Finite Group Databasebeta

← C2×C3×#560⋊C3 #11919770

Subgroups — C2×C3×#560⋊C3 #11919770

Order 1152.

704 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 132 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
1C3×#560⋊C3C214yesmaximal, direct product factor
2C2×C3×#127⋊C3C211yesmaximal, characteristic, complemented
3C2×C3×#126⋊C3C212yesmaximal, complemented
4C2×C3×#560C311yesmaximal, characteristic, complemented
5C2×#560⋊C3C313yesmaximal, direct product factor
6C23×C3×A4—41nomaximal
7C3×#127⋊C3C2212yescomplemented
8C3×#126⋊C3C2214yescomplemented
9C2×C3×C42⋊C3C2211yescharacteristic, complemented
10C3×#560—34no—
11C3×#560C2×C314yescomplemented
12C2×C3×#127C2×C311yescharacteristic, complemented
13C2×C3×#126C2×C312yescomplemented
14C2×C3×#123—31no—
15C2×C3×#121—32no—
16C2×C3×#120—31no—
17#560⋊C3C2×C3112yesdirect product factor
18C2×#127⋊C3C2×C313yescomplemented
19C2×#126⋊C3C2×C316yescomplemented
20C22×C3×A4—42no—
21C22×C3×A4—41no—
22C22×C3×A4—44no—
23C3×C42⋊C3C2311yescharacteristic, complemented
24C2×#560C3211yescharacteristic, complemented, Sylow
25C25×C3A411yescharacteristic
26C23×A4—43no—
27C22×C3×D4—31no—
28C22×C3×D4—32no—
29C22×C3×D4—31no—
30C23×C3×C4—31no—
31C3×#127—32no—
32C3×#127C22×C312yescomplemented
33C3×#126—34no—
34C3×#126C22×C314yescomplemented
35C3×#123—34no—
36C3×#123—34no—
37C3×#121—38no—
38C3×#121—38no—
39C3×#120—34no—
40C3×#120—34no—
41C2×C3×C4⋊C4—32no—
42C2×C3×C22⋊C4—31no—
43C2×C3×C22⋊C4—32no—
44C2×C3×C22⋊C4—31no—
45C2×C3×C42C22×C311yescharacteristic, complemented
46#127⋊C3C22×C316yescomplemented
47#126⋊C3C22×C3112yescomplemented
48C2×C42⋊C3C22×C313yescomplemented
49C23×C32—161no—
50C2×C3×A4—41no—
51C2×C3×A4—44no—
52C2×C3×A4—42no—
53#560—34no—
54#560C2×C3214yescomplemented
55C2×#127C2×C3211yescharacteristic, complemented
56C2×#126C2×C3212yescomplemented
57C2×#123—31no—
58C2×#121—32no—
59C2×#120—31no—
60C24×C3—31no—
61C24×C3—121no—
62C24×C3—121no—
63C24×C3C2×A412yes—
64C24×C3C2×A411yescharacteristic
65C24×C3C2×A414yes—
66C22×A4—46no—
67C22×A4—43no—
68C22×A4—412no—
69C2×C3×D4—32no—
70C2×C3×D4—32no—
71C2×C3×D4—38no—
72C2×C3×D4—32no—
73C2×C3×D4—32no—
74C2×C3×D4—122no—
75C2×C3×D4—122no—
76C2×C3×D4—121no—
77C2×C3×D4—121no—
78C2×C3×D4—121no—
79C2×C3×D4—121no—
80C22×C3×C4—31no—
81C22×C3×C4—32no—
82C22×C3×C4—34no—
83C22×C3×C4—121no—
84C22×C3×C4—121no—
85C3×C4⋊C4—34no—
86C3×C4⋊C4—34no—
87C3×C22⋊C4—34no—
88C3×C22⋊C4—34no—
89C3×C22⋊C4—34no—
90C3×C22⋊C4—34no—
91C3×C42—31no—
92C3×C42C23×C311yescharacteristic, complemented
93C42⋊C3C23×C313yescomplemented
94C22×C32—164no—
95C22×C32—162no—
96C22×C32—161no—
97C3×A4—41no—
98C25C3×A411yescharacteristic
99C22×D4—31no—
100C22×D4—32no—
101C22×D4—31no—
102C23×C4—31no—
103#127—32no—
104#127C22×C3212yescomplemented
105#126—34no—
106#126C22×C3214yescomplemented
107#123—34no—
108#123—34no—
109#121—38no—
110#121—38no—
111#120—34no—
112#120—34no—
113C2×C4⋊C4—32no—
114C2×C22⋊C4—31no—
115C2×C22⋊C4—32no—
116C2×C22⋊C4—31no—
117C2×C42C22×C3211yescharacteristic, complemented
118C23×C3—32no—
119C23×C3—41no—
120C23×C3—62no—
121C23×C3—61no—
122C23×C3—62no—
123C23×C3—61no—
124C23×C3—61no—
125C23×C3—61no—
126C23×C3—62no—
127C23×C3—62no—
128C23×C3—121no—
129C23×C3—124no—
130C23×C3—124no—
131C23×C3—163no—
132C23×C3C22×A411yescharacteristic
133C23×C3C22×A414yes—
134C23×C3C22×A412yes—
135C2×A4—43no—
136C2×A4—412no—
137C2×A4—46no—
138C3×D4—128no—
139C3×D4—122no—
140C3×D4—122no—
141C3×D4—122no—
142C3×D4—122no—
143C2×C3×C4—31no—
144C2×C3×C4—31no—
145C2×C3×C4—34no—
146C2×C3×C4—62no—
147C2×C3×C4—62no—
148C2×C3×C4—61no—
149C2×C3×C4—61no—
150C2×C3×C4—124no—
151C2×C32—164no—
152C2×C32—162no—
153C2×C32—161no—
154C24—31no—
155C24—121no—
156C24—121no—
157C24C2×C3×A412yes—
158C24C2×C3×A411yescharacteristic
159C24C2×C3×A414yes—
160C2×D4—32no—
161C2×D4—32no—
162C2×D4—38no—
163C2×D4—32no—
164C2×D4—32no—
165C2×D4—122no—
166C2×D4—122no—
167C2×D4—121no—
168C2×D4—121no—
169C2×D4—121no—
170C2×D4—121no—
171C22×C4—31no—
172C22×C4—32no—
173C22×C4—34no—
174C22×C4—121no—
175C22×C4—121no—
176C4⋊C4—34no—
177C4⋊C4—34no—
178C22⋊C4—34no—
179C22⋊C4—34no—
180C22⋊C4—34no—
181C22⋊C4—34no—
182C42—31no—
183C42C23×C3211yescharacteristic, complemented
184C22×C3—31no—
185C22×C3—31no—
186C22×C3—44no—
187C22×C3—42no—
188C22×C3—41no—
189C22×C3—64no—
190C22×C3—62no—
191C22×C3—62no—
192C22×C3—64no—
193C22×C3—122no—
194C22×C3—121no—
195C22×C3—122no—
196C22×C3—121no—
197C22×C3—124no—
198C22×C3—121no—
199C22×C3—1612no—
200C22×C3—166no—
201C22×C3—163no—
202C22×C3C23×A411yescharacteristic
203A4—43no—
204C3×C4—64no—
205C3×C4—61no—
206C3×C4—61no—
207C32—161noSylow
208C23—32no—
209C23—41no—
210C23—62no—
211C23—61no—
212C23—62no—
213C23—61no—
214C23—61no—
215C23—61no—
216C23—62no—
217C23—62no—
218C23—121no—
219C23—124no—
220C23—124no—
221C23C22×C3×A411yescharacteristic
222C23C22×C3×A414yes—
223C23C22×C3×A412yes—
224D4—128no—
225D4—122no—
226D4—122no—
227D4—122no—
228D4—122no—
229C2×C4—31no—
230C2×C4—31no—
231C2×C4—34no—
232C2×C4—62no—
233C2×C4—62no—
234C2×C4—61no—
235C2×C4—61no—
236C2×C4—124no—
237C2×C3—31no—
238C2×C3—31no—
239C2×C3—44no—
240C2×C3—42no—
241C2×C3—122no—
242C2×C3—1612no—
243C2×C3—166no—
244C2×C3—163no—
245C2×C3#560⋊C311yescharacteristic, direct product factor
246C22—31no—
247C22—31no—
248C22—44no—
249C22—42no—
250C22—41no—
251C22—64no—
252C22—62no—
253C22—62no—
254C22—64no—
255C22—122no—
256C22—121no—
257C22—122no—
258C22—121no—
259C22—124no—
260C22—121no—
261C22C23×C3×A411yescharacteristic
262C4—64no—
263C4—61no—
264C4—61no—
265C3—163no—
266C3C2×#560⋊C311yescharacteristic, direct product factor
267C2—31no—
268C2—31no—
269C2—44no—
270C2—42no—
271C2—122no—
272C2C3×#560⋊C311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3#560⋊C3
commutator, cosocle, hypocenterC42C23×C32
FrattiniC22C23×C3×A4
FittingC2×C3×#560C3
socleC23×C3C22×A4
LCSG ▷ C42
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C3×#560⋊C3C2direct (N × H)
2. C2×C3×#127⋊C3C2semidirect
3. C2×C3×#126⋊C3C2semidirect
4. C2×C3×#560C3semidirect
5. C2×#560⋊C3C3direct (N × H)
7. C3×#127⋊C3C22semidirect
8. C3×#126⋊C3C22semidirect
9. C2×C3×C42⋊C3C22semidirect
11. C3×#560C2×C3semidirect
12. C2×C3×#127C2×C3semidirect
13. C2×C3×#126C2×C3semidirect
17. #560⋊C3C2×C3direct (N × H)
18. C2×#127⋊C3C2×C3semidirect
19. C2×#126⋊C3C2×C3semidirect
23. C3×C42⋊C3C23semidirect
24. C2×#560C32semidirect
32. C3×#127C22×C3semidirect
34. C3×#126C22×C3semidirect
45. C2×C3×C42C22×C3semidirect
46. #127⋊C3C22×C3semidirect
47. #126⋊C3C22×C3semidirect
48. C2×C42⋊C3C22×C3semidirect
54. #560C2×C32semidirect
55. C2×#127C2×C32semidirect
56. C2×#126C2×C32semidirect
92. C3×C42C23×C3semidirect
93. C42⋊C3C23×C3semidirect
104. #127C22×C32semidirect
106. #126C22×C32semidirect
117. C2×C42C22×C32semidirect
183. C42C23×C32semidirect
245. C2×C3#560⋊C3direct (N × H)
266. C3C2×#560⋊C3direct (N × H)
272. C2C3×#560⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#560⋊C32
C2×C3×#127⋊C32
C2×C3×#126⋊C32
C2×C3×#5603
C2×#560⋊C33
C23×C3×A44