Finite Group Databasebeta

← C2×C4⋊(C9⋊S4) #13164451

Subgroups — C2×C4⋊(C9⋊S4) #13164451

Order 1728.

915 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 915 conjugacy classes of subgroups — too many to draw. Only its 115 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×C9⋊S4C212yesmaximal, complemented
2C2×C4×C9×A4C211yesmaximal, characteristic, complemented
3C4⋊(C9⋊S4)C214yesmaximal, direct product factor
4C2×C4⋊(C3⋊S4)—31nomaximal
5C2×C4⋊((C22⋊C9)⋊C2)—32nomaximal
6C2×#63492—31nomaximal
7C22×C9×A4C2211yescharacteristic
8C2×C9⋊S4—24no—
9C2×C9⋊S4C2214yescomplemented
10C2×C4⋊(C3⋊D9)—41nomaximal
11C4×C9×A4C2212yescomplemented
12C22×C3⋊S4—32no—
13C2×C3×C4×A4D311yescharacteristic
14C4⋊(C3⋊S4)—34no—
15C22×(C22⋊C9)⋊C2—34no—
16C23×C4×C9D311yescharacteristic, complemented
17C22×#611—32no—
18C22×D36—31no—
19C2×C4×C22⋊C9D312yescomplemented
20C4⋊((C22⋊C9)⋊C2)—38no—
21#63492—34no—
22#63492—34no—
23C2×(C2×D9)⋊C4—32no—
24C2×C4⋊Dic9—31no—
25C22×C3⋊D9—42no—
26C2×C9×A4D412yescomplemented
27C2×C9×A4C2311yescharacteristic
28C9⋊S4—24no—
29C2×C3×C4×C9—41no—
30C4⋊(C3⋊D9)—44no—
31C2×C4⋊S4—93no—
32C2×#930—91no—
33C22×C3×A4C2×D311yescharacteristic
34C2×C3⋊S4—34no—
35C2×C3⋊S4—64no—
36C2×C4⋊(C3⋊D3)—121no—
37C3×C4×A4C2×D312yes—
38C24×C9C2×D311yescharacteristic
39C23×D9—32no—
40C22×C22⋊C9C2×D312yes—
41C2×(C22⋊C9)⋊C2—38no—
42C2×(C22⋊C9)⋊C2—68no—
43C22×C4×C9—31no—
44C22×C4×C9—31no—
45C22×C4×C9—32no—
46C22×C4×C9C2×D312yescomplemented
47C2×#611—34no—
48C2×#611—34no—
49C2×#611—64no—
50C2×#611—64no—
51C22×Dic9—32no—
52C2×D36—34no—
53C2×D36—61no—
54C2×D36—62no—
55C2×D36—61no—
56C2×D36—122no—
57C4×C22⋊C9C2×D314yescomplemented
58(C2×D9)⋊C4—38no—
59C4⋊Dic9—34no—
60C22×C3×C9—41no—
61C2×C3⋊D9—44no—
62C2×C3⋊D9—84no—
63C9×A4C2×D411yescharacteristic, complemented
64C3×C4×C9—42no—
65C22×S4—96no—
66C23×C3×C4C3⋊D311yescharacteristic
67C22×#67—92no—
68C22×D12—91no—
69C2×C4×A4D913yescomplemented
70C4⋊S4—912no—
71#930—94no—
72#930—94no—
73C2×(C2×D3)⋊C4—92no—
74C2×C4⋊Dic3—91no—
75C22×C3⋊D3—122no—
76C2×C3×A4D1212yes—
77C2×C3×A4C22×D311yescharacteristic
78C3⋊S4—64no—
79C2×C32×C4—41no—
80C4⋊(C3⋊D3)—124no—
81C23×C9—31no—
82C23×C9—31no—
83C23×C9—32no—
84C23×C9D1212yescomplemented
85C23×C9C22×D311yescharacteristic
86C22×D9—34no—
87C22×D9—62no—
88C22×D9—64no—
89C22×D9—64no—
90C22×D9—62no—
91C22×D9—124no—
92C2×C22⋊C9D1214yescomplemented
93C2×C22⋊C9C22×D312yes—
94(C22⋊C9)⋊C2—68no—
95C2×C4×C9—31no—
96C2×C4×C9—32no—
97C2×C4×C9—32no—
98C2×C4×C9—32no—
99C2×C4×C9—42no—
100C2×C4×C9—61no—
101C2×C4×C9S411yescharacteristic, complemented
102#611—68no—
103#611—64no—
104#611—64no—
105C2×Dic9—34no—
106C2×Dic9—64no—
107D36—64no—
108D36—64no—
109D36—128no—
110C2×#121—271noSylow
111C2×C3×C9—42no—
112C2×C3×C9—41no—
113C3⋊D9—84no—
114C24×C3C2×C3⋊D311yescharacteristic
115C23×D3—92no—
116C22×A4C2×D913yes—
117C2×S4—912no—
118C2×S4—1812no—
119C22×C3×C4—31no—
120C22×C3×C4—31no—
121C22×C3×C4—32no—
122C22×C3×C4C2×C3⋊D312yes—
123C2×#67—94no—
124C2×#67—94no—
125C2×#67—184no—
126C2×#67—184no—
127C22×Dic3—92no—
128C2×D12—94no—
129C2×D12—181no—
130C2×D12—182no—
131C2×D12—181no—
132C2×D12—363no—
133C4×A4C2×D916yescomplemented
134(C2×D3)⋊C4—98no—
135C4⋊Dic3—94no—
136C22×C32—41no—
137C2×C3⋊D3—124no—
138C2×C3⋊D3—244no—
139C3×A4C2×D1211yescharacteristic
140C32×C4—42no—
141C22×C9—32no—
142C22×C9—31no—
143C22×C9—32no—
144C22×C9—32no—
145C22×C9—31no—
146C22×C9—31no—
147C22×C9—42no—
148C22×C9—61no—
149C22×C9C2×D1211yescharacteristic, complemented
150C22×C9C2×S411yescharacteristic
151C2×D9—64no—
152C2×D9—64no—
153C2×D9—64no—
154C2×D9—128no—
155C2×D9—124no—
156C2×D9—124no—
157C2×D9—248no—
158C22⋊C9C2×D1212yescomplemented
159C4×C9—32no—
160C4×C9—44no—
161C4×C9C2×S412yescomplemented
162Dic9—64no—
163C22×D4—271no—
164C22×D4—272no—
165C23×C4C3⋊D911yescharacteristic, complemented
166#121—274no—
167#121—274no—
168C2×C4⋊C4—271no—
169C2×C22⋊C4—272no—
170C3×C9—41noSylow
171C23×C3—31no—
172C23×C3—31no—
173C23×C3—32no—
174C23×C3C4⋊(C3⋊D3)12yes—
175C23×C3C22×C3⋊D311yescharacteristic
176C22×D3—94no—
177C22×D3—182no—
178C22×D3—184no—
179C22×D3—184no—
180C22×D3—182no—
181C22×D3—366no—
182C2×A4D3616yescomplemented
183C2×A4C22×D913yes—
184S4—1812no—
185C2×C3×C4—31no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—32no—
189C2×C3×C4—43no—
190C2×C3×C4—61no—
191C2×C3×C4C3⋊S411yescharacteristic
192#67—188no—
193#67—184no—
194#67—184no—
195C2×Dic3—94no—
196C2×Dic3—184no—
197D12—184no—
198D12—184no—
199D12—3612no—
200C2×C32—42no—
201C2×C32—41no—
202C3⋊D3—244no—
203C2×C9—31no—
204C2×C9—31no—
205C2×C9—32no—
206C2×C9—44no—
207C2×C9—42no—
208C2×C9C4⋊S412yescomplemented
209C2×C9C22×S411yescharacteristic
210D9—124no—
211D9—248no—
212C24—272no—
213C24C2×C3⋊D911yescharacteristic
214C2×D4—274no—
215C2×D4—274no—
216C2×D4—274no—
217C2×D4—541no—
218C2×D4—542no—
219C2×D4—544no—
220C2×D4—544no—
221C2×D4—541no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—32no—
225C22×C4—272no—
226C22×C4C2×C3⋊D912yescomplemented
227C4⋊C4—274no—
228C22⋊C4—278no—
229C22×C3—32no—
230C22×C3—31no—
231C22×C3—32no—
232C22×C3—32no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—43no—
236C22×C3—61no—
237C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
238C22×C3C2×C3⋊S411yescharacteristic
239C2×D3—184no—
240C2×D3—184no—
241C2×D3—184no—
242C2×D3—3612no—
243C2×D3—364no—
244C2×D3—364no—
245C2×D3—7212no—
246A4C2×D3613yescomplemented
247C3×C4—32no—
248C3×C4—46no—
249C3×C4C2×C3⋊S412yes—
250Dic3—184no—
251C32—41no—
252C9—42no—
253C9C2×C4⋊S411yescharacteristic, complemented
254C23—31no—
255C23—31no—
256C23—32no—
257C23—274no—
258C23—542no—
259C23—544no—
260C23—544no—
261C23—542no—
262C23C4⋊(C3⋊D9)12yescomplemented
263C23C22×C3⋊D911yescharacteristic
264D4—544no—
265D4—548no—
266D4—544no—
267D4—544no—
268D4—544no—
269C2×C4—31no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4—61no—
274C2×C4—274no—
275C2×C4—544no—
276C2×C4C9⋊S411yescharacteristic, complemented
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—32no—
280C2×C3—43no—
281C2×C3—46no—
282C2×C3C4⋊(C3⋊S4)12yes—
283C2×C3C22×C3⋊S411yescharacteristic
284D3—364no—
285D3—7212no—
286C22—32no—
287C22—31no—
288C22—31no—
289C22—31no—
290C22—32no—
291C22—32no—
292C22—61no—
293C22—544no—
294C22—544no—
295C22—544no—
296C22—1084no—
297C22—1084no—
298C22C2×C4⋊(C3⋊D9)11yescharacteristic, complemented
299C22C2×C9⋊S411yescharacteristic
300C4—32no—
301C4—544no—
302C4C2×C9⋊S412yescomplemented
303C3—43no—
304C3C2×C4⋊(C3⋊S4)11yescharacteristic
305C2—31no—
306C2—31no—
307C2—32no—
308C2—1084no—
309C2C4⋊(C9⋊S4)12yesdirect product factor
310C2C22×C9⋊S411yescharacteristic
subgroupG/H
centerC22C2×C9⋊S4
commutator, cosocleC2×C9×A4C23
FrattiniC2×C3C22×C3⋊S4
FittingC23×C4×C9D3
socleC24×C3C2×C3⋊D3
hypercenterC2×C4C9⋊S4
hypocenterC9×A4C2×D4
LCSG ▷ C2×C9×A4 ▷ C9×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C22×C9⋊S4C2semidirect
2. C2×C4×C9×A4C2semidirect
3. C4⋊(C9⋊S4)C2direct (N × H)
9. C2×C9⋊S4C22semidirect
11. C4×C9×A4C22semidirect
16. C23×C4×C9D3semidirect
19. C2×C4×C22⋊C9D3semidirect
26. C2×C9×A4D4semidirect
46. C22×C4×C9C2×D3semidirect
57. C4×C22⋊C9C2×D3semidirect
63. C9×A4C2×D4semidirect
69. C2×C4×A4D9semidirect
84. C23×C9D12semidirect
92. C2×C22⋊C9D12semidirect
101. C2×C4×C9S4semidirect
133. C4×A4C2×D9semidirect
149. C22×C9C2×D12semidirect
158. C22⋊C9C2×D12semidirect
161. C4×C9C2×S4semidirect
165. C23×C4C3⋊D9semidirect
182. C2×A4D36semidirect
208. C2×C9C4⋊S4semidirect
226. C22×C4C2×C3⋊D9semidirect
246. A4C2×D36semidirect
253. C9C2×C4⋊S4semidirect
262. C23C4⋊(C3⋊D9)semidirect
276. C2×C4C9⋊S4semidirect
298. C22C2×C4⋊(C3⋊D9)semidirect
302. C4C2×C9⋊S4semidirect
309. C2C4⋊(C9⋊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×C9⋊S42
C2×C4×C9×A42
C4⋊(C9⋊S4)2
C2×C4⋊(C3⋊S4)3
C2×C4⋊((C22⋊C9)⋊C2)3
C2×#634923
C2×C4⋊(C3⋊D9)4