Finite Group Databasebeta

← C2×C3×C3⋊(#127⋊C3) #13173084

Subgroups — C2×C3×C3⋊(#127⋊C3) #13173084

Order 1728.

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

normal subgroups only

This group has 556 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⋊(#127⋊C3)C212yesmaximal, direct product factor
3C2×C3×#874C311yesmaximal, characteristic, complemented
4C2×C3×#127⋊C3—31nomaximal
5C2×C3⋊(#127⋊C3)C313yesmaximal, direct product factor
6C2×C3×D3×A4—41nomaximal
7C32×C42⋊C3C2211yescharacteristic, complemented
8C22×C3×D12—31no—
9C22×C3×D12—31no—
10C2×C32×C42C2×C311yescharacteristic, complemented
11C3×#874—32no—
12C3×#874C2×C312yescomplemented
13C3×#127⋊C3—32no—
14C2×C3×C42⋊C3—21no—
15C2×C3×C42⋊C3—23no—
16C2×C3×C42⋊C3D311yescharacteristic, complemented
17C2×C3×C42⋊C3C2×C313yescomplemented
18C3⋊(#127⋊C3)C2×C316yesdirect product factor
19C2×C32×A4—41no—
20C3×D3×A4—42no—
21C2×C3×#127—31no—
22C2×#874C3211yescharacteristic, complemented
23C2×#127⋊C3—33no—
24C23×C3×D3—31no—
25C23×C3×D3A411yescharacteristic
26C22×C3×A4—121no—
27C2×D3×A4—43no—
28C22×C32×C4—31no—
29C2×C3×D12—32no—
30C2×C3×D12—32no—
31C2×C3×D12—32no—
32C2×C3×D12—32no—
33C2×C3×D12—62no—
34C2×C3×D12—62no—
35C2×C3×D12—62no—
36C2×C3×D12—62no—
37C32×C42—31no—
38C32×C42C22×C311yescharacteristic, complemented
39C3×C42⋊C3—21no—
40C3×C42⋊C3—23no—
41C3×C42⋊C3C2×D311yescharacteristic, complemented
42C3×C42⋊C3C22×C313yescomplemented
43C2×C32×D3—161no—
44C32×A4—41no—
45C22×C3×D4—91no—
46C22×C3×D4—91no—
47C22×D12—31no—
48C22×D12—31no—
49C3×#127—32no—
50C3×#127—92no—
51C2×C3×C42—21no—
52C2×C3×C42C3×D311yescharacteristic, complemented
53C2×C3×C42C2×C3211yescharacteristic, complemented
54#874—32no—
55#874C2×C3212yescomplemented
56#127⋊C3—36no—
57C2×C42⋊C3—23no—
58C2×C42⋊C3C3×D313yescomplemented
59C23×C32C2×A411yescharacteristic
60C22×C3×D3—32no—
61C22×C3×D3—61no—
62C22×C3×D3—62no—
63C22×C3×D3—61no—
64C22×C3×D3—61no—
65C22×C3×D3—62no—
66C22×C3×D3—61no—
67C22×C3×D3C2×A412yes—
68C2×C3×A4—41no—
69C2×C3×A4—43no—
70C2×C3×A4—83no—
71C2×C3×A4—81no—
72C2×C3×A4—122no—
73D3×A4—46no—
74C2×C32×C4—32no—
75C2×C32×C4—32no—
76C2×C32×C4—31no—
77C2×C32×C4—31no—
78C3×D12—64no—
79C3×D12—64no—
80C3×D12—64no—
81C3×D12—64no—
82C2×#127—31noSylow
83C2×C33—161no—
84C32×D3—162no—
85C24×C3—31no—
86C24×C3—91no—
87C23×D3—31no—
88C23×D3C3×A411yescharacteristic
89C22×A4—123no—
90C2×C3×D4—92no—
91C2×C3×D4—92no—
92C2×C3×D4—92no—
93C2×C3×D4—92no—
94C2×C3×D4—182no—
95C2×C3×D4—182no—
96C2×C3×D4—182no—
97C2×C3×D4—182no—
98C22×C3×C4—31no—
99C22×C3×C4—31no—
100C22×C3×C4—61no—
101C2×D12—32no—
102C2×D12—32no—
103C2×D12—32no—
104C2×D12—32no—
105C2×D12—62no—
106C2×D12—62no—
107C2×D12—62no—
108C2×D12—62no—
109C3×C42—21no—
110C3×C42—31no—
111C3×C42—31no—
112C3×C42—61no—
113C3×C42C2×C3×D311yescharacteristic, complemented
114C3×C42C22×C3211yescharacteristic, complemented
115C42⋊C3—23no—
116C42⋊C3C2×C3×D313yescomplemented
117C22×C32—31no—
118C22×C32—31no—
119C22×C32—481no—
120C22×C32C22×A411yescharacteristic
121C2×C3×D3—41no—
122C2×C3×D3—62no—
123C2×C3×D3—62no—
124C2×C3×D3—64no—
125C2×C3×D3—121no—
126C2×C3×D3—121no—
127C2×C3×D3—122no—
128C2×C3×D3—121no—
129C2×C3×D3—163no—
130C3×A4—41no—
131C3×A4—43no—
132C3×A4—83no—
133C3×A4—81no—
134C32×C4—32no—
135C32×C4—32no—
136C22×D4—91no—
137C22×D4—91no—
138#127—32no—
139#127—92no—
140C2×C42C32×D311yescharacteristic, complemented
141C33—161noSylow
142C23×C3—21no—
143C23×C3—32no—
144C23×C3—92no—
145C23×C3—181no—
146C23×C3—181no—
147C23×C3—181no—
148C23×C3—182no—
149C23×C3—181no—
150C23×C3—182no—
151C23×C3D3×A411yescharacteristic
152C23×C3C2×C3×A411yescharacteristic
153C22×D3—32no—
154C22×D3—61no—
155C22×D3—62no—
156C22×D3—61no—
157C22×D3—61no—
158C22×D3—62no—
159C22×D3—61no—
160C22×D3C2×C3×A412yes—
161C2×A4—43no—
162C2×A4—83no—
163C2×A4—126no—
164C3×D4—184no—
165C3×D4—184no—
166C3×D4—184no—
167C3×D4—184no—
168C2×C3×C4—32no—
169C2×C3×C4—32no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—62no—
179C2×C3×C4—62no—
180D12—64no—
181D12—64no—
182D12—64no—
183D12—64no—
184C2×C32—31no—
185C2×C32—31no—
186C2×C32—161no—
187C2×C32—163no—
188C2×C32—323no—
189C2×C32—321no—
190C2×C32—482no—
191C2×C32#127⋊C311yescharacteristic, complemented
192C3×D3—42no—
193C3×D3—122no—
194C3×D3—166no—
195C24—31no—
196C24—91no—
197C2×D4—92no—
198C2×D4—92no—
199C2×D4—92no—
200C2×D4—92no—
201C2×D4—182no—
202C2×D4—182no—
203C2×D4—182no—
204C2×D4—182no—
205C22×C4—31no—
206C42—31no—
207C42C2×C32×D311yescharacteristic, complemented
208C22×C3—21no—
209C22×C3—31no—
210C22×C3—31no—
211C22×C3—31no—
212C22×C3—31no—
213C22×C3—61no—
214C22×C3—61no—
215C22×C3—121no—
216C22×C3—182no—
217C22×C3—182no—
218C22×C3—184no—
219C22×C3—361no—
220C22×C3—362no—
221C22×C3—361no—
222C22×C3—361no—
223C22×C3—483no—
224C22×C3C2×D3×A411yescharacteristic
225C22×C3C22×C3×A411yescharacteristic
226C2×D3—41no—
227C2×D3—62no—
228C2×D3—62no—
229C2×D3—64no—
230C2×D3—121no—
231C2×D3—121no—
232C2×D3—122no—
233C2×D3—121no—
234A4—43no—
235A4—83no—
236C3×C4—32no—
237C3×C4—32no—
238C3×C4—32no—
239C3×C4—32no—
240C3×C4—62no—
241C3×C4—62no—
242C32—161no—
243C32—163no—
244C32—323no—
245C32—321no—
246C32C2×#127⋊C311yescharacteristic, complemented
247C23—32no—
248C23—92no—
249C23—181no—
250C23—181no—
251C23—181no—
252C23—182no—
253C23—181no—
254C23—182no—
255C23C3×D3×A411yescharacteristic
256D4—184no—
257D4—184no—
258D4—184no—
259D4—184no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C3—21no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—31no—
268C2×C3—31no—
269C2×C3—61no—
270C2×C3—61no—
271C2×C3—122no—
272C2×C3—163no—
273C2×C3—323no—
274C2×C3—362no—
275C2×C3—486no—
276C2×C3C3⋊(#127⋊C3)11yescharacteristic, direct product factor
277C2×C3C3×#127⋊C311yescharacteristic, complemented
278D3—42no—
279D3—122no—
280C22—31no—
281C22—31no—
282C22—121no—
283C22—182no—
284C22—182no—
285C22—184no—
286C22—361no—
287C22—362no—
288C22—361no—
289C22—361no—
290C22C2×C3×D3×A411yescharacteristic
291C4—32no—
292C4—32no—
293C3—21no—
294C3—163no—
295C3—323no—
296C3C2×C3⋊(#127⋊C3)11yescharacteristic, direct product factor
297C3C2×C3×#127⋊C311yescharacteristic, complemented
298C2—31no—
299C2—31no—
300C2—122no—
301C2—362no—
302C2C3×C3⋊(#127⋊C3)11yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3C3⋊(#127⋊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⋊(#127⋊C3)C2direct (N × H)
3. C2×C3×#874C3semidirect
5. C2×C3⋊(#127⋊C3)C3direct (N × H)
7. C32×C42⋊C3C22semidirect
10. C2×C32×C42C2×C3semidirect
12. C3×#874C2×C3semidirect
16. C2×C3×C42⋊C3D3semidirect
17. C2×C3×C42⋊C3C2×C3semidirect
18. C3⋊(#127⋊C3)C2×C3direct (N × H)
22. C2×#874C32semidirect
38. C32×C42C22×C3semidirect
41. C3×C42⋊C3C2×D3semidirect
42. C3×C42⋊C3C22×C3semidirect
52. C2×C3×C42C3×D3semidirect
53. C2×C3×C42C2×C32semidirect
55. #874C2×C32semidirect
58. C2×C42⋊C3C3×D3semidirect
113. C3×C42C2×C3×D3semidirect
114. C3×C42C22×C32semidirect
116. C42⋊C3C2×C3×D3semidirect
140. C2×C42C32×D3semidirect
191. C2×C32#127⋊C3semidirect
207. C42C2×C32×D3semidirect
246. C32C2×#127⋊C3semidirect
276. C2×C3C3⋊(#127⋊C3)direct (N × H)
277. C2×C3C3×#127⋊C3semidirect
296. C3C2×C3⋊(#127⋊C3)direct (N × H)
297. C3C2×C3×#127⋊C3semidirect
302. C2C3×C3⋊(#127⋊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⋊(#127⋊C3)2
C2×C3×#8743
C2×C3×#127⋊C33
C2×C3⋊(#127⋊C3)3
C2×C3×D3×A44