Finite Group Databasebeta

← C2×C3×#6233 #13048939

Subgroups — C2×C3×#6233 #13048939

Order 1296.

780 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 182 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
1C23×C32×C9C211yesmaximal, characteristic, complemented
2C22×C3×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×#6233C214yesmaximal, direct product factor
4C2×C3×C3⋊Dic9C211yesmaximal, characteristic, complemented
5C2×C3×#638—31nomaximal
6C2×#6233C311yesmaximal, characteristic, direct product factor
7C2×C3×#611—33nomaximal
8C22×C32×C9—22no—
9C22×C32×C9C2211yescharacteristic
10C22×C32×C9C2212yescomplemented
11C2×C3×C3⋊D9—22no—
12C2×C3×C3⋊D9C2212yescomplemented
13C3×C3⋊Dic9C2212yescomplemented
14C23×C33D311yescharacteristic
15C22×C3×C3⋊D3—31no—
16C3×#638—34no—
17C2×C3×C3⋊Dic3—31no—
18C23×C3×C9—21no—
19C23×C3×C9—23no—
20C23×C3×C9D313yescomplemented
21C23×C3×C9C2×C311yescharacteristic, complemented
22C22×C3⋊D9C2×C311yescharacteristic, complemented
23C22×C3×D9—33no—
24#6233C2×C314yesdirect product factor
25C2×C3⋊Dic9C2×C311yescharacteristic, complemented
26C3×#611—312no—
27C2×C3×Dic9—33no—
28C2×C32×C9—22no—
29C2×C32×C9D412yescomplemented
30C2×C32×C9C2311yescharacteristic
31C3×C3⋊D9—22no—
32C2×#638—31no—
33C2×C3×#67—93no—
34C2×C3×#67—91no—
35C2×#611—33no—
36C22×C33—22no—
37C22×C33C2×D312yes—
38C22×C33C2×D311yescharacteristic
39C2×C3×C3⋊D3—32no—
40C2×C3×C3⋊D3—62no—
41C3×C3⋊Dic3—32no—
42C22×C3×C9—24no—
43C22×C3×C9—212no—
44C22×C3×C9—21no—
45C22×C3×C9—22no—
46C22×C3×C9—23no—
47C22×C3×C9—26no—
48C22×C3×C9—26no—
49C22×C3×C9—22no—
50C22×C3×C9C2×D313yes—
51C22×C3×C9C2×D316yescomplemented
52C22×C3×C9C22×C311yescharacteristic
53C22×C3×C9C22×C312yescomplemented
54C2×C3⋊D9—22no—
55C2×C3⋊D9C22×C312yescomplemented
56C2×C3×D9—36no—
57C2×C3×D9—66no—
58C3⋊Dic9C22×C312yescomplemented
59C3×Dic9—36no—
60C32×C9C2×D411yescharacteristic, complemented, Sylow
61C23×C32—23no—
62C23×C32—21no—
63C23×C32D913yescomplemented
64C23×C32C3×D311yescharacteristic
65C23×C32C3⋊D311yescharacteristic
66C22×C3⋊D3—31no—
67C22×C3×D3—93no—
68C22×C3×D3—91no—
69#638—34no—
70C2×C3⋊Dic3—31no—
71C3×#67—912no—
72C3×#67—94no—
73C2×C3×Dic3—93no—
74C2×C3×Dic3—91no—
75C23×C9—23no—
76C23×C9C3×D313yescomplemented
77C22×D9—33no—
78#611—312no—
79C2×Dic9—33no—
80C2×C33—22no—
81C2×C33#6712yes—
82C2×C33C22×D311yescharacteristic
83C3×C3⋊D3—62no—
84C2×C3×C9—24no—
85C2×C3×C9—212no—
86C2×C3×C9—26no—
87C2×C3×C9—22no—
88C2×C3×C9—22no—
89C2×C3×C9—21no—
90C2×C3×C9—26no—
91C2×C3×C9—23no—
92C2×C3×C9#6716yescomplemented
93C2×C3×C9C3×D412yescomplemented
94C2×C3×C9C22×D313yes—
95C2×C3×C9C23×C311yescharacteristic
96C3⋊D9—22no—
97C3×D9—66no—
98C2×C3×D4—271no—
99C2×#67—93no—
100C2×#67—91no—
101C22×C32—24no—
102C22×C32—212no—
103C22×C32—26no—
104C22×C32—22no—
105C22×C32—22no—
106C22×C32—23no—
107C22×C32—26no—
108C22×C32—21no—
109C22×C32—22no—
110C22×C32C2×D913yes—
111C22×C32C2×D916yescomplemented
112C22×C32C2×C3×D311yescharacteristic
113C22×C32C2×C3×D312yes—
114C22×C32C2×C3⋊D311yescharacteristic
115C22×C32C2×C3⋊D312yes—
116C2×C3⋊D3—32no—
117C2×C3⋊D3—62no—
118C2×C3×D3—96no—
119C2×C3×D3—92no—
120C2×C3×D3—186no—
121C2×C3×D3—182no—
122C3⋊Dic3—32no—
123C3×Dic3—96no—
124C3×Dic3—92no—
125C22×C9—212no—
126C22×C9—26no—
127C22×C9—23no—
128C22×C9—26no—
129C22×C9C2×C3×D313yes—
130C22×C9C2×C3×D316yescomplemented
131C2×D9—36no—
132C2×D9—66no—
133Dic9—36no—
134C33C2×#6711yescharacteristic
135C3×C9—21no—
136C3×C9—23no—
137C3×C9C2×#6713yescomplemented
138C3×C9C2×C3×D411yescharacteristic, complemented
139C23×C3—23no—
140C23×C3—21no—
141C23×C3—271no—
142C23×C3C3×D913yescomplemented
143C23×C3C3⋊D911yescharacteristic, complemented
144C23×C3C3×C3⋊D311yescharacteristic
145C22×D3—93no—
146C22×D3—91no—
147C3×D4—274no—
148C2×C3×C4—271no—
149#67—912no—
150#67—94no—
151C2×Dic3—93no—
152C2×Dic3—91no—
153C2×C32—24no—
154C2×C32—212no—
155C2×C32—26no—
156C2×C32—22no—
157C2×C32—22no—
158C2×C32—26no—
159C2×C32—23no—
160C2×C32—22no—
161C2×C32—21no—
162C2×C32#61116yescomplemented
163C2×C32C22×D913yes—
164C2×C32C3×#6712yes—
165C2×C32#63812yes—
166C2×C32C22×C3×D311yescharacteristic
167C2×C32C22×C3⋊D311yescharacteristic
168C3⋊D3—62no—
169C3×D3—186no—
170C3×D3—182no—
171C2×C9—212no—
172C2×C9—26no—
173C2×C9—26no—
174C2×C9—23no—
175C2×C9C3×#6716yescomplemented
176C2×C9C22×C3×D313yes—
177D9—66no—
178C2×D4—271noSylow
179C22×C3—212no—
180C22×C3—24no—
181C22×C3—26no—
182C22×C3—22no—
183C22×C3—22no—
184C22×C3—23no—
185C22×C3—21no—
186C22×C3—26no—
187C22×C3—22no—
188C22×C3—272no—
189C22×C3—542no—
190C22×C3C2×C3×D913yes—
191C22×C3C2×C3×D916yescomplemented
192C22×C3C2×C3⋊D911yescharacteristic
193C22×C3C2×C3⋊D912yescomplemented
194C22×C3C2×C3×C3⋊D311yescharacteristic
195C22×C3C2×C3×C3⋊D312yes—
196C2×D3—96no—
197C2×D3—92no—
198C2×D3—186no—
199C2×D3—182no—
200C3×C4—272no—
201Dic3—96no—
202Dic3—92no—
203C32—23no—
204C32—21no—
205C32C2×#61113yescomplemented
206C32C2×C3×#6711yescharacteristic
207C32C2×#63811yescharacteristic
208C9—23no—
209C9C2×C3×#6713yescomplemented
210C23—271no—
211C23C3×C3⋊D911yescharacteristic, complemented
212D4—274no—
213C2×C4—271no—
214C2×C3—212no—
215C2×C3—24no—
216C2×C3—26no—
217C2×C3—22no—
218C2×C3—22no—
219C2×C3—26no—
220C2×C3—22no—
221C2×C3—23no—
222C2×C3—21no—
223C2×C3—542no—
224C2×C3C3×#61116yescomplemented
225C2×C3#623312yesdirect product factor
226C2×C3C22×C3×D913yes—
227C2×C3C22×C3⋊D911yescharacteristic
228C2×C3C3×#63812yes—
229C2×C3C22×C3×C3⋊D311yescharacteristic
230D3—186no—
231D3—182no—
232C22—22no—
233C22—272no—
234C22—542no—
235C22C2×C3×C3⋊D911yescharacteristic
236C22C2×C3×C3⋊D912yescomplemented
237C4—272no—
238C3—23no—
239C3—21no—
240C3C2×C3×#61113yescomplemented
241C3C2×#623311yescharacteristic, direct product factor
242C3C2×C3×#63811yescharacteristic
243C2—22no—
244C2—542no—
245C2C3×#623312yesdirect product factor
246C2C22×C3×C3⋊D911yescharacteristic
subgroupG/H
centerC22×C3C2×C3⋊D9
commutator, cosocleC2×C3×C9C23×C3
FrattiniC2×C3C22×C3×C3⋊D3
FittingC23×C32×C9C2
socleC22×C33C2×D3
hypercenterC23×C3C3⋊D9
hypocenterC3×C9C2×C3×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C23×C32×C9C2semidirect
2. C22×C3×C3⋊D9C2semidirect
3. C3×#6233C2direct (N × H)
4. C2×C3×C3⋊Dic9C2semidirect
6. C2×#6233C3direct (N × H)
10. C22×C32×C9C22semidirect
12. C2×C3×C3⋊D9C22semidirect
13. C3×C3⋊Dic9C22semidirect
20. C23×C3×C9D3semidirect
21. C23×C3×C9C2×C3semidirect
22. C22×C3⋊D9C2×C3semidirect
24. #6233C2×C3direct (N × H)
25. C2×C3⋊Dic9C2×C3semidirect
29. C2×C32×C9D4semidirect
51. C22×C3×C9C2×D3semidirect
53. C22×C3×C9C22×C3semidirect
55. C2×C3⋊D9C22×C3semidirect
58. C3⋊Dic9C22×C3semidirect
60. C32×C9C2×D4semidirect
63. C23×C32D9semidirect
76. C23×C9C3×D3semidirect
92. C2×C3×C9#67semidirect
93. C2×C3×C9C3×D4semidirect
111. C22×C32C2×D9semidirect
130. C22×C9C2×C3×D3semidirect
137. C3×C9C2×#67semidirect
138. C3×C9C2×C3×D4semidirect
142. C23×C3C3×D9semidirect
143. C23×C3C3⋊D9semidirect
162. C2×C32#611semidirect
175. C2×C9C3×#67semidirect
191. C22×C3C2×C3×D9semidirect
193. C22×C3C2×C3⋊D9semidirect
205. C32C2×#611semidirect
209. C9C2×C3×#67semidirect
211. C23C3×C3⋊D9semidirect
224. C2×C3C3×#611semidirect
225. C2×C3#6233direct (N × H)
236. C22C2×C3×C3⋊D9semidirect
240. C3C2×C3×#611semidirect
241. C3C2×#6233direct (N × H)
245. C2C3×#6233direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C32×C92
C22×C3×C3⋊D92
C3×#62332
C2×C3×C3⋊Dic92
C2×C3×#6383
C2×#62333
C2×C3×#6113