Finite Group Databasebeta

← #13048328

Subgroups — #13048328

Order 1296.

520 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 55 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
1C2×#66432C211yesmaximal, characteristic, complemented
2C2×C3×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
3Dic3×#272C211yesmaximal, characteristic, complemented
4#10619884C212yesmaximal, complemented
5#6292⋊C3C212yesmaximal, complemented
6D3×#626—32nomaximal
7#88655—31nomaximal
8C2×C3×#272C2211yescharacteristic
9C2×C3⋊((C3⋊D3)⋊C3)C2212yescomplemented
10C2×C3×(C3⋊D3)⋊C3C2212yescomplemented
11#66432—22no—
12C3×((C3⋊D3)⋊C3)⋊C2—22no—
13C3⋊#1084C2211yescharacteristic, complemented
14Dic3×H(3,𝔽3)C2211yescharacteristic, complemented
15C2×D3×C3⋊D3—32no—
16C2×C3×D32—32no—
17#6295—32no—
18#6292D312yescomplemented
19#6291—32no—
20C3×#626—32no—
21C3×D3×Dic3—32no—
22C3×D3×Dic3—61no—
23C2×((C3⋊D3)⋊C3)⋊C2—31no—
24C2×((C3⋊D3)⋊C3)⋊C2D311yescharacteristic, complemented
25C4×#272—31no—
26C4⋊((C3⋊D3)⋊C3)—32no—
27#6198—32no—
28#6198—62no—
29C2×C3×H(3,𝔽3)C2311yescharacteristic
30C3×#272D412yescomplemented
31C3⋊((C3⋊D3)⋊C3)—22no—
32C3×(C3⋊D3)⋊C3—22no—
33D3×#67—92no—
34C2×#626—91no—
35D3×D12—92no—
36C2×C3⋊(C3⋊D3)C2×D312yescomplemented
37C2×C3×C3⋊D3—32no—
38C2×C3×C3⋊D3C2×D312yescomplemented
39C2×C32×D3—32no—
40C2×C32×D3—32no—
41C2×C32×D3—61no—
42D3×C3⋊D3—64no—
43C3×D32—64no—
44C3×C3⋊Dic3—32no—
45C3×C3⋊Dic3—61no—
46C32×Dic3—21no—
47C32×Dic3C2×D312yescomplemented
48C2×#272C2×D311yescharacteristic
49C2×(C3⋊D3)⋊C3—22no—
50C2×(C3⋊D3)⋊C3—32no—
51C2×(C3⋊D3)⋊C3—62no—
52C2×(C3⋊D3)⋊C3C2×D312yescomplemented
53((C3⋊D3)⋊C3)⋊C2—22no—
54((C3⋊D3)⋊C3)⋊C2—62no—
55C4×H(3,𝔽3)—31no—
56#1084—31no—
57#1084—62no—
58#1084—121no—
59C3×H(3,𝔽3)C2×D411yescharacteristic, complemented, Sylow
60C22×C3⋊D3—91no—
61C22×C3×D3—91no—
62C22×C3×D3—92no—
63C2×D32—32no—
64C2×D32—92no—
65C2×D32—92no—
66C2×D32—182no—
67#638—92no—
68C4⋊(C3⋊D3)—32no—
69C3×#67—92no—
70C2×C3×Dic3—91no—
71C3×D12—92no—
72C3×C4×D3—92no—
73C3×C4×D3—181no—
74#626—32no—
75#626—92no—
76#626—92no—
77#626—92no—
78#626—92no—
79#626—182no—
80#625—92no—
81D3×Dic3—32no—
82D3×Dic3—61no—
83C2×C33—21no—
84C2×C33C22×D312yes—
85C3⋊(C3⋊D3)—22no—
86C3×C3⋊D3—22no—
87C3×C3⋊D3—62no—
88C32×D3—34no—
89C32×D3—62no—
90C32×D3—62no—
91C2×H(3,𝔽3)—22no—
92C2×H(3,𝔽3)—41no—
93C2×H(3,𝔽3)C22×D311yescharacteristic
94#272#6712yescomplemented
95(C3⋊D3)⋊C3—22no—
96(C3⋊D3)⋊C3—42no—
97(C3⋊D3)⋊C3—62no—
98(C3⋊D3)⋊C3—122no—
99C2×#67—271no—
100D3×D4—272no—
101C2×D12—271no—
102C22×C32—92no—
103C22×C32—91no—
104C2×C3⋊D3—32no—
105C2×C3⋊D3—32no—
106C2×C3⋊D3—62no—
107C2×C3⋊D3—92no—
108C2×C3⋊D3—182no—
109C2×C3⋊D3—182no—
110C2×C3⋊D3D3212yescomplemented
111C2×C3×D3—32no—
112C2×C3×D3—32no—
113C2×C3×D3—32no—
114C2×C3×D3—32no—
115C2×C3×D3—62no—
116C2×C3×D3—62no—
117C2×C3×D3—61no—
118C2×C3×D3—61no—
119C2×C3×D3—92no—
120C2×C3×D3—92no—
121C2×C3×D3—92no—
122C2×C3×D3—121no—
123C2×C3×D3—182no—
124C2×C3×D3—184no—
125C2×C3×D3—182no—
126D32—64no—
127D32—184no—
128D32—184no—
129D32—188no—
130C32×C4—32no—
131C32×C4—61no—
132C3⋊Dic3—32no—
133C3⋊Dic3—61no—
134C3×Dic3—32no—
135C3×Dic3—61no—
136C3×Dic3—92no—
137C3×Dic3—91no—
138C3×Dic3—181no—
139C3×Dic3—182no—
140C3×Dic3—361no—
141C3×Dic3D3211yescharacteristic
142C33—21no—
143C33D3×D412yescomplemented
144H(3,𝔽3)—22no—
145H(3,𝔽3)—41no—
146H(3,𝔽3)C2×#6711yescharacteristic, complemented
147C23×C3—271no—
148C22×D3—91no—
149C22×D3—92no—
150C22×D3—272no—
151C22×D3—271no—
152C22×D3—271no—
153C22×D3—541no—
154C3×D4—272no—
155C2×C3×C4—271no—
156#67—92no—
157#67—272no—
158#67—272no—
159#67—272no—
160#67—542no—
161C2×Dic3—91no—
162D12—92no—
163D12—272no—
164D12—272no—
165C4×D3—92no—
166C4×D3—181no—
167C2×C32—22no—
168C2×C32—21no—
169C2×C32—32no—
170C2×C32—41no—
171C2×C32—62no—
172C2×C32—61no—
173C2×C32—92no—
174C2×C32—121no—
175C2×C32—182no—
176C2×C32C2×D3211yescharacteristic
177C2×C32C2×D3212yes—
178C3⋊D3—22no—
179C3⋊D3—62no—
180C3⋊D3—62no—
181C3⋊D3—122no—
182C3⋊D3—182no—
183C3⋊D3—184no—
184C3×D3—34no—
185C3×D3—34no—
186C3×D3—64no—
187C3×D3—62no—
188C3×D3—62no—
189C3×D3—62no—
190C3×D3—62no—
191C3×D3—122no—
192C3×D3—122no—
193C3×D3—182no—
194C3×D3—182no—
195C3×D3—184no—
196C3×D3—182no—
197C2×D4—811noSylow
198C22×C3—92no—
199C22×C3—91no—
200C22×C3—91no—
201C22×C3—92no—
202C22×C3—181no—
203C22×C3—182no—
204C22×C3—272no—
205C22×C3—542no—
206C2×D3—32no—
207C2×D3—32no—
208C2×D3—61no—
209C2×D3—92no—
210C2×D3—92no—
211C2×D3—92no—
212C2×D3—182no—
213C2×D3—184no—
214C2×D3—182no—
215C2×D3—272no—
216C2×D3—542no—
217C2×D3—544no—
218C2×D3—542no—
219C2×D3—542no—
220C2×D3—544no—
221C3×C4—31no—
222C3×C4—92no—
223C3×C4—181no—
224C3×C4—271no—
225Dic3—92no—
226Dic3—91no—
227Dic3—181no—
228Dic3—182no—
229Dic3—361no—
230Dic3((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
231C32—21no—
232C32—22no—
233C32—32no—
234C32—41no—
235C32—62no—
236C32—61no—
237C32—121no—
238C32#380211yescharacteristic
239C32D3×#6712yescomplemented
240C23—271no—
241C23—811no—
242D4—272no—
243D4—812no—
244C2×C4—271no—
245C2×C3—21no—
246C2×C3—32no—
247C2×C3—61no—
248C2×C3—62no—
249C2×C3—92no—
250C2×C3—92no—
251C2×C3—121no—
252C2×C3—182no—
253C2×C3—182no—
254C2×C3—182no—
255C2×C3—362no—
256C2×C3—542no—
257C2×C3C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
258C2×C3D3311yescharacteristic
259D3—34no—
260D3—62no—
261D3—62no—
262D3—182no—
263D3—184no—
264D3—182no—
265D3—182no—
266D3—542no—
267D3—1082no—
268C22—91no—
269C22—92no—
270C22—272no—
271C22—542no—
272C22—1622no—
273C4—31no—
274C4—271no—
275C3—21no—
276C3—32no—
277C3—61no—
278C3—62no—
279C3—121no—
280C3#8865511yescharacteristic, complemented
281C3#8895311yescharacteristic
282C2—92no—
283C2—182no—
284C2—542no—
285C2D3×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
center, hypercenterC2D3×((C3⋊D3)⋊C3)⋊C2
commutator, Fitting, cosocleC2×C3×H(3,𝔽3)C23
FrattiniC2×C3D33
socleC2×C32C2×D32
hypocenterC3×H(3,𝔽3)C2×D4
LCSG ▷ C2×C3×H(3,𝔽3) ▷ C3×H(3,𝔽3)
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×#66432C2semidirect
2. C2×C3×((C3⋊D3)⋊C3)⋊C2C2semidirect
3. Dic3×#272C2semidirect
4. #10619884C2semidirect
5. #6292⋊C3C2semidirect
9. C2×C3⋊((C3⋊D3)⋊C3)C22semidirect
10. C2×C3×(C3⋊D3)⋊C3C22semidirect
13. C3⋊#1084C22semidirect
14. Dic3×H(3,𝔽3)C22semidirect
18. #6292D3semidirect
24. C2×((C3⋊D3)⋊C3)⋊C2D3semidirect
30. C3×#272D4semidirect
36. C2×C3⋊(C3⋊D3)C2×D3semidirect
38. C2×C3×C3⋊D3C2×D3semidirect
47. C32×Dic3C2×D3semidirect
52. C2×(C3⋊D3)⋊C3C2×D3semidirect
59. C3×H(3,𝔽3)C2×D4semidirect
94. #272#67semidirect
110. C2×C3⋊D3D32semidirect
143. C33D3×D4semidirect
146. H(3,𝔽3)C2×#67semidirect
230. Dic3((C3⋊D3)⋊C3)⋊C2semidirect
239. C32D3×#67semidirect
280. C3#88655semidirect

Maximal subgroups

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

subgroupindex
C2×#664322
C2×C3×((C3⋊D3)⋊C3)⋊C22
Dic3×#2722
#106198842
#6292⋊C32
D3×#6263
#886553