Finite Group Databasebeta

← C2×C3×#6195 #13048581

Subgroups — C2×C3×#6195 #13048581

Order 1296.

496 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C22×C3×C3⋊D9C211yesmaximal, characteristic, complemented
2C22×C3×D3×C9C211yesmaximal, characteristic, complemented
3C2×C32×Dic9C211yesmaximal, characteristic, complemented
4C3×#6195C214yesmaximal, direct product factor
5C2×C3×#626—31nomaximal
6C2×C3×#611—31nomaximal
7C2×#6195C311yesmaximal, characteristic, direct product factor
8C22×C32×C9C2211yescharacteristic
9C2×C3×C3⋊D9—22no—
10C2×C3×C3⋊D9C2212yescomplemented
11C2×C3×D3×C9—22no—
12C2×C3×D3×C9C2212yescomplemented
13C32×Dic9C2212yescomplemented
14C22×C3×C3⋊D3—31no—
15C22×C32×D3D311yescharacteristic
16C2×C32×Dic3—31no—
17C3×#626—34no—
18C23×C3×C9—31no—
19C22×C3⋊D9C2×C311yescharacteristic, complemented
20C22×D3×C9—21no—
21C22×D3×C9C2×C311yescharacteristic, complemented
22C22×C3×D9—31no—
23C22×C3×D9—61no—
24C3×#611—34no—
25C2×C3×Dic9—21no—
26C2×C3×Dic9D311yescharacteristic, complemented
27C2×C3×Dic9C2×C311yescharacteristic, complemented
28#6195C2×C314yesdirect product factor
29C2×C32×C9D412yescomplemented
30C2×C32×C9C2311yescharacteristic
31C3×C3⋊D9—22no—
32C3×D3×C9—22no—
33C2×C3×#67—91no—
34C2×C3×D12—91no—
35C2×#626—31no—
36C2×#611—31no—
37C22×C33C2×D311yescharacteristic
38C2×C3×C3⋊D3—32no—
39C2×C3×C3⋊D3—62no—
40C2×C32×D3—22no—
41C2×C32×D3C2×D312yes—
42C32×Dic3—32no—
43C22×C3×C9—21no—
44C22×C3×C9—21no—
45C22×C3×C9—21no—
46C22×C3×C9—32no—
47C22×C3×C9—41no—
48C22×C3×C9—62no—
49C22×C3×C9C2×D311yescharacteristic
50C22×C3×C9C22×C311yescharacteristic
51C2×C3⋊D9—22no—
52C2×C3⋊D9C22×C312yescomplemented
53C2×D3×C9—24no—
54C2×D3×C9—22no—
55C2×D3×C9—22no—
56C2×D3×C9C22×C312yescomplemented
57C2×C3×D9—32no—
58C2×C3×D9—62no—
59C2×C3×D9—64no—
60C2×C3×D9—62no—
61C3×Dic9—22no—
62C3×Dic9C2×D312yescomplemented
63C3×Dic9C22×C312yescomplemented
64C32×C9C2×D411yescharacteristic, complemented, Sylow
65C23×C32—31no—
66C22×C3⋊D3—31no—
67C22×C3×D3—21no—
68C22×C3×D3—91no—
69C22×C3×D3—91no—
70C22×C3×D3—181no—
71C22×C3×D3D911yescharacteristic, complemented
72C22×C3×D3C3×D311yescharacteristic
73C2×C32×C4—91no—
74C3×#67—94no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—61no—
78C3×D12—94no—
79#626—34no—
80C23×C9—31no—
81C23×C9—61no—
82C22×D9—31no—
83C22×D9—61no—
84#611—34no—
85C2×Dic9C3×D311yescharacteristic, complemented
86C2×C33#6712yes—
87C2×C33C22×D311yescharacteristic
88C3×C3⋊D3—62no—
89C32×D3—22no—
90C2×C3×C9—22no—
91C2×C3×C9—22no—
92C2×C3×C9—21no—
93C2×C3×C9—21no—
94C2×C3×C9—22no—
95C2×C3×C9—21no—
96C2×C3×C9—42no—
97C2×C3×C9—41no—
98C2×C3×C9—62no—
99C2×C3×C9D1212yescomplemented
100C2×C3×C9C3×D412yescomplemented
101C2×C3×C9C22×D311yescharacteristic
102C2×C3×C9C23×C311yescharacteristic
103C3⋊D9—22no—
104D3×C9—24no—
105D3×C9—22no—
106C3×D9—64no—
107C3×D9—62no—
108C2×C3×D4—271no—
109C2×#67—91no—
110C2×D12—91no—
111C22×C32—21no—
112C22×C32—21no—
113C22×C32—21no—
114C22×C32—32no—
115C22×C32—41no—
116C22×C32—62no—
117C22×C32C2×D911yescharacteristic
118C22×C32D3211yescharacteristic
119C22×C32C2×C3×D311yescharacteristic
120C2×C3⋊D3—32no—
121C2×C3⋊D3—62no—
122C2×C3×D3—24no—
123C2×C3×D3—22no—
124C2×C3×D3—22no—
125C2×C3×D3—22no—
126C2×C3×D3—92no—
127C2×C3×D3—92no—
128C2×C3×D3—182no—
129C2×C3×D3—184no—
130C2×C3×D3—182no—
131C2×C3×D3—182no—
132C2×C3×D3C2×D912yescomplemented
133C2×C3×D3C2×C3×D312yes—
134C32×C4—92no—
135C3×Dic3—32no—
136C3×Dic3—32no—
137C3×Dic3—62no—
138C22×C9—21no—
139C22×C9—21no—
140C22×C9—32no—
141C22×C9—41no—
142C22×C9—62no—
143C22×C9—64no—
144C22×C9—62no—
145C22×C9C2×C3×D311yescharacteristic
146C2×D9—32no—
147C2×D9—62no—
148C2×D9—64no—
149C2×D9—62no—
150Dic9C2×C3×D312yescomplemented
151C33C2×#6711yescharacteristic
152C3×C9—21no—
153C3×C9—21no—
154C3×C9—21no—
155C3×C9—41no—
156C3×C9C2×D1211yescharacteristic, complemented
157C3×C9C2×C3×D411yescharacteristic, complemented
158C23×C3—31no—
159C23×C3—31no—
160C23×C3—61no—
161C23×C3—271no—
162C22×D3—91no—
163C22×D3—91no—
164C22×D3—181no—
165C22×D3C3×D911yescharacteristic, complemented
166C3×D4—274no—
167C2×C3×C4—91no—
168C2×C3×C4—91no—
169C2×C3×C4—181no—
170#67—94no—
171C2×Dic3—31no—
172D12—94no—
173C2×C32—22no—
174C2×C32—22no—
175C2×C32—21no—
176C2×C32—21no—
177C2×C32—22no—
178C2×C32—21no—
179C2×C32—42no—
180C2×C32—41no—
181C2×C32—62no—
182C2×C32#61112yescomplemented
183C2×C32C22×D911yescharacteristic
184C2×C32#62612yes—
185C2×C32C3×#6712yes—
186C2×C32C2×D3211yescharacteristic
187C2×C32C22×C3×D311yescharacteristic
188C3⋊D3—62no—
189C3×D3—24no—
190C3×D3—22no—
191C3×D3—22no—
192C3×D3—184no—
193C3×D3—182no—
194C3×D3—182no—
195C2×C9—22no—
196C2×C9—22no—
197C2×C9—21no—
198C2×C9—21no—
199C2×C9—42no—
200C2×C9—41no—
201C2×C9—64no—
202C2×C9—62no—
203C2×C9C3×D1212yescomplemented
204C2×C9C22×C3×D311yescharacteristic
205D9—64no—
206D9—62no—
207C2×D4—271noSylow
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—32no—
212C22×C3—32no—
213C22×C3—41no—
214C22×C3—62no—
215C22×C3—64no—
216C22×C3—62no—
217C22×C3—62no—
218C22×C3—272no—
219C22×C3—542no—
220C22×C3D3×D911yescharacteristic
221C22×C3C2×C3×D911yescharacteristic
222C22×C3C3×D3211yescharacteristic
223C2×D3—22no—
224C2×D3—92no—
225C2×D3—92no—
226C2×D3—182no—
227C2×D3—184no—
228C2×D3—182no—
229C2×D3—182no—
230C2×D3C2×C3×D912yescomplemented
231C3×C4—92no—
232C3×C4—92no—
233C3×C4—182no—
234Dic3—32no—
235C32—21no—
236C32—21no—
237C32—21no—
238C32—41no—
239C32C2×#61111yescharacteristic, complemented
240C32C2×#62611yescharacteristic
241C32C2×C3×#6711yescharacteristic
242C9—21no—
243C9—21no—
244C9—41no—
245C9C2×C3×D1211yescharacteristic, complemented
246C23—31no—
247C23—271no—
248D4—274no—
249C2×C4—91no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—21no—
253C2×C3—22no—
254C2×C3—22no—
255C2×C3—22no—
256C2×C3—41no—
257C2×C3—42no—
258C2×C3—64no—
259C2×C3—62no—
260C2×C3—62no—
261C2×C3—542no—
262C2×C3#619512yesdirect product factor
263C2×C3C3×#61112yescomplemented
264C2×C3C2×D3×D911yescharacteristic
265C2×C3C22×C3×D911yescharacteristic
266C2×C3C3×#62612yes—
267C2×C3C2×C3×D3211yescharacteristic
268D3—22no—
269D3—184no—
270D3—182no—
271D3—182no—
272C22—32no—
273C22—62no—
274C22—272no—
275C22—542no—
276C22C3×D3×D911yescharacteristic
277C4—92no—
278C3—21no—
279C3—21no—
280C3—21no—
281C3—41no—
282C3C2×#619511yescharacteristic, direct product factor
283C3C2×C3×#61111yescharacteristic, complemented
284C3C2×C3×#62611yescharacteristic
285C2—62no—
286C2—542no—
287C2C3×#619512yesdirect product factor
288C2C2×C3×D3×D911yescharacteristic
subgroupG/H
center, hypercenterC22×C3D3×D9
commutator, cosocleC2×C3×C9C23×C3
FrattiniC2×C3C2×C3×D32
FittingC22×C32×C9C22
socleC22×C33C2×D3
hypocenterC3×C9C2×C3×D4
LCSG ▷ C2×C3×C9 ▷ C3×C9
UCSC1 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C22×C3×C3⋊D9C2semidirect
2. C22×C3×D3×C9C2semidirect
3. C2×C32×Dic9C2semidirect
4. C3×#6195C2direct (N × H)
7. C2×#6195C3direct (N × H)
10. C2×C3×C3⋊D9C22semidirect
12. C2×C3×D3×C9C22semidirect
13. C32×Dic9C22semidirect
19. C22×C3⋊D9C2×C3semidirect
21. C22×D3×C9C2×C3semidirect
26. C2×C3×Dic9D3semidirect
27. C2×C3×Dic9C2×C3semidirect
28. #6195C2×C3direct (N × H)
29. C2×C32×C9D4semidirect
52. C2×C3⋊D9C22×C3semidirect
56. C2×D3×C9C22×C3semidirect
62. C3×Dic9C2×D3semidirect
63. C3×Dic9C22×C3semidirect
64. C32×C9C2×D4semidirect
71. C22×C3×D3D9semidirect
85. C2×Dic9C3×D3semidirect
99. C2×C3×C9D12semidirect
100. C2×C3×C9C3×D4semidirect
132. C2×C3×D3C2×D9semidirect
150. Dic9C2×C3×D3semidirect
156. C3×C9C2×D12semidirect
157. C3×C9C2×C3×D4semidirect
165. C22×D3C3×D9semidirect
182. C2×C32#611semidirect
203. C2×C9C3×D12semidirect
230. C2×D3C2×C3×D9semidirect
239. C32C2×#611semidirect
245. C9C2×C3×D12semidirect
262. C2×C3#6195direct (N × H)
263. C2×C3C3×#611semidirect
282. C3C2×#6195direct (N × H)
283. C3C2×C3×#611semidirect
287. C2C3×#6195direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C3⋊D92
C22×C3×D3×C92
C2×C32×Dic92
C3×#61952
C2×C3×#6263
C2×C3×#6113
C2×#61953