Finite Group Databasebeta

← #67×#272 #13048734

Subgroups — #67×#272 #13048734

Order 1296.

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

normal subgroups only

This group has 770 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
1C22×C3×#272C211yesmaximal, characteristic, complemented
2C2×D3×#272C211yesmaximal, characteristic, complemented
3#10619994C211yesmaximal, characteristic, complemented
4#67×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5#10619893C211yesmaximal, characteristic, complemented
6#10619891C211yesmaximal, characteristic, complemented
7Dic3×#272C211yesmaximal, characteristic, complemented
8C3×D3×#67—34nomaximal
9D4×#272—31nomaximal
10C22×C3×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×C3⋊#272C2211yescharacteristic, complemented
12C2×C3×#272—22no—
13C2×C3×#272C2211yescharacteristic
14C2×C3×#272C2211yescharacteristic, complemented
15C2×D3×H(3,𝔽3)C2211yescharacteristic, complemented
16D3×#272—22no—
17C3⋊#1084C2211yescharacteristic, complemented
18Dic3×H(3,𝔽3)C2211yescharacteristic, complemented
19C22×C32×D3—34no—
20C2×C3×D32—34no—
21C3×#638—34no—
22C32×#67D314yescomplemented
23C3×#626—34no—
24C3×#625—34no—
25C3×D3×Dic3—34no—
26C22×#272—31no—
27C22×#272D311yescharacteristic, complemented
28D4×H(3,𝔽3)—31no—
29#6235—31no—
30#6235—31no—
31#6235—64no—
32C4⋊#272—31no—
33C4×#272—31no—
34C2×C3×H(3,𝔽3)—21no—
35C2×C3×H(3,𝔽3)C2311yescharacteristic
36C3⋊#272—21no—
37C3×#272—21no—
38C3×#272D412yescomplemented
39D3×H(3,𝔽3)—21no—
40C2×C3×#67—91no—
41C3×D3×D4—94no—
42D3×#67—34no—
43C22×C33C2×D314yescomplemented
44C2×C3×C3⋊D3—34no—
45C2×C32×D3—34no—
46C2×C32×D3—34no—
47C2×C32×D3—68no—
48C2×C32×D3C2×D314yescomplemented
49C3×D32—68no—
50C3×C3⋊Dic3—34no—
51C32×Dic3C2×D314yescomplemented
52C22×H(3,𝔽3)—24no—
53C22×H(3,𝔽3)—31no—
54C22×H(3,𝔽3)C2×D311yescharacteristic, complemented
55C2×#272—22no—
56C2×#272—31no—
57C2×#272—62no—
58C2×#272—64no—
59C2×#272C2×D311yescharacteristic
60C2×#272C2×D311yescharacteristic, complemented
61C4×H(3,𝔽3)—31no—
62#1084—31no—
63#1084—64no—
64C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
65C23×C32—91no—
66C22×C3×D3—34no—
67C22×C3×D3—34no—
68C22×C3×D3—64no—
69C22×C3×D3—94no—
70C22×C3×D3—91no—
71C2×D32—34no—
72C32×D4—34no—
73#638—34no—
74C3×#67—34no—
75C3×#67—94no—
76C3×#67—94no—
77C3×#67—91no—
78C3×#67—91no—
79C3×#67—91no—
80C3×#67—184no—
81C3×#67C3⋊D311yescharacteristic
82C2×C3×Dic3—91no—
83C3×D12—94no—
84C3×C4×D3—94no—
85#626—34no—
86#625—34no—
87D3×Dic3—34no—
88C2×C33—24no—
89C2×C33C22×D314yes—
90C3×C3⋊D3—64no—
91C32×D3—24no—
92C32×D3—38no—
93C32×D3—64no—
94C2×H(3,𝔽3)—21no—
95C2×H(3,𝔽3)—24no—
96C2×H(3,𝔽3)—44no—
97C2×H(3,𝔽3)—61no—
98C2×H(3,𝔽3)C22×D311yescharacteristic
99#272—21no—
100#272—61no—
101#272—124no—
102#272#6712yesdirect product factor
103C2×C3×D4—271no—
104C2×#67—91no—
105D3×D4—94no—
106C22×C32—24no—
107C22×C32—34no—
108C22×C32—34no—
109C22×C32—64no—
110C22×C32—91no—
111C22×C32—91no—
112C22×C32—182no—
113C22×C32D3214yescomplemented
114C22×C32C2×C3⋊D311yescharacteristic
115C2×C3⋊D3—34no—
116C2×C3×D3—34no—
117C2×C3×D3—34no—
118C2×C3×D3—34no—
119C2×C3×D3—34no—
120C2×C3×D3—34no—
121C2×C3×D3—64no—
122C2×C3×D3—64no—
123C2×C3×D3—68no—
124C2×C3×D3—68no—
125C2×C3×D3—616no—
126C2×C3×D3—94no—
127C2×C3×D3—91no—
128C2×C3×D3—184no—
129C2×C3×D3—182no—
130C2×C3×D3—188no—
131C2×C3×D3C2×C3⋊D311yescharacteristic
132D32—68no—
133C32×C4—34no—
134C3⋊Dic3—34no—
135C3×Dic3—34no—
136C3×Dic3—94no—
137C3×Dic3—91no—
138C3×Dic3—184no—
139C3×Dic3C2×C3⋊D311yescharacteristic
140C33D3×D414yescomplemented
141H(3,𝔽3)—24no—
142H(3,𝔽3)C2×#6711yescharacteristic, complemented
143C23×C3—91no—
144C23×C3—91no—
145C23×C3—181no—
146C23×C3—271no—
147C22×D3—34no—
148C22×D3—94no—
149C22×D3—91no—
150C3×D4—31no—
151C3×D4—94no—
152C3×D4—271no—
153C3×D4—271no—
154C3×D4—271no—
155C2×C3×C4—271no—
156#67—94no—
157#67—94no—
158#67—91no—
159#67—91no—
160#67—91no—
161#67—184no—
162#67#27211yescharacteristic, direct product factor
163C2×Dic3—91no—
164D12—94no—
165C4×D3—94no—
166C2×C32—21no—
167C2×C32—24no—
168C2×C32—24no—
169C2×C32—34no—
170C2×C32—44no—
171C2×C32—64no—
172C2×C32—64no—
173C2×C32—68no—
174C2×C32—64no—
175C2×C32—92no—
176C2×C32—181no—
177C2×C32C2×D3214yes—
178C2×C32C22×C3⋊D311yescharacteristic
179C3⋊D3—64no—
180C3×D3—21no—
181C3×D3—38no—
182C3×D3—38no—
183C3×D3—64no—
184C3×D3—68no—
185C3×D3—64no—
186C3×D3—68no—
187C3×D3—64no—
188C3×D3—184no—
189C3×D3—181no—
190C3×D3—364no—
191C2×D4—271noSylow
192C22×C3—21no—
193C22×C3—31no—
194C22×C3—34no—
195C22×C3—64no—
196C22×C3—94no—
197C22×C3—91no—
198C22×C3—91no—
199C22×C3—91no—
200C22×C3—91no—
201C22×C3—181no—
202C22×C3—181no—
203C22×C3—182no—
204C22×C3—182no—
205C22×C3—184no—
206C22×C3—271no—
207C22×C3—542no—
208C22×C3C2×#27211yescharacteristic, complemented
209C22×C3D3×C3⋊D311yescharacteristic
210C2×D3—34no—
211C2×D3—34no—
212C2×D3—68no—
213C2×D3—94no—
214C2×D3—91no—
215C2×D3—184no—
216C2×D3—182no—
217C2×D3—188no—
218C2×D3C2×#27211yescharacteristic, complemented
219C3×C4—31no—
220C3×C4—94no—
221C3×C4—271no—
222Dic3—94no—
223Dic3—91no—
224Dic3—184no—
225Dic3C2×#27211yescharacteristic, complemented
226C32—24no—
227C32—34no—
228C32—64no—
229C32D3×#6714yescomplemented
230C32D4×C3⋊D311yescharacteristic
231C23—91no—
232C23—271no—
233D4—31no—
234D4—271no—
235D4—271no—
236D4—271no—
237C2×C4—271no—
238C2×C3—22no—
239C2×C3—21no—
240C2×C3—21no—
241C2×C3—21no—
242C2×C3—34no—
243C2×C3—64no—
244C2×C3—61no—
245C2×C3—64no—
246C2×C3—92no—
247C2×C3—92no—
248C2×C3—124no—
249C2×C3—184no—
250C2×C3—182no—
251C2×C3—182no—
252C2×C3—181no—
253C2×C3—181no—
254C2×C3—541no—
255C2×C3C22×#27211yescharacteristic
256C2×C3C2×D3×C3⋊D311yescharacteristic
257D3—21no—
258D3—38no—
259D3—64no—
260D3—184no—
261D3—181no—
262D3—364no—
263C22—31no—
264C22—91no—
265C22—91no—
266C22—182no—
267C22—271no—
268C22—542no—
269C22D3×#27211yescharacteristic, complemented
270C4—31no—
271C4—271no—
272C3—21no—
273C3—34no—
274C3—64no—
275C3D4×#27211yescharacteristic, complemented
276C3C3⋊D3×#6711yescharacteristic
277C2—21no—
278C2—61no—
279C2—92no—
280C2—181no—
281C2—541no—
282C2C2×D3×#27211yescharacteristic
subgroupG/H
center, FrattiniC2×C3C2×D3×C3⋊D3
commutator, cosocleC2×C3×H(3,𝔽3)C23
FittingC22×C3×H(3,𝔽3)C22
socleC2×C32C22×C3⋊D3
hypercenterC22×C3D3×C3⋊D3
hypocenterC3×H(3,𝔽3)C2×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×#272C2semidirect
2. C2×D3×#272C2semidirect
3. #10619994C2semidirect
4. #67×H(3,𝔽3)C2semidirect
5. #10619893C2semidirect
6. #10619891C2semidirect
7. Dic3×#272C2semidirect
10. C22×C3×H(3,𝔽3)C22semidirect
11. C2×C3⋊#272C22semidirect
14. C2×C3×#272C22semidirect
15. C2×D3×H(3,𝔽3)C22semidirect
17. C3⋊#1084C22semidirect
18. Dic3×H(3,𝔽3)C22semidirect
22. C32×#67D3semidirect
27. C22×#272D3semidirect
38. C3×#272D4semidirect
43. C22×C33C2×D3semidirect
48. C2×C32×D3C2×D3semidirect
51. C32×Dic3C2×D3semidirect
54. C22×H(3,𝔽3)C2×D3semidirect
60. C2×#272C2×D3semidirect
64. C3×H(3,𝔽3)C2×D4semidirect
102. #272#67direct (N × H)
113. C22×C32D32semidirect
140. C33D3×D4semidirect
142. H(3,𝔽3)C2×#67semidirect
162. #67#272direct (N × H)
208. C22×C3C2×#272semidirect
218. C2×D3C2×#272semidirect
225. Dic3C2×#272semidirect
229. C32D3×#67semidirect
269. C22D3×#272semidirect
275. C3D4×#272semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#2722
C2×D3×#2722
#106199942
#67×H(3,𝔽3)2
#106198932
#106198912
Dic3×#2722
C3×D3×#673
D4×#2723