Finite Group Databasebeta

← C2×C3×#64225 #13181017

Subgroups — C2×C3×#64225 #13181017

Order 1728.

1256 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1256 conjugacy classes of subgroups — too many to draw. Only its 194 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×#3802C212yesmaximal, complemented
2C2×C3×#3787C211yesmaximal, characteristic, complemented
3C2×C3×#3774C214yesmaximal, complemented
4C3×#64225C218yesmaximal, direct product factor
5C2×#64225C311yesmaximal, characteristic, direct product factor
6C22×C3×D32—22no—
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×D12—22no—
9C2×C3×#625C2214yescomplemented
10C3×#3802—24no—
11C3×#3802C2218yescomplemented
12C2×C3×#622C2212yescomplemented
13C3×#3787C2214yescomplemented
14C3×#3774C22116yescomplemented
15C2×C3×D3×D4—62no—
16C2×#3802C2×C312yescomplemented
17C2×#3787C2×C311yescharacteristic, complemented
18C2×#3774C2×C314yescomplemented
19#64225C2×C318yesdirect product factor
20C22×C3×C3⋊D3D411yescharacteristic
21C22×C32×D3—24no—
22C2×C3×D32—28no—
23C2×C3×D32—48no—
24C2×C33×C4D411yescharacteristic
25C2×C3×C3⋊Dic3C2311yescharacteristic
26C3×C4×C3⋊D3D412yes—
27C3×C4×C3⋊D3C2312yes—
28C32×D12—28no—
29C3×#625—24no—
30C3×#625C2318yescomplemented
31C3×#622C2314yescomplemented
32C22×C3×D8—91nomaximal
33C23×C3×D3—64no—
34C22×D32—22no—
35C2×C32×D4—62no—
36C2×C4×C3⋊D3C22×C311yescharacteristic
37C2×C3×#67—64no—
38C3×D3×D4—68no—
39C3×D3×D4—68no—
40C2×C3×D12—22no—
41C2×C3×D12—22no—
42C2×C3×D12—42no—
43C2×C3×C4×D3—62no—
44C2×#625C22×C314yescomplemented
45#3802—24no—
46#3802C22×C318yescomplemented
47C2×#622C22×C312yescomplemented
48#3787C22×C314yescomplemented
49#3774C22×C3116yescomplemented
50C22×C33C2×D411yescharacteristic
51C2×C3×C3⋊D3D814yescomplemented
52C2×C3×C3⋊D3C2×D412yes—
53C2×C32×D3—28no—
54C2×C32×D3—48no—
55C3×D32—416no—
56C33×C4C2×D412yes—
57C3×C3⋊Dic3C2×D411yescharacteristic
58C3×C3⋊Dic3C2411yescharacteristic
59C22×C3×D4—92no—
60C2×D3×D4—62no—
61C2×C3×D8—94no—
62C2×C3×D8—98no—
63C22×C3×C8—91no—
64C23×C32—64no—
65C22×C3⋊D3C3×D411yescharacteristic
66C22×C3×D3—24no—
67C22×C3×D3—24no—
68C22×C3×D3—44no—
69C22×C3×D3—62no—
70C22×C3×D3—616no—
71C22×C3×D3—1216no—
72C2×D32—28no—
73C2×D32—48no—
74C32×D4—68no—
75C2×C32×C4—22no—
76C2×C32×C4—42no—
77C2×C32×C4C3×D411yescharacteristic
78C2×C3⋊Dic3C23×C311yescharacteristic
79C4×C3⋊D3C3×D412yes—
80C4×C3⋊D3C23×C312yes—
81C3×#67—68no—
82C3×#67—68no—
83C2×C3×Dic3—62no—
84C3×D12—28no—
85C3×D12—28no—
86C3×D12—48no—
87C3×C4×D3—64no—
88C3×C4×D3—64no—
89#625—24no—
90#625C23×C318yescomplemented
91#622C23×C314yescomplemented
92C22×D8—91noSylow
93C2×C33C2×D812yescomplemented
94C2×C33C22×D411yescharacteristic
95C3×C3⋊D3C2×D814yescomplemented
96C32×D3—48no—
97C24×C3—182no—
98C23×D3—64no—
99C2×C3×D4—62no—
100C2×C3×D4—62no—
101C2×C3×D4—98no—
102C2×C3×D4—94no—
103C2×C3×D4—122no—
104C2×C3×D4—184no—
105C22×C3×C4—91no—
106C2×#67—64no—
107D3×D4—68no—
108D3×D4—68no—
109C2×D12—22no—
110C2×C4×D3—62no—
111C3×D8—916no—
112C2×C3×C8—94no—
113C2×C3×C8—92no—
114C22×C32—22no—
115C22×C32—42no—
116C22×C32—68no—
117C22×C32—128no—
118C22×C32C2×C3×D411yescharacteristic
119C2×C3⋊D3C3×D814yescomplemented
120C2×C3⋊D3C2×C3×D412yes—
121C2×C3×D3—28no—
122C2×C3×D3—28no—
123C2×C3×D3—48no—
124C2×C3×D3—48no—
125C2×C3×D3—48no—
126C2×C3×D3—68no—
127C2×C3×D3—64no—
128C2×C3×D3—88no—
129C2×C3×D3—1232no—
130D32—416no—
131C32×C4—24no—
132C32×C4—44no—
133C32×C4C2×C3×D412yes—
134C3⋊Dic3C2×C3×D411yescharacteristic
135C3⋊Dic3C24×C311yescharacteristic
136C3×Dic3—62no—
137C3×Dic3—62no—
138C22×D4—92no—
139C2×D8—98no—
140C2×D8—94no—
141C22×C8—91no—
142C33C22×D811yescharacteristic, complemented, Sylow
143C23×C3—64no—
144C23×C3—64no—
145C23×C3—91no—
146C23×C3—124no—
147C23×C3—188no—
148C23×C3—368no—
149C22×D3—24no—
150C22×D3—62no—
151C22×D3—616no—
152C22×D3—1216no—
153C3×D4—68no—
154C3×D4—68no—
155C3×D4—98no—
156C3×D4—128no—
157C3×D4—184no—
158C2×C3×C4—22no—
159C2×C3×C4—42no—
160C2×C3×C4—91no—
161C2×C3×C4—92no—
162C2×C3×C4—92no—
163C2×C3×C4GO+(4,𝔽2)11yescharacteristic
164#67—68no—
165#67—68no—
166C2×Dic3—62no—
167D12—28no—
168C4×D3—64no—
169C4×D3—64no—
170C3×C8—94no—
171C2×C32—22no—
172C2×C32—24no—
173C2×C32—44no—
174C2×C32—42no—
175C2×C32—128no—
176C2×C32C2×C3×D812yescomplemented
177C2×C32C22×C3×D411yescharacteristic
178C3⋊D3C2×C3×D814yescomplemented
179C3×D3—48no—
180C3×D3—48no—
181C3×D3—68no—
182C3×D3—88no—
183C24—182no—
184C2×D4—62no—
185C2×D4—98no—
186C2×D4—94no—
187C2×D4—184no—
188C22×C4—91no—
189D8—916no—
190C2×C8—94no—
191C2×C8—92no—
192C22×C3—22no—
193C22×C3—42no—
194C22×C3—68no—
195C22×C3—68no—
196C22×C3—92no—
197C22×C3—94no—
198C22×C3—128no—
199C22×C3—128no—
200C22×C3—128no—
201C22×C3—248no—
202C22×C3—3616no—
203C22×C3C2×GO+(4,𝔽2)11yescharacteristic
204C2×D3—28no—
205C2×D3—48no—
206C2×D3—64no—
207C2×D3—68no—
208C2×D3—1232no—
209C3×C4—24no—
210C3×C4—44no—
211C3×C4—91no—
212C3×C4—91no—
213C3×C4C2×GO+(4,𝔽2)12yes—
214Dic3—62no—
215Dic3—62no—
216C32—22no—
217C32—42no—
218C32C22×C3×D811yescharacteristic, complemented
219C23—64no—
220C23—91no—
221C23—188no—
222C23—368no—
223D4—68no—
224D4—98no—
225D4—184no—
226C2×C4—91no—
227C2×C4—92no—
228C2×C4—92no—
229C2×C4C3×GO+(4,𝔽2)11yescharacteristic
230C8—94no—
231C2×C3—22no—
232C2×C3—24no—
233C2×C3—42no—
234C2×C3—44no—
235C2×C3—94no—
236C2×C3—128no—
237C2×C3—128no—
238C2×C3—248no—
239C2×C3#6422512yesdirect product factor
240C2×C3C22×GO+(4,𝔽2)11yescharacteristic
241D3—48no—
242D3—68no—
243C22—68no—
244C22—92no—
245C22—94no—
246C22—128no—
247C22—3616no—
248C22C2×C3×GO+(4,𝔽2)11yescharacteristic
249C4—91no—
250C4—91no—
251C4C2×C3×GO+(4,𝔽2)12yes—
252C3—22no—
253C3—42no—
254C3C2×#6422511yescharacteristic, direct product factor
255C2—94no—
256C2—128no—
257C2C3×#6422512yesdirect product factor
258C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC3⋊Dic3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C22×C3×D8
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3802C2semidirect
2. C2×C3×#3787C2semidirect
3. C2×C3×#3774C2semidirect
4. C3×#64225C2direct (N × H)
5. C2×#64225C3direct (N × H)
9. C2×C3×#625C22semidirect
11. C3×#3802C22semidirect
12. C2×C3×#622C22semidirect
13. C3×#3787C22semidirect
14. C3×#3774C22semidirect
16. C2×#3802C2×C3semidirect
17. C2×#3787C2×C3semidirect
18. C2×#3774C2×C3semidirect
19. #64225C2×C3direct (N × H)
30. C3×#625C23semidirect
31. C3×#622C23semidirect
44. C2×#625C22×C3semidirect
46. #3802C22×C3semidirect
47. C2×#622C22×C3semidirect
48. #3787C22×C3semidirect
49. #3774C22×C3semidirect
51. C2×C3×C3⋊D3D8semidirect
90. #625C23×C3semidirect
91. #622C23×C3semidirect
93. C2×C33C2×D8semidirect
95. C3×C3⋊D3C2×D8semidirect
119. C2×C3⋊D3C3×D8semidirect
142. C33C22×D8semidirect
176. C2×C32C2×C3×D8semidirect
178. C3⋊D3C2×C3×D8semidirect
218. C32C22×C3×D8semidirect
239. C2×C3#64225direct (N × H)
254. C3C2×#64225direct (N × H)
257. C2C3×#64225direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#38022
C2×C3×#37872
C2×C3×#37742
C3×#642252
C2×#642253
C22×C3×D89