×D8—31no—
84C2×C3×C8—31no—
85#216C2×C911yescharacteristic, complemented
86#213C2×C911yescharacteristic, complemented
87#204C2×C911yescharacteristic, complemented
88D3×C8C2×C911yescharacteristic, complemented
89C22×C32—21no—
90C2×C3×D3—21no—
91C2×C3×D3C3×D411yescharacteristic
92C32×C4—21no—
93C32×C4C23×C311yescharacteristic
94C3×Dic3—21no—
95C3×Dic3C3×D411yescharacteristic
96C22×C9—21no—
97C22×C9—31no—
98C22×C9—41no—
99C22×C9—61no—
100C4×C9—21no—
101C4×C9—21no—
102C4×C9—31no—
103C4×C9—41no—
104C4×C9—61no—
105C4×C9C22×D311yescharacteristic
106#135—31noSylow
107C3×C9#13511yescharacteristic, complemented, Sylow
108C3×Q8—21no—
109C3×Q8—31no—
110C3×Q8C22×C911yescharacteristic
111C3×Q8C2×C3×D311yescharacteristic
112C3×D4—21no—
113C3×D4—31no—
114C3×D4—61no—
115C3×D4—61no—
116C3×D4C22×C911yescharacteristic
117C3×D4C2×C3×D311yescharacteristic
118C2×C3×C4—31no—
119C2×C3×C4—61no—
120C2×C3×C4—61no—
121#67—21no—
122C2×Dic3—21no—
123D12—21no—
124D12C22×C911yescharacteristic
125C4×D3—21no—
126C4×D3C22×C911yescharacteristic
127Dic6C22×C911yescharacteristic
128C3×C8—21no—
129C3×C8—31no—
130C3×C8C22×C911yescharacteristic
131C3×C8C2×C3×D311yescharacteristic
132C3⋊C8C22×C911yescharacteristic
133C2×C32—41no—
134C2×C32C2×C3×D411yescharacteristic
135C3×D3—21no—
136C3×D3—41no—
137C2×C9—21no—
138C2×C9—41no—
139C2×C9—61no—
140C2×C9—81no—
141C2×C9—121no—
142C2×C9D3×D411yescharacteristic
143#41—31no—
144#41—31no—
145Q16—31no—
146SD16—31no—
147SD16D3×C911yescharacteristic, complemented
148D8—31no—
149C2×C8—31no—
150C22×C3—21no—
151C22×C3—21no—
152C22×C3—31no—
153C22×C3—41no—
154C22×C3—61no—
155C2×D3—21no—
156C2×D3D4×C911yescharacteristic
157C3×C4—21no—
158C3×C4—21no—
159C3×C4—21no—
160C3×C4—31no—
161C3×C4—41no—
162C3×C4—61no—
163C3×C4C23×C911yescharacteristic
164C3×C4C22×C3×D311yescharacteristic
165Dic3—21no—
166Dic3D4×C911yescharacteristic
167C32C3×#13511yescharacteristic
168C9—21no—
169C9#91611yescharacteristic, direct product factor
170Q8—31no—
171Q8C2×D3×C911yescharacteristic
172D4—31no—
173D4—61no—
174D4—61no—
175D4C2×D3×C911yescharacteristic
176C2×C4—31no—
177C2×C4—61no—
178C2×C4—61no—
179C8—31no—
180C8C2×D3×C911yescharacteristic
181C2×C3—21no—
182C2×C3—41no—
183C2×C3—41no—
184C2×C3—61no—
185C2×C3—81no—
186C2×C3—121no—
187C2×C3C2×D4×C911yescharacteristic
188C2×C3C3×D3×D411yescharacteristic
189D3—21no—
190D3—41no—
191C22—21no—
192C22—31no—
193C22—61no—
194C4—21no—
195C4—31no—
196C4—61no—
197C4C22×D3×C911yescharacteristic
198C3—21no—
199C3C9×#13511yescharacteristic, complemented
200C3C3×#91611yescharacteristic
201C2—41no—
202C2—61no—
203C2—121no—
204C2D3×D4×C911yescharacteristic
subgroupG/H
centerC2×C9D3×D4
commutatorC3×C4C23×C9
FrattiniC3×C4C22×C3×D3
FittingC3×C9×SD16C2
socleC2×C32C2×C3×D4
prime radicalC3×#916C3
cosocleC32×C4C23×C3
hypercenterC9×SD16D3
hypocenterC3C9×#135
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. C9×#237C2semidirect
3. C3×C9×SD16C2semidirect
4. C9×#216C2semidirect
5. C9×#213C2semidirect
6. C9×#204C2semidirect
7. D3×C8×C9C2semidirect
33. C9×SD16D3semidirect
44. #916C9direct (N × H)
76. #239C2×C9semidirect
77. #237C2×C9semidirect
81. C3×SD16C2×C9semidirect
85. #216C2×C9semidirect
86. #213C2×C9semidirect
87. #204C2×C9semidirect
88. D3×C8C2×C9semidirect
107. C3×C9#135semidirect
147. SD16D3×C9semidirect
169. C9#916direct (N × H)
199. C3C9×#135semidirect

Maximal subgroups

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

subgroupindex
C9×#2392
C9×#2372
C3×C9×SD162
C9×#2162
C9×#2132
C9×#2042
D3×C8×C92
C3×#9163
C9×#1353