Finite Group Databasebeta

← C9×#5968 #13164958

Subgroups — C9×#5968 #13164958

Order 1728.

2576 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C9×2−1+4C211yesmaximal, characteristic, complemented
2C9×#1009C2110yesmaximal, complemented
3D3×C9×#41C2110yesmaximal, complemented
4C9×#1007C215yesmaximal, complemented
5C2×C9×#239C215yesmaximal, complemented
6C3×#5968C311yesmaximal, characteristic
7C9×#585—31nomaximal
8C3×C9×#41C22110yescomplemented
9C2×C3×Q8×C9C2215yescomplemented
10C9×#239C22110yescomplemented
11C9×#239C22120yescomplemented
12D3×Q8×C9C22110yescomplemented
13C9×#237C22110yescomplemented
14D3×D4×C9C22130yescomplemented
15C9×#235C22130yescomplemented
16C2×C9×D12C22115yescomplemented
17C2×C4×D3×C9C22115yescomplemented
18C32×2−1+4C2×C311yescharacteristic
19C3×#1009C2×C3110yes—
20C3×D3×#41C2×C3110yes—
21C3×#1007C2×C315yes—
22C2×C3×#239C2×C315yes—
23C9×2−1+4—21no—
24C9×2−1+4—35no—
25C9×2−1+4D311yescharacteristic, complemented
26C9×H(5,𝔽2)—310no—
27C2×C9×#41—310no—
28C2×C9×#41—35no—
29C22×D3×C9C23115yes—
30C3×Q8×C9C23110yes—
31C3×D4×C9C23110yes—
32C2×C3×C4×C9C23115yes—
33C9×#67C23120yes—
34C2×C9×Dic3C2315yes—
35C9×D12C23130yes—
36C4×D3×C9C23110yes—
37C4×D3×C9C23130yes—
38C9×Dic6C23110yes—
39C3×#585—31no—
40#5968C911yescharacteristic, direct product factor
41C32×#41C22×C3110yes—
42C2×C32×Q8C22×C315yes—
43C3×#239C22×C3110yes—
44C3×#239C22×C3120yes—
45C3×D3×Q8C22×C3110yes—
46C3×#237C22×C3110yes—
47C3×D3×D4C22×C3130yes—
48C3×#235C22×C3130yes—
49C2×C3×D12C22×C3115yes—
50C2×C3×C4×D3C22×C3115yes—
51C9×#41—210no—
52C9×#41—310no—
53C9×#41—320no—
54C9×#41—310no—
55C9×#41—330no—
56C9×#41C2×D3110yescomplemented
57C2×Q8×C9—25no—
58C2×Q8×C9—310no—
59C2×Q8×C9C2×D315yescomplemented
60C2×D4×C9—330no—
61C2×D4×C9—315no—
62C22×C4×C9—315no—
63C22×C3×C9C2415yes—
64C2×D3×C9—415no—
65C2×D3×C9C24110yes—
66C3×C4×C9C24110yes—
67C9×Dic3C2415yes—
68C9×Dic3C2411yescharacteristic
69C3×2−1+4—21no—
70C3×2−1+4—35no—
71C3×2−1+4C2×C911yescharacteristic, complemented
72C3×2−1+4C3×D311yescharacteristic
73C3×H(5,𝔽2)—310no—
74C2×C3×#41—310no—
75C2×C3×#41—35no—
76#1009C2×C9110yescomplemented
77D3×#41C2×C9110yescomplemented
78#1007C2×C915yescomplemented
79C2×#239C2×C915yescomplemented
80C22×C3×D3C23×C3115yes—
81C32×Q8C23×C3110yes—
82C32×D4C23×C3110yes—
83C2×C32×C4C23×C3115yes—
84C3×#67C23×C3120yes—
85C2×C3×Dic3C23×C315yes—
86C3×D12C23×C3130yes—
87C3×C4×D3C23×C3110yes—
88C3×C4×D3C23×C3130yes—
89C3×Dic6C23×C3110yes—
90C23×C9—315no—
91Q8×C9—210no—
92Q8×C9—310no—
93Q8×C9C22×D3110yes—
94D4×C9—210no—
95D4×C9—320no—
96D4×C9—330no—
97D4×C9C22×D3110yes—
98C2×C4×C9—215no—
99C2×C4×C9—35no—
100C2×C4×C9—310no—
101C2×C4×C9—330no—
102C2×C4×C9C22×D3115yes—
103#585—31noSylow
104C2×C3×C9—25no—
105C2×C3×C9C2511yescharacteristic
106D3×C9—210no—
107C3×#41—210no—
108C3×#41—310no—
109C3×#41—320no—
110C3×#41—310no—
111C3×#41—330no—
112C3×#41C22×C9110yescomplemented
113C3×#41C2×C3×D3110yes—
114C2×C3×Q8—25no—
115C2×C3×Q8—310no—
116C2×C3×Q8C22×C915yescomplemented
117C2×C3×Q8C2×C3×D315yes—
118C2×C3×D4—330no—
119C2×C3×D4—315no—
120C22×C3×C4—315no—
121#239C22×C9110yescomplemented
122#239C22×C9120yescomplemented
123D3×Q8C22×C9110yescomplemented
124#237C22×C9110yescomplemented
125D3×D4C22×C9130yescomplemented
126#235C22×C9130yescomplemented
127C2×D12C22×C9115yescomplemented
128C2×C4×D3C22×C9115yescomplemented
129C22×C32C24×C315yes—
130C2×C3×D3—415no—
131C2×C3×D3C24×C3110yes—
132C32×C4C24×C3110yes—
133C3×Dic3C24×C315yes—
134C3×Dic3C24×C311yescharacteristic
135C22×C9—25no—
136C22×C9—310no—
137C22×C9—1215no—
138C22×C9C23×D315yes—
139C4×C9—210no—
140C4×C9—31no—
141C4×C9—35no—
142C4×C9C23×D3110yes—
1432−1+4—35no—
1442−1+4D3×C911yescharacteristic, complemented
145H(5,𝔽2)—310no—
146C2×#41—310no—
147C2×#41—35no—
148C3×C9#58511yescharacteristic, complemented, Sylow
149C23×C3—315no—
150C22×D3C23×C9115yes—
151C3×Q8—210no—
152C3×Q8—310no—
153C3×Q8C23×C9110yes—
154C3×Q8C22×C3×D3110yes—
155C3×D4—210no—
156C3×D4—320no—
157C3×D4—330no—
158C3×D4C23×C9110yes—
159C3×D4C22×C3×D3110yes—
160C2×C3×C4—215no—
161C2×C3×C4—35no—
162C2×C3×C4—310no—
163C2×C3×C4—330no—
164C2×C3×C4C23×C9115yes—
165C2×C3×C4C22×C3×D3115yes—
166#67C23×C9120yes—
167C2×Dic3C23×C915yes—
168D12C23×C9130yes—
169C4×D3C23×C9110yes—
170C4×D3C23×C9130yes—
171Dic6C23×C9110yes—
172C2×C32—25no—
173C2×C32C25×C311yescharacteristic
174C3×D3—210no—
175C2×C9—25no—
176C2×C9—21no—
177C2×C9—45no—
178C2×C9—610no—
179C2×C9C24×D311yescharacteristic
180#41—310no—
181#41—320no—
182#41—310no—
183#41—330no—
184#41C2×D3×C9110yescomplemented
185C2×Q8—310no—
186C2×Q8C2×D3×C915yescomplemented
187C2×D4—330no—
188C2×D4—315no—
189C22×C4—315no—
190C22×C3—25no—
191C22×C3—310no—
192C22×C3—1215no—
193C22×C3C24×C915yes—
194C22×C3C23×C3×D315yes—
195C2×D3—415no—
196C2×D3C24×C9110yes—
197C3×C4—210no—
198C3×C4—31no—
199C3×C4—35no—
200C3×C4C24×C9110yes—
201C3×C4C23×C3×D3110yes—
202Dic3C24×C915yes—
203Dic3C24×C911yescharacteristic
204C32C3×#58511yescharacteristic
205C9—21no—
206C9#596811yescharacteristic, direct product factor
207C23—315no—
208Q8—310no—
209Q8C22×D3×C9110yes—
210D4—320no—
211D4—330no—
212D4C22×D3×C9110yes—
213C2×C4—35no—
214C2×C4—310no—
215C2×C4—330no—
216C2×C4C22×D3×C9115yes—
217C2×C3—25no—
218C2×C3—25no—
219C2×C3—21no—
220C2×C3—45no—
221C2×C3—610no—
222C2×C3C25×C911yescharacteristic
223C2×C3C24×C3×D311yescharacteristic
224D3—210no—
225C22—310no—
226C22—1215no—
227C22C23×D3×C915yes—
228C4—31no—
229C4—35no—
230C4C23×D3×C9110yes—
231C3—21no—
232C3C9×#58511yescharacteristic, complemented
233C3C3×#596811yescharacteristic
234C2—25no—
235C2—610no—
236C2C24×D3×C911yescharacteristic
subgroupG/H
centerC2×C9C24×D3
commutatorC2×C3C25×C9
FrattiniC2×C3C24×C3×D3
FittingC3×C9×2−1+4C2
socle, cosocleC2×C32C25×C3
prime radicalC3×#5968C3
hypercenterC9×2−1+4D3
hypocenterC3C9×#585
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C9 ◁ C9×2−1+4

Complements

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

NH ≅ G/Ntype
1. C3×C9×2−1+4C2semidirect
2. C9×#1009C2semidirect
3. D3×C9×#41C2semidirect
4. C9×#1007C2semidirect
5. C2×C9×#239C2semidirect
8. C3×C9×#41C22semidirect
9. C2×C3×Q8×C9C22semidirect
10. C9×#239C22semidirect
11. C9×#239C22semidirect
12. D3×Q8×C9C22semidirect
13. C9×#237C22semidirect
14. D3×D4×C9C22semidirect
15. C9×#235C22semidirect
16. C2×C9×D12C22semidirect
17. C2×C4×D3×C9C22semidirect
25. C9×2−1+4D3semidirect
40. #5968C9direct (N × H)
56. C9×#41C2×D3semidirect
59. C2×Q8×C9C2×D3semidirect
71. C3×2−1+4C2×C9semidirect
76. #1009C2×C9semidirect
77. D3×#41C2×C9semidirect
78. #1007C2×C9semidirect
79. C2×#239C2×C9semidirect
112. C3×#41C22×C9semidirect
116. C2×C3×Q8C22×C9semidirect
121. #239C22×C9semidirect
122. #239C22×C9semidirect
123. D3×Q8C22×C9semidirect
124. #237C22×C9semidirect
125. D3×D4C22×C9semidirect
126. #235C22×C9semidirect
127. C2×D12C22×C9semidirect
128. C2×C4×D3C22×C9semidirect
144. 2−1+4D3×C9semidirect
148. C3×C9#585semidirect
184. #41C2×D3×C9semidirect
186. C2×Q8C2×D3×C9semidirect
206. C9#5968direct (N × H)
232. C3C9×#585semidirect

Maximal subgroups

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

subgroupindex
C3×C9×2−1+42
C9×#10092
D3×C9×#412
C9×#10072
C2×C9×#2392
C3×#59683
C9×#5853