Finite Group Databasebeta

← C2×C3×C3⋊(#126⋊C3) #13173085

Subgroups — C2×C3×C3⋊(#126⋊C3) #13173085

Order 1728.

388 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 64 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×C32×C42⋊C3C211yesmaximal, characteristic, complemented
2C3×C3⋊(#126⋊C3)C212yesmaximal, direct product factor
3C2×C3×#876C311yesmaximal, characteristic, complemented
4C2×C3×#126⋊C3—31nomaximal
5C2×C3⋊(#126⋊C3)C313yesmaximal, direct product factor
6C2×C3×D3×A4—41nomaximal
7C32×C42⋊C3C2211yescharacteristic, complemented
8C2×C32×C42C2×C311yescharacteristic, complemented
9C2×C3×(C2×D3)⋊C4—31no—
10C2×C3×Dic3⋊C4—31no—
11C3×#876—32no—
12C3×#876C2×C312yescomplemented
13C3×#126⋊C3—32no—
14C2×C3×C42⋊C3—21no—
15C2×C3×C42⋊C3—23no—
16C2×C3×C42⋊C3D311yescharacteristic, complemented
17C2×C3×C42⋊C3C2×C313yescomplemented
18C3⋊(#126⋊C3)C2×C316yesdirect product factor
19C2×C32×A4—41no—
20C3×D3×A4—42no—
21C2×C3×#126—31no—
22C2×#876C3211yescharacteristic, complemented
23C2×#126⋊C3—33no—
24C23×C3×D3A411yescharacteristic
25C22×C3×A4—121no—
26C2×D3×A4—43no—
27C22×C32×C4—31no—
28C22×C3×Dic3—31no—
29C32×C42—31no—
30C32×C42C22×C311yescharacteristic, complemented
31C3×(C2×D3)⋊C4—32no—
32C3×(C2×D3)⋊C4—32no—
33C3×Dic3⋊C4—32no—
34C3×Dic3⋊C4—32no—
35C3×C42⋊C3—21no—
36C3×C42⋊C3—23no—
37C3×C42⋊C3C2×D311yescharacteristic, complemented
38C3×C42⋊C3C22×C313yescomplemented
39C2×C32×D3—161no—
40C32×A4—41no—
41C3×#126—32no—
42C3×#126—92no—
43C2×C3×C4⋊C4—91no—
44C2×C3×C22⋊C4—91no—
45C2×C3×C42—21no—
46C2×C3×C42C3×D311yescharacteristic, complemented
47C2×C3×C42C2×C3211yescharacteristic, complemented
48C2×(C2×D3)⋊C4—31no—
49C2×Dic3⋊C4—31no—
50#876—32no—
51#876C2×C3212yescomplemented
52#126⋊C3—36no—
53C2×C42⋊C3—23no—
54C2×C42⋊C3C3×D313yescomplemented
55C23×C32C2×A411yescharacteristic
56C22×C3×D3—61no—
57C22×C3×D3—61no—
58C22×C3×D3C2×A412yes—
59C2×C3×A4—41no—
60C2×C3×A4—43no—
61C2×C3×A4—83no—
62C2×C3×A4—81no—
63C2×C3×A4—122no—
64D3×A4—46no—
65C2×C32×C4—31no—
66C2×C32×C4—31no—
67C2×C32×C4—61no—
68C2×C32×C4—61no—
69C2×C3×Dic3—32no—
70C2×C3×Dic3—61no—
71C2×C3×Dic3—61no—
72C2×#126—31noSylow
73C2×C33—161no—
74C32×D3—162no—
75C24×C3—31no—
76C23×D3C3×A411yescharacteristic
77C22×A4—123no—
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—61no—
81C22×C3×C4—91no—
82C22×Dic3—31no—
83C3×C4⋊C4—92no—
84C3×C4⋊C4—92no—
85C3×C22⋊C4—92no—
86C3×C22⋊C4—92no—
87C3×C42—21no—
88C3×C42—31no—
89C3×C42—31no—
90C3×C42—61no—
91C3×C42C2×C3×D311yescharacteristic, complemented
92C3×C42C22×C3211yescharacteristic, complemented
93(C2×D3)⋊C4—32no—
94(C2×D3)⋊C4—32no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97C42⋊C3—23no—
98C42⋊C3C2×C3×D313yescomplemented
99C22×C32—31no—
100C22×C32—31no—
101C22×C32—481no—
102C22×C32C22×A411yescharacteristic
103C2×C3×D3—41no—
104C2×C3×D3—62no—
105C2×C3×D3—121no—
106C2×C3×D3—163no—
107C3×A4—41no—
108C3×A4—43no—
109C3×A4—83no—
110C3×A4—81no—
111C32×C4—61no—
112C32×C4—61no—
113C3×Dic3—62no—
114#126—32no—
115#126—92no—
116C2×C4⋊C4—91no—
117C2×C22⋊C4—91no—
118C2×C42C32×D311yescharacteristic, complemented
119C33—161noSylow
120C23×C3—21no—
121C23×C3—32no—
122C23×C3—181no—
123C23×C3—181no—
124C23×C3D3×A411yescharacteristic
125C23×C3C2×C3×A411yescharacteristic
126C22×D3—61no—
127C22×D3—61no—
128C22×D3C2×C3×A412yes—
129C2×A4—43no—
130C2×A4—83no—
131C2×A4—126no—
132C2×C3×C4—31no—
133C2×C3×C4—31no—
134C2×C3×C4—31no—
135C2×C3×C4—31no—
136C2×C3×C4—61no—
137C2×C3×C4—61no—
138C2×C3×C4—61no—
139C2×C3×C4—61no—
140C2×C3×C4—61no—
141C2×C3×C4—61no—
142C2×C3×C4—62no—
143C2×C3×C4—62no—
144C2×C3×C4—92no—
145C2×C3×C4—181no—
146C2×C3×C4—181no—
147C2×Dic3—32no—
148C2×Dic3—61no—
149C2×Dic3—61no—
150C2×C32—31no—
151C2×C32—31no—
152C2×C32—161no—
153C2×C32—163no—
154C2×C32—323no—
155C2×C32—321no—
156C2×C32—482no—
157C2×C32#126⋊C311yescharacteristic, complemented
158C3×D3—42no—
159C3×D3—166no—
160C24—31no—
161C22×C4—31no—
162C22×C4—91no—
163C4⋊C4—92no—
164C4⋊C4—92no—
165C22⋊C4—92no—
166C22⋊C4—92no—
167C42—31no—
168C42C2×C32×D311yescharacteristic, complemented
169C22×C3—21no—
170C22×C3—31no—
171C22×C3—31no—
172C22×C3—31no—
173C22×C3—31no—
174C22×C3—61no—
175C22×C3—61no—
176C22×C3—121no—
177C22×C3—182no—
178C22×C3—361no—
179C22×C3—483no—
180C22×C3C2×D3×A411yescharacteristic
181C22×C3C22×C3×A411yescharacteristic
182C2×D3—41no—
183C2×D3—62no—
184C2×D3—121no—
185A4—43no—
186A4—83no—
187C3×C4—61no—
188C3×C4—61no—
189C3×C4—61no—
190C3×C4—61no—
191C3×C4—62no—
192C3×C4—62no—
193C3×C4—182no—
194Dic3—62no—
195C32—161no—
196C32—163no—
197C32—323no—
198C32—321no—
199C32C2×#126⋊C311yescharacteristic, complemented
200C23—32no—
201C23—181no—
202C23—181no—
203C23C3×D3×A411yescharacteristic
204C2×C4—31no—
205C2×C4—31no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—92no—
209C2×C4—181no—
210C2×C4—181no—
211C2×C3—21no—
212C2×C3—31no—
213C2×C3—31no—
214C2×C3—31no—
215C2×C3—31no—
216C2×C3—61no—
217C2×C3—61no—
218C2×C3—122no—
219C2×C3—163no—
220C2×C3—323no—
221C2×C3—486no—
222C2×C3C3⋊(#126⋊C3)11yescharacteristic, direct product factor
223C2×C3C3×#126⋊C311yescharacteristic, complemented
224D3—42no—
225C22—31no—
226C22—31no—
227C22—121no—
228C22—182no—
229C22—361no—
230C22C2×C3×D3×A411yescharacteristic
231C4—61no—
232C4—61no—
233C4—182no—
234C3—21no—
235C3—163no—
236C3—323no—
237C3C2×C3⋊(#126⋊C3)11yescharacteristic, direct product factor
238C3C2×C3×#126⋊C311yescharacteristic, complemented
239C2—31no—
240C2—31no—
241C2—122no—
242C2C3×C3⋊(#126⋊C3)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊(#126⋊C3)
commutator, cosocle, hypocenterC3×C42C22×C32
FrattiniC22C2×C3×D3×A4
FittingC2×C32×C42C2×C3
socleC23×C32C2×A4
LCSG ▷ C3×C42
UCSC1 ◁ C2×C3

Complements

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

NH ≅ G/Ntype
1. C2×C32×C42⋊C3C2semidirect
2. C3×C3⋊(#126⋊C3)C2direct (N × H)
3. C2×C3×#876C3semidirect
5. C2×C3⋊(#126⋊C3)C3direct (N × H)
7. C32×C42⋊C3C22semidirect
8. C2×C32×C42C2×C3semidirect
12. C3×#876C2×C3semidirect
16. C2×C3×C42⋊C3D3semidirect
17. C2×C3×C42⋊C3C2×C3semidirect
18. C3⋊(#126⋊C3)C2×C3direct (N × H)
22. C2×#876C32semidirect
30. C32×C42C22×C3semidirect
37. C3×C42⋊C3C2×D3semidirect
38. C3×C42⋊C3C22×C3semidirect
46. C2×C3×C42C3×D3semidirect
47. C2×C3×C42C2×C32semidirect
51. #876C2×C32semidirect
54. C2×C42⋊C3C3×D3semidirect
91. C3×C42C2×C3×D3semidirect
92. C3×C42C22×C32semidirect
98. C42⋊C3C2×C3×D3semidirect
118. C2×C42C32×D3semidirect
157. C2×C32#126⋊C3semidirect
168. C42C2×C32×D3semidirect
199. C32C2×#126⋊C3semidirect
222. C2×C3C3⋊(#126⋊C3)direct (N × H)
223. C2×C3C3×#126⋊C3semidirect
237. C3C2×C3⋊(#126⋊C3)direct (N × H)
238. C3C2×C3×#126⋊C3semidirect
242. C2C3×C3⋊(#126⋊C3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C42⋊C32
C3×C3⋊(#126⋊C3)2
C2×C3×#8763
C2×C3×#126⋊C33
C2×C3⋊(#126⋊C3)3
C2×C3×D3×A44