Finite Group Databasebeta

← #13048736

Subgroups — #13048736

Order 1296.

826 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 79 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
3C3×#6235C211yesmaximal, characteristic, complemented
4#67×H(3,𝔽3)C211yesmaximal, characteristic, complemented
5#10619893C211yesmaximal, characteristic, complemented
6#10619892C211yesmaximal, characteristic, complemented
7#10619890C211yesmaximal, characteristic, complemented
8C3×#3811—34nomaximal
9D4×#272—31nomaximal
10C22×C3×H(3,𝔽3)C2211yescharacteristic, complemented
11C2×C3⋊#272—22no—
12C2×C3⋊#272C2211yescharacteristic
13C2×C3⋊#272C2211yescharacteristic, complemented
14C2×C3×#272C2211yescharacteristic, complemented
15C2×D3×H(3,𝔽3)C2211yescharacteristic, complemented
16D3×#272—22no—
17C3×#1084C2211yescharacteristic, complemented
18Dic3×H(3,𝔽3)C2211yescharacteristic, complemented
19C22×C3×C3⋊D3—34no—
20C2×C3×D32—34no—
21C32×#67—34no—
22C32×#67D314yescomplemented
23C3×#626—34no—
24C3×#626—34no—
25C3×#624—34no—
26C22×#272—31no—
27C22×#272—31no—
28C22×#272—64no—
29D4×H(3,𝔽3)—31no—
30#6235—31no—
31#6235D311yescharacteristic, complemented
32C4⋊#272—31no—
33C4×#272—31no—
34C2×C3×H(3,𝔽3)—21no—
35C2×C3×H(3,𝔽3)C2311yescharacteristic
36C3⋊#272—21no—
37C3⋊#272D412yescomplemented
38C3×#272—21no—
39D3×H(3,𝔽3)—21no—
40C3×D3×D4—94no—
41C3×D3×D4—91no—
42#3811—34no—
43C22×C33C2×D314yescomplemented
44C2×C3×C3⋊D3—34no—
45C2×C3×C3⋊D3—34no—
46C2×C3×C3⋊D3—68no—
47C2×C32×D3—34no—
48C2×C32×D3C2×D314yescomplemented
49C3×D32—68no—
50C32×Dic3—34no—
51C32×Dic3C2×D314yescomplemented
52C22×H(3,𝔽3)—24no—
53C22×H(3,𝔽3)—31no—
54C22×H(3,𝔽3)C2×D311yescharacteristic, complemented
55C2×#272—31no—
56C2×#272—31no—
57C2×#272—62no—
58C2×#272—62no—
59C2×#272—64no—
60C2×#272—64no—
61C2×#272—616no—
62C2×#272C2×D311yescharacteristic, complemented
63C4×H(3,𝔽3)—31no—
64#1084C2×D311yescharacteristic, complemented
65C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
66C22×C3⋊D3—34no—
67C22×C3×D3—94no—
68C22×C3×D3—94no—
69C22×C3×D3—91no—
70C22×C3×D3—91no—
71C22×C3×D3—184no—
72C2×D32—34no—
73C32×D4—34no—
74C32×D4—91no—
75C3×#67—34no—
76C3×#67—34no—
77C3×#67—34no—
78C3×#67—64no—
79C3×#67—94no—
80C3×#67—91no—
81C3×#67C3⋊D311yescharacteristic
82C3×D12—94no—
83C3×D12—91no—
84C3×C4×D3—94no—
85C3×C4×D3—91no—
86#626—34no—
87#626—34no—
88#624—34no—
89C2×C33—24no—
90C2×C33C22×D314yes—
91C3×C3⋊D3—38no—
92C3×C3⋊D3—64no—
93C32×D3—24no—
94C32×D3—64no—
95C2×H(3,𝔽3)—28no—
96C2×H(3,𝔽3)—21no—
97C2×H(3,𝔽3)—24no—
98C2×H(3,𝔽3)—61no—
99C2×H(3,𝔽3)C22×D311yescharacteristic
100#272—21no—
101#272—32no—
102#272—68no—
103#272—68no—
104#272—61no—
105C2×C3×D4—271no—
106D3×D4—94no—
107D3×D4—91no—
108C22×C32—24no—
109C22×C32—34no—
110C22×C32—34no—
111C22×C32—64no—
112C22×C32—91no—
113C22×C32D3214yescomplemented
114C22×C32C2×C3⋊D311yescharacteristic
115C2×C3⋊D3—34no—
116C2×C3⋊D3—34no—
117C2×C3⋊D3—68no—
118C2×C3×D3—34no—
119C2×C3×D3—34no—
120C2×C3×D3—34no—
121C2×C3×D3—64no—
122C2×C3×D3—94no—
123C2×C3×D3—91no—
124C2×C3×D3—94no—
125C2×C3×D3—91no—
126C2×C3×D3—184no—
127C2×C3×D3—184no—
128C2×C3×D3—182no—
129C2×C3×D3—188no—
130C2×C3×D3—188no—
131C2×C3×D3—182no—
132C2×C3×D3—1816no—
133C2×C3×D3C2×C3⋊D311yescharacteristic
134D32—68no—
135C32×C4—34no—
136C32×C4—91no—
137C3×Dic3—34no—
138C3×Dic3—34no—
139C3×Dic3—34no—
140C3×Dic3—64no—
141C3×Dic3C2×C3⋊D311yescharacteristic
142C33D3×D414yescomplemented
143H(3,𝔽3)—24no—
144H(3,𝔽3)D3×D411yescharacteristic, complemented
145C23×C3—271no—
146C23×C3—271no—
147C22×D3—94no—
148C22×D3—94no—
149C22×D3—91no—
150C22×D3—91no—
151C22×D3—184no—
152C3×D4—31no—
153C3×D4—94no—
154C3×D4—91no—
155C3×D4—91no—
156C3×D4—181no—
157C3×D4—271no—
158C3×D4—271no—
159C2×C3×C4—271no—
160#67—34no—
161#67—94no—
162#67—91no—
163#67#27211yescharacteristic, complemented
164D12—94no—
165D12—91no—
166C4×D3—94no—
167C4×D3—91no—
168C2×C32—21no—
169C2×C32—28no—
170C2×C32—24no—
171C2×C32—24no—
172C2×C32—34no—
173C2×C32—64no—
174C2×C32—64no—
175C2×C32—64no—
176C2×C32—124no—
177C2×C32—181no—
178C2×C32C2×D3214yes—
179C2×C32C22×C3⋊D311yescharacteristic
180C3⋊D3—38no—
181C3⋊D3—64no—
182C3×D3—21no—
183C3×D3—64no—
184C3×D3—64no—
185C3×D3—64no—
186C3×D3—98no—
187C3×D3—92no—
188C3×D3—124no—
189C3×D3—188no—
190C3×D3—188no—
191C3×D3—184no—
192C3×D3—181no—
193C2×D4—271noSylow
194C22×C3—21no—
195C22×C3—31no—
196C22×C3—34no—
197C22×C3—64no—
198C22×C3—94no—
199C22×C3—91no—
200C22×C3—91no—
201C22×C3—181no—
202C22×C3—271no—
203C22×C3—271no—
204C22×C3—542no—
205C22×C3—542no—
206C22×C3C2×#27211yescharacteristic, complemented
207C22×C3D3×C3⋊D311yescharacteristic
208C2×D3—34no—
209C2×D3—94no—
210C2×D3—91no—
211C2×D3—94no—
212C2×D3—91no—
213C2×D3—184no—
214C2×D3—184no—
215C2×D3—182no—
216C2×D3—188no—
217C2×D3—182no—
218C2×D3—188no—
219C2×D3—1816no—
220C2×D3C2×#27211yescharacteristic, complemented
221C3×C4—31no—
222C3×C4—94no—
223C3×C4—91no—
224C3×C4—91no—
225C3×C4—181no—
226Dic3—34no—
227Dic3C2×#27211yescharacteristic, complemented
228C32—24no—
229C32—34no—
230C32—64no—
231C32#381114yescomplemented
232C32D4×C3⋊D311yescharacteristic
233C23—271no—
234C23—271no—
235D4—31no—
236D4—91no—
237D4—271no—
238D4—271no—
239C2×C4—271no—
240C2×C3—21no—
241C2×C3—21no—
242C2×C3—21no—
243C2×C3—34no—
244C2×C3—41no—
245C2×C3—64no—
246C2×C3—68no—
247C2×C3—61no—
248C2×C3—64no—
249C2×C3—184no—
250C2×C3—181no—
251C2×C3—181no—
252C2×C3—272no—
253C2×C3—361no—
254C2×C3—541no—
255C2×C3C22×#27211yescharacteristic
256C2×C3C2×D3×C3⋊D311yescharacteristic
257D3—21no—
258D3—64no—
259D3—98no—
260D3—92no—
261D3—188no—
262D3—184no—
263D3—188no—
264D3—181no—
265C22—31no—
266C22—91no—
267C22—271no—
268C22—271no—
269C22—542no—
270C22—542no—
271C22D3×#27211yescharacteristic, complemented
272C4—31no—
273C4—91no—
274C3—21no—
275C3—34no—
276C3—64no—
277C3D4×#27211yescharacteristic, complemented
278C3#8903911yescharacteristic
279C2—21no—
280C2—61no—
281C2—181no—
282C2—272no—
283C2—541no—
284C2C2×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. C3×#6235C2semidirect
4. #67×H(3,𝔽3)C2semidirect
5. #10619893C2semidirect
6. #10619892C2semidirect
7. #10619890C2semidirect
10. C22×C3×H(3,𝔽3)C22semidirect
13. 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
31. #6235D3semidirect
37. C3⋊#272D4semidirect
43. C22×C33C2×D3semidirect
48. C2×C32×D3C2×D3semidirect
51. C32×Dic3C2×D3semidirect
54. C22×H(3,𝔽3)C2×D3semidirect
62. C2×#272C2×D3semidirect
64. #1084C2×D3semidirect
65. C3×H(3,𝔽3)C2×D4semidirect
113. C22×C32D32semidirect
142. C33D3×D4semidirect
144. H(3,𝔽3)D3×D4semidirect
163. #67#272semidirect
206. C22×C3C2×#272semidirect
220. C2×D3C2×#272semidirect
227. Dic3C2×#272semidirect
231. C32#3811semidirect
271. C22D3×#272semidirect
277. 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
C3×#62352
#67×H(3,𝔽3)2
#106198932
#106198922
#106198902
C3×#38113
D4×#2723