Finite Group Databasebeta

← #13049504

Subgroups — #13049504

Order 1296.

1140 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 114 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×C3⋊#1113C211yesmaximal, characteristic, complemented
2C2×C3⋊D32C211yesmaximal, characteristic, complemented
3#10620147C212yesmaximal, complemented
4C3×#6292C212yesmaximal, complemented
5C32×#624C211yesmaximal, characteristic, complemented
6D3×C4⋊(C3⋊D3)—32nomaximal
7#88955—34nomaximal
8C2×C3⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
9C2×C3×C3⋊(C3⋊D3)C2212yescomplemented
10C2×C32×C3⋊D3C2211yescharacteristic
11C2×C32×C3⋊D3C2211yescharacteristic, complemented
12C3⋊#1113—22no—
13C3⋊D32—22no—
14C33×Dic3C2212yescomplemented
15C2×#1113—34no—
16C2×D3×C3⋊D3—32no—
17C2×D3×C3⋊D3—34no—
18C2×D3×C3⋊D3—32no—
19C2×D3×C3⋊D3—61no—
20C4⋊(C3⋊(C3⋊D3))—32no—
21C3×C4⋊(C3⋊D3)—32no—
22C32×C4×D3—32no—
23#6292—32no—
24#6292—38no—
25#6292—68no—
26#6292D312yescomplemented
27C3×#626—38no—
28C3×#624D314yescomplemented
29C2×C34C2311yescharacteristic
30C3⋊(C3⋊(C3⋊D3))—21no—
31C3×C3⋊(C3⋊D3)—22no—
32C32×C3⋊D3—21no—
33C32×C3⋊D3D412yescomplemented
34C2×C4⋊(C3⋊D3)—91no—
35#3811—91no—
36D3×D12—98no—
37C2×C3⋊(C3⋊D3)—32no—
38C2×C3⋊(C3⋊D3)—34no—
39C2×C3⋊(C3⋊D3)—61no—
40C2×C3⋊(C3⋊D3)—68no—
41C2×C3⋊(C3⋊D3)—124no—
42C2×C3⋊(C3⋊D3)C2×D312yescomplemented
43C2×C3×C3⋊D3—21no—
44C2×C3×C3⋊D3—32no—
45C2×C3×C3⋊D3—38no—
46C2×C3×C3⋊D3—68no—
47C2×C3×C3⋊D3C2×D314yes—
48C2×C3×C3⋊D3C2×D312yescomplemented
49C2×C32×D3—32no—
50C2×C32×D3—34no—
51C2×C32×D3—61no—
52#1113—68no—
53D3×C3⋊D3—64no—
54D3×C3⋊D3—64no—
55D3×C3⋊D3—64no—
56D3×C3⋊D3—68no—
57C33×C4—32no—
58C32×Dic3—28no—
59C32×Dic3C2×D318yescomplemented
60C32×Dic3C2×D312yescomplemented
61C34C2×D411yescharacteristic, complemented, Sylow
62C22×C3⋊D3—91no—
63C22×C3⋊D3—91no—
64C22×C3⋊D3—91no—
65C2×D32—98no—
66C2×D32—98no—
67C2×D32—91no—
68C2×D32—184no—
69C2×C32×C4—91no—
70C4⋊(C3⋊D3)—32no—
71C4⋊(C3⋊D3)—92no—
72C4⋊(C3⋊D3)—98no—
73C4⋊(C3⋊D3)—188no—
74C3×#67—92no—
75C3×D12—98no—
76C3×C4×D3—38no—
77#626—38no—
78#626—98no—
79#626—92no—
80#626—188no—
81#624C3⋊D311yescharacteristic, complemented
82C2×C33—28no—
83C2×C33—21no—
84C2×C33—44no—
85C2×C33C22×D312yes—
86C2×C33C22×D314yes—
87C3⋊(C3⋊D3)—22no—
88C3⋊(C3⋊D3)—62no—
89C3⋊(C3⋊D3)—64no—
90C3⋊(C3⋊D3)—62no—
91C3⋊(C3⋊D3)—128no—
92C3⋊(C3⋊D3)—128no—
93C3×C3⋊D3—22no—
94C3×C3⋊D3—22no—
95C3×C3⋊D3—62no—
96C3×C3⋊D3—68no—
97C3×C3⋊D3—128no—
98C3×C3⋊D3D1218yescomplemented
99C32×D3—34no—
100C32×D3—62no—
101C32×D3—64no—
102D3×D4—272no—
103C2×D12—274no—
104C22×C32—91no—
105C22×C32—91no—
106C2×C3⋊D3—32no—
107C2×C3⋊D3—38no—
108C2×C3⋊D3—68no—
109C2×C3⋊D3—91no—
110C2×C3⋊D3—98no—
111C2×C3⋊D3—91no—
112C2×C3⋊D3—188no—
113C2×C3⋊D3—184no—
114C2×C3⋊D3—188no—
115C2×C3⋊D3—182no—
116C2×C3⋊D3—182no—
117C2×C3⋊D3—182no—
118C2×C3⋊D3—364no—
119C2×C3⋊D3—364no—
120C2×C3⋊D3—3612no—
121C2×C3⋊D3D3211yescharacteristic, complemented
122C2×C3⋊D3C2×C3⋊D311yescharacteristic
123C2×C3×D3—38no—
124C2×C3×D3—38no—
125C2×C3×D3—64no—
126C2×C3×D3—64no—
127C2×C3×D3—98no—
128C2×C3×D3—92no—
129C2×C3×D3—188no—
130D32—1816no—
131D32—1816no—
132D32—1816no—
133D32—182no—
134C32×C4—32no—
135C32×C4—38no—
136C32×C4—68no—
137C3×Dic3—28no—
138C3×Dic3D3218yescomplemented
139C3×Dic3C2×C3⋊D312yescomplemented
140C33—28no—
141C33—21no—
142C33—44no—
143C33C2×D1214yescomplemented
144C33D3×D412yescomplemented
145C22×D3—272no—
146C22×D3—272no—
147C22×D3—274no—
148C22×D3—274no—
149C22×D3—541no—
150C3×D4—272no—
151C2×C3×C4—94no—
152#67—92no—
153#67—272no—
154D12—98no—
155D12—272no—
156D12—278no—
157D12—548no—
158C4×D3—32no—
159C2×C32—28no—
160C2×C32—24no—
161C2×C32—28no—
162C2×C32—44no—
163C2×C32—412no—
164C2×C32—44no—
165C2×C32—92no—
166C2×C32—181no—
167C2×C32C2×D3218yes—
168C2×C32C2×D3211yescharacteristic
169C2×C32C22×C3⋊D311yescharacteristic
170C3⋊D3—21no—
171C3⋊D3—62no—
172C3⋊D3—68no—
173C3⋊D3—128no—
174C3⋊D3—181no—
175C3⋊D3—181no—
176C3⋊D3—188no—
177C3⋊D3—188no—
178C3⋊D3—368no—
179C3⋊D3—368no—
180C3⋊D3—368no—
181C3⋊D3—368no—
182C3⋊D3—3624no—
183C3⋊D3C4⋊(C3⋊D3)12yescomplemented
184C3×D3—316no—
185C3×D3—68no—
186C3×D3—68no—
187C3×D3—68no—
188C3×D3—188no—
189C3×D3—182no—
190C3×D3—368no—
191C2×D4—811noSylow
192C22×C3—94no—
193C22×C3—92no—
194C22×C3—181no—
195C22×C3—272no—
196C2×D3—32no—
197C2×D3—34no—
198C2×D3—61no—
199C2×D3—98no—
200C2×D3—92no—
201C2×D3—188no—
202C2×D3—274no—
203C2×D3—272no—
204C2×D3—541no—
205C2×D3—548no—
206C2×D3—544no—
207C2×D3—544no—
208C2×D3—548no—
209C2×D3—544no—
210C2×D3—548no—
211C2×D3—1084no—
212C3×C4—38no—
213C3×C4—32no—
214C3×C4—68no—
215Dic3D3×C3⋊D312yescomplemented
216C32—28no—
217C32—24no—
218C32—28no—
219C32—44no—
220C32—412no—
221C32—44no—
222C32D3×D1218yescomplemented
223C32#381111yescharacteristic, complemented
224C32C2×C4⋊(C3⋊D3)11yescharacteristic, complemented
225C23—811no—
226C23—811no—
227D4—272no—
228D4—812no—
229C2×C4—91no—
230C2×C3—28no—
231C2×C3—21no—
232C2×C3—44no—
233C2×C3—98no—
234C2×C3—182no—
235C2×C3—182no—
236C2×C3—542no—
237C2×C3D3314yes—
238C2×C3C2×D3×C3⋊D312yes—
239D3—34no—
240D3—62no—
241D3—64no—
242D3—188no—
243D3—182no—
244D3—368no—
245D3—542no—
246D3—544no—
247D3—542no—
248D3—1088no—
249D3—1088no—
250C22—91no—
251C22—91no—
252C22—272no—
253C22—811no—
254C22—1622no—
255C22—1622no—
256C4—32no—
257C3—21no—
258C3—28no—
259C3—44no—
260C3#8895514yescomplemented
261C3D3×C4⋊(C3⋊D3)12yescomplemented
262C2—92no—
263C2—181no—
264C2—542no—
265C2—1621no—
266C2D32×C3⋊D311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×C3⋊D3
commutator, Fitting, socle, cosocleC2×C34C23
hypocenterC34C2×D4
LCSG ▷ C2×C34 ▷ C34
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊#1113C2semidirect
2. C2×C3⋊D32C2semidirect
3. #10620147C2semidirect
4. C3×#6292C2semidirect
5. C32×#624C2semidirect
8. C2×C3⋊(C3⋊(C3⋊D3))C22semidirect
9. C2×C3×C3⋊(C3⋊D3)C22semidirect
11. C2×C32×C3⋊D3C22semidirect
14. C33×Dic3C22semidirect
26. #6292D3semidirect
28. C3×#624D3semidirect
33. C32×C3⋊D3D4semidirect
42. C2×C3⋊(C3⋊D3)C2×D3semidirect
48. C2×C3×C3⋊D3C2×D3semidirect
59. C32×Dic3C2×D3semidirect
60. C32×Dic3C2×D3semidirect
61. C34C2×D4semidirect
81. #624C3⋊D3semidirect
98. C3×C3⋊D3D12semidirect
121. C2×C3⋊D3D32semidirect
138. C3×Dic3D32semidirect
139. C3×Dic3C2×C3⋊D3semidirect
143. C33C2×D12semidirect
144. C33D3×D4semidirect
183. C3⋊D3C4⋊(C3⋊D3)semidirect
215. Dic3D3×C3⋊D3semidirect
222. C32D3×D12semidirect
223. C32#3811semidirect
224. C32C2×C4⋊(C3⋊D3)semidirect
260. C3#88955semidirect
261. C3D3×C4⋊(C3⋊D3)semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊#11132
C2×C3⋊D322
#106201472
C3×#62922
C32×#6242
D3×C4⋊(C3⋊D3)3
#889553