Finite Group Databasebeta

← C2×C4⋊(C3⋊(C3⋊S4)) #13181395

Subgroups — C2×C4⋊(C3⋊(C3⋊S4)) #13181395

Order 1728.

2262 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 2262 conjugacy classes of subgroups — too many to draw. Only its 283 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⋊S4)C212yesmaximal, complemented
2C2×C32×C4×A4C211yesmaximal, characteristic, complemented
3C4⋊(C3⋊(C3⋊S4))C214yesmaximal, direct product factor
4C2×C4⋊(C3⋊S4)—312nomaximal
5C2×#64139—31nomaximal
6C22×C32×A4C2211yescharacteristic
7C2×C3⋊(C3⋊S4)—24no—
8C2×C3⋊(C3⋊S4)C2214yescomplemented
9C2×C4⋊(C3⋊(C3⋊D3))—41nomaximal
10C32×C4×A4C2212yescomplemented
11C22×C3⋊S4—324no—
12C23×C32×C4D311yescharacteristic, complemented
13C22×#638—32no—
14C22×C4⋊(C3⋊D3)—31no—
15C2×C3×C4×A4D3112yescomplemented
16C4⋊(C3⋊S4)—348no—
17#64139—34no—
18#64139—34no—
19C2×#3752—32no—
20C2×C4⋊(C3⋊Dic3)—31no—
21C22×C3⋊(C3⋊D3)—42no—
22C2×C32×A4D412yescomplemented
23C2×C32×A4C2311yescharacteristic
24C3⋊(C3⋊S4)—24no—
25C2×C33×C4—41no—
26C4⋊(C3⋊(C3⋊D3))—44no—
27C2×C4⋊S4—99no—
28C2×#930—94no—
29C24×C32C2×D311yescharacteristic
30C23×C3⋊D3—32no—
31C22×C3×A4C2×D3112yes—
32C2×C3⋊S4—348no—
33C2×C3⋊S4—648no—
34C22×C32×C4—31no—
35C22×C32×C4—31no—
36C22×C32×C4—32no—
37C22×C32×C4C2×D312yescomplemented
38C2×#638—34no—
39C2×#638—34no—
40C2×#638—64no—
41C2×#638—64no—
42C22×C3⋊Dic3—32no—
43C2×C4⋊(C3⋊D3)—34no—
44C2×C4⋊(C3⋊D3)—61no—
45C2×C4⋊(C3⋊D3)—62no—
46C2×C4⋊(C3⋊D3)—61no—
47C2×C4⋊(C3⋊D3)—1212no—
48C3×C4×A4C2×D3124yescomplemented
49#3752—38no—
50C4⋊(C3⋊Dic3)—34no—
51C22×C33—41no—
52C2×C3⋊(C3⋊D3)—44no—
53C2×C3⋊(C3⋊D3)—84no—
54C32×A4C2×D411yescharacteristic, complemented
55C33×C4—42no—
56C22×S4—918no—
57C23×C3×C4C3⋊D314yescomplemented
58C22×#67—98no—
59C22×D12—94no—
60C2×C4×A4C3⋊D319yescomplemented
61C4⋊S4—936no—
62#930—916no—
63#930—916no—
64C2×(C2×D3)⋊C4—98no—
65C2×C4⋊Dic3—94no—
66C23×C32—31no—
67C23×C32—31no—
68C23×C32—32no—
69C23×C32D1212yescomplemented
70C23×C32C22×D311yescharacteristic
71C22×C3⋊D3—34no—
72C22×C3⋊D3—64no—
73C22×C3⋊D3—62no—
74C22×C3⋊D3—64no—
75C22×C3⋊D3—62no—
76C22×C3⋊D3—1224no—
77C2×C3×A4D12124yescomplemented
78C2×C3×A4C22×D3112yes—
79C3⋊S4—648no—
80C2×C32×C4—31no—
81C2×C32×C4—32no—
82C2×C32×C4—32no—
83C2×C32×C4—32no—
84C2×C32×C4—412no—
85C2×C32×C4—61no—
86C2×C32×C4S411yescharacteristic, complemented
87#638—68no—
88#638—64no—
89#638—64no—
90C2×C3⋊Dic3—34no—
91C2×C3⋊Dic3—64no—
92C4⋊(C3⋊D3)—64no—
93C4⋊(C3⋊D3)—64no—
94C4⋊(C3⋊D3)—1248no—
95C2×#121—271noSylow
96C2×C33—42no—
97C2×C33—41no—
98C3⋊(C3⋊D3)—84no—
99C24×C3C2×C3⋊D314yes—
100C23×D3—98no—
101C22×A4C2×C3⋊D319yes—
102C2×S4—936no—
103C2×S4—1836no—
104C22×C3×C4—34no—
105C22×C3×C4—34no—
106C22×C3×C4—38no—
107C22×C3×C4C2×C3⋊D318yescomplemented
108C2×#67—916no—
109C2×#67—916no—
110C2×#67—1816no—
111C2×#67—1816no—
112C22×Dic3—98no—
113C2×D12—916no—
114C2×D12—184no—
115C2×D12—188no—
116C2×D12—184no—
117C2×D12—369no—
118C4×A4C2×C3⋊D3118yescomplemented
119(C2×D3)⋊C4—932no—
120C4⋊Dic3—916no—
121C22×C32—32no—
122C22×C32—31no—
123C22×C32—32no—
124C22×C32—32no—
125C22×C32—31no—
126C22×C32—31no—
127C22×C32—412no—
128C22×C32—61no—
129C22×C32C2×D1211yescharacteristic, complemented
130C22×C32C2×S411yescharacteristic
131C2×C3⋊D3—64no—
132C2×C3⋊D3—64no—
133C2×C3⋊D3—64no—
134C2×C3⋊D3—1248no—
135C2×C3⋊D3—124no—
136C2×C3⋊D3—124no—
137C2×C3⋊D3—2448no—
138C3×A4C2×D12112yescomplemented
139C32×C4—32no—
140C32×C4—424no—
141C32×C4C2×S412yescomplemented
142C3⋊Dic3—64no—
143C22×D4—271no—
144C22×D4—272no—
145C23×C4C3⋊(C3⋊D3)11yescharacteristic, complemented
146#121—274no—
147#121—274no—
148C2×C4⋊C4—271no—
149C2×C22⋊C4—272no—
150C33—41noSylow
151C23×C3—34no—
152C23×C3—34no—
153C23×C3—38no—
154C23×C3C4⋊(C3⋊D3)18yescomplemented
155C23×C3C22×C3⋊D314yes—
156C22×D3—916no—
157C22×D3—188no—
158C22×D3—1816no—
159C22×D3—1816no—
160C22×D3—188no—
161C22×D3—3618no—
162C2×A4C4⋊(C3⋊D3)118yescomplemented
163C2×A4C22×C3⋊D319yes—
164S4—1836no—
165C2×C3×C4—34no—
166C2×C3×C4—38no—
167C2×C3×C4—38no—
168C2×C3×C4—38no—
169C2×C3×C4—49no—
170C2×C3×C4—64no—
171C2×C3×C4C3⋊S414yescomplemented
172#67—1832no—
173#67—1816no—
174#67—1816no—
175C2×Dic3—916no—
176C2×Dic3—1816no—
177D12—1816no—
178D12—1816no—
179D12—3636no—
180C2×C32—31no—
181C2×C32—31no—
182C2×C32—32no—
183C2×C32—424no—
184C2×C32—412no—
185C2×C32C4⋊S412yescomplemented
186C2×C32C22×S411yescharacteristic
187C3⋊D3—124no—
188C3⋊D3—2448no—
189C24—272no—
190C24C2×C3⋊(C3⋊D3)11yescharacteristic
191C2×D4—274no—
192C2×D4—274no—
193C2×D4—274no—
194C2×D4—541no—
195C2×D4—542no—
196C2×D4—544no—
197C2×D4—544no—
198C2×D4—541no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—32no—
202C22×C4—272no—
203C22×C4C2×C3⋊(C3⋊D3)12yescomplemented
204C4⋊C4—274no—
205C22⋊C4—278no—
206C22×C3—38no—
207C22×C3—34no—
208C22×C3—38no—
209C22×C3—38no—
210C22×C3—34no—
211C22×C3—34no—
212C22×C3—49no—
213C22×C3—64no—
214C22×C3C2×C4⋊(C3⋊D3)14yescomplemented
215C22×C3C2×C3⋊S414yes—
216C2×D3—1816no—
217C2×D3—1816no—
218C2×D3—1816no—
219C2×D3—3636no—
220C2×D3—3616no—
221C2×D3—3616no—
222C2×D3—7236no—
223A4C2×C4⋊(C3⋊D3)19yescomplemented
224C3×C4—38no—
225C3×C4—418no—
226C3×C4C2×C3⋊S418yescomplemented
227Dic3—1816no—
228C32—412no—
229C32C2×C4⋊S411yescharacteristic, complemented
230C23—31no—
231C23—31no—
232C23—32no—
233C23—274no—
234C23—544no—
235C23—542no—
236C23—544no—
237C23—542no—
238C23C4⋊(C3⋊(C3⋊D3))12yescomplemented
239C23C22×C3⋊(C3⋊D3)11yescharacteristic
240D4—544no—
241D4—548no—
242D4—544no—
243D4—544no—
244D4—544no—
245C2×C4—31no—
246C2×C4—32no—
247C2×C4—32no—
248C2×C4—32no—
249C2×C4—61no—
250C2×C4—274no—
251C2×C4—544no—
252C2×C4C3⋊(C3⋊S4)11yescharacteristic, complemented
253C2×C3—34no—
254C2×C3—34no—
255C2×C3—38no—
256C2×C3—418no—
257C2×C3—49no—
258C2×C3C4⋊(C3⋊S4)18yescomplemented
259C2×C3C22×C3⋊S414yes—
260D3—3616no—
261D3—7236no—
262C22—32no—
263C22—31no—
264C22—32no—
265C22—32no—
266C22—31no—
267C22—31no—
268C22—61no—
269C22—544no—
270C22—544no—
271C22—544no—
272C22—1084no—
273C22—1084no—
274C22C2×C4⋊(C3⋊(C3⋊D3))11yescharacteristic, complemented
275C22C2×C3⋊(C3⋊S4)11yescharacteristic
276C4—32no—
277C4—544no—
278C4C2×C3⋊(C3⋊S4)12yescomplemented
279C3—49no—
280C3C2×C4⋊(C3⋊S4)14yescomplemented
281C2—31no—
282C2—31no—
283C2—32no—
284C2—1084no—
285C2C4⋊(C3⋊(C3⋊S4))12yesdirect product factor
286C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
centerC22C2×C3⋊(C3⋊S4)
commutator, cosocleC2×C32×A4C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC23×C32×C4D3
socleC24×C32C2×D3
hypercenterC2×C4C3⋊(C3⋊S4)
hypocenterC32×A4C2×D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊(C3⋊S4)C2semidirect
2. C2×C32×C4×A4C2semidirect
3. C4⋊(C3⋊(C3⋊S4))C2direct (N × H)
8. C2×C3⋊(C3⋊S4)C22semidirect
10. C32×C4×A4C22semidirect
12. C23×C32×C4D3semidirect
15. C2×C3×C4×A4D3semidirect
22. C2×C32×A4D4semidirect
37. C22×C32×C4C2×D3semidirect
48. C3×C4×A4C2×D3semidirect
54. C32×A4C2×D4semidirect
57. C23×C3×C4C3⋊D3semidirect
60. C2×C4×A4C3⋊D3semidirect
69. C23×C32D12semidirect
77. C2×C3×A4D12semidirect
86. C2×C32×C4S4semidirect
107. C22×C3×C4C2×C3⋊D3semidirect
118. C4×A4C2×C3⋊D3semidirect
129. C22×C32C2×D12semidirect
138. C3×A4C2×D12semidirect
141. C32×C4C2×S4semidirect
145. C23×C4C3⋊(C3⋊D3)semidirect
154. C23×C3C4⋊(C3⋊D3)semidirect
162. C2×A4C4⋊(C3⋊D3)semidirect
171. C2×C3×C4C3⋊S4semidirect
185. C2×C32C4⋊S4semidirect
203. C22×C4C2×C3⋊(C3⋊D3)semidirect
214. C22×C3C2×C4⋊(C3⋊D3)semidirect
223. A4C2×C4⋊(C3⋊D3)semidirect
226. C3×C4C2×C3⋊S4semidirect
229. C32C2×C4⋊S4semidirect
238. C23C4⋊(C3⋊(C3⋊D3))semidirect
252. C2×C4C3⋊(C3⋊S4)semidirect
258. C2×C3C4⋊(C3⋊S4)semidirect
274. C22C2×C4⋊(C3⋊(C3⋊D3))semidirect
278. C4C2×C3⋊(C3⋊S4)semidirect
280. C3C2×C4⋊(C3⋊S4)semidirect
285. C2C4⋊(C3⋊(C3⋊S4))direct (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⋊S4)2
C2×C32×C4×A42
C4⋊(C3⋊(C3⋊S4))2
C2×C4⋊(C3⋊S4)3
C2×#641393
C2×C4⋊(C3⋊(C3⋊D3))4