Finite Group Databasebeta

← C2×(C22⋊A4)⋊H(3,𝔽3) #72220570

Subgroups — C2×(C22⋊A4)⋊H(3,𝔽3) #72220570

Order 2592.

906 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 98 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
1(C22⋊A4)⋊H(3,𝔽3)C211yesmaximal, characteristic, direct product factor
2C2×C32×C22⋊A4C311yesmaximal, characteristic, complemented
3C2×C3×A42C313yesmaximal, complemented
4C2×C32⋊(C22⋊A4)C311yesmaximal, characteristic, complemented
5C23×C22⋊H(3,𝔽3)C312yesmaximal, complemented
6C2×(C3×A4)⋊A4C316yesmaximal, complemented
7C2×A4×H(3,𝔽3)—42nomaximal
8C32×C22⋊A4C2×C311yescharacteristic, complemented
9C3×A42C2×C313yescomplemented
10C32⋊(C22⋊A4)C2×C311yescharacteristic, complemented
11C22×C22⋊H(3,𝔽3)—32no—
12C22×C22⋊H(3,𝔽3)—32no—
13C22×C22⋊H(3,𝔽3)C2×C312yescomplemented
14(C3×A4)⋊A4C2×C316yescomplemented
15A4×H(3,𝔽3)—42no—
16C25×C32C3211yescharacteristic, complemented
17C2×C3×C22⋊A4—33no—
18C2×C3×C22⋊A4C3211yescharacteristic
19C2×C3×C22⋊A4C3212yescomplemented
20C2×C3×C22⋊A4C3213yescomplemented
21C23×C3×A4C3216yescomplemented
22C2×A42—39no—
23C2×C32×A4—42no—
24C2×C32×A4—46no—
25C2×C32×A4—121no—
26C23×H(3,𝔽3)—42no—
27C2×C22⋊H(3,𝔽3)—32no—
28C2×C22⋊H(3,𝔽3)—32no—
29C2×C22⋊H(3,𝔽3)—42no—
30C2×C22⋊H(3,𝔽3)—412no—
31C2×C22⋊H(3,𝔽3)—121no—
32C2×C22⋊H(3,𝔽3)A412yescomplemented
33C2×C3×H(3,𝔽3)—161no—
34C24×C32—32no—
35C24×C32—32no—
36C24×C32—91no—
37C24×C32—91no—
38C24×C32C2×C3211yescharacteristic, complemented
39C3×C22⋊A4—33no—
40C3×C22⋊A4C2×C3211yescharacteristic
41C3×C22⋊A4C2×C3213yescomplemented
42C3×C22⋊A4C2×C3212yescomplemented
43C22×C3×A4—36no—
44C22×C3×A4—36no—
45C22×C3×A4C2×C3216yescomplemented
46A42—39no—
47C32×A4—42no—
48C32×A4—46no—
49C32×A4—121no—
50C22×H(3,𝔽3)—42no—
51C22×H(3,𝔽3)—122no—
52C22×H(3,𝔽3)—122no—
53C22⋊H(3,𝔽3)—42no—
54C22⋊H(3,𝔽3)—412no—
55C22⋊H(3,𝔽3)—121no—
56C22⋊H(3,𝔽3)C2×A412yescomplemented
57C25×C3—31no—
58C25×C3C3311yescharacteristic
59C2×C22⋊A4—32no—
60C2×C22⋊A4—33no—
61C2×C22⋊A4H(3,𝔽3)13yescomplemented
62C23×A4—36no—
63C3×H(3,𝔽3)—161noSylow
64C23×C32—31no—
65C23×C32—31no—
66C23×C32—32no—
67C23×C32—32no—
68C23×C32—46no—
69C23×C32—91no—
70C23×C32—91no—
71C23×C32—92no—
72C23×C32—92no—
73C23×C32—92no—
74C23×C32—91no—
75C23×C32—91no—
76C23×C32—91no—
77C23×C32—94no—
78C23×C32C3×A412yescomplemented
79C2×C3×A4—36no—
80C2×C3×A4—36no—
81C2×C3×A4—42no—
82C2×C3×A4—46no—
83C2×C3×A4—44no—
84C2×C3×A4—121no—
85C2×C3×A4—122no—
86C2×C3×A4—123no—
87C2×C3×A4—1218no—
88C2×C3×A4—126no—
89C2×C3×A4—129no—
90C2×C3×A4C3×A416yescomplemented
91C2×C33—163no—
92C2×C33—161no—
93C2×H(3,𝔽3)—42no—
94C2×H(3,𝔽3)—122no—
95C2×H(3,𝔽3)—122no—
96C2×H(3,𝔽3)—166no—
97C2×H(3,𝔽3)—161no—
98C24×C3—32no—
99C24×C3—32no—
100C24×C3—31no—
101C24×C3—92no—
102C24×C3—92no—
103C24×C3—91no—
104C24×C3—93no—
105C24×C3—91no—
106C24×C3—93no—
107C24×C3C2×C3311yescharacteristic
108C22⋊A4—32no—
109C22⋊A4—33no—
110C22⋊A4C2×H(3,𝔽3)13yescomplemented
111C22×A4—36no—
112C22×A4—96no—
113C22×A4—96no—
114C22×C32—31no—
115C22×C32—32no—
116C22×C32—32no—
117C22×C32—31no—
118C22×C32—46no—
119C22×C32—91no—
120C22×C32—92no—
121C22×C32—91no—
122C22×C32—92no—
123C22×C32—92no—
124C22×C32—94no—
125C22×C32—91no—
126C22×C32—91no—
127C22×C32—91no—
128C22×C32—126no—
129C22×C32—126no—
130C22×C32C2×C3×A412yescomplemented
131C3×A4—42no—
132C3×A4—46no—
133C3×A4—44no—
134C3×A4—121no—
135C3×A4—122no—
136C3×A4—123no—
137C3×A4—1218no—
138C3×A4—126no—
139C3×A4—129no—
140C3×A4C2×C3×A416yescomplemented
141C25C3×H(3,𝔽3)11yescharacteristic, complemented, Sylow
142C33—163no—
143C33—161no—
144H(3,𝔽3)—42no—
145H(3,𝔽3)—166no—
146H(3,𝔽3)—161no—
147C23×C3—33no—
148C23×C3—31no—
149C23×C3—31no—
150C23×C3—32no—
151C23×C3—32no—
152C23×C3—32no—
153C23×C3—92no—
154C23×C3—92no—
155C23×C3—91no—
156C23×C3—91no—
157C23×C3—91no—
158C23×C3—92no—
159C23×C3—92no—
160C23×C3—92no—
161C23×C3—93no—
162C23×C3—91no—
163C23×C3—91no—
164C23×C3—93no—
165C23×C3—91no—
166C23×C3—94no—
167C23×C3—93no—
168C23×C3—96no—
169C23×C3—96no—
170C23×C3—96no—
171C23×C3—93no—
172C23×C3—93no—
173C23×C3—912no—
174C23×C3—126no—
175C23×C3C32×A412yes—
176C2×A4—36no—
177C2×A4—46no—
178C2×A4—96no—
179C2×A4—96no—
180C2×A4—123no—
181C2×A4—126no—
182C2×A4—124no—
183C2×A4—126no—
184C2×A4—363no—
185C2×C32—32no—
186C2×C32—32no—
187C2×C32—46no—
188C2×C32—91no—
189C2×C32—91no—
190C2×C32—126no—
191C2×C32—126no—
192C2×C32—161no—
193C2×C32—162no—
194C2×C32—163no—
195C2×C32—483no—
196C2×C32—489no—
197C2×C32A4211yescharacteristic, complemented
198C24—32no—
199C24—32no—
200C24—91no—
201C24—91no—
202C24C2×C3×H(3,𝔽3)11yescharacteristic, complemented
203C22×C3—32no—
204C22×C3—32no—
205C22×C3—33no—
206C22×C3—31no—
207C22×C3—31no—
208C22×C3—32no—
209C22×C3—92no—
210C22×C3—92no—
211C22×C3—91no—
212C22×C3—91no—
213C22×C3—92no—
214C22×C3—91no—
215C22×C3—92no—
216C22×C3—92no—
217C22×C3—93no—
218C22×C3—94no—
219C22×C3—93no—
220C22×C3—91no—
221C22×C3—91no—
222C22×C3—93no—
223C22×C3—96no—
224C22×C3—93no—
225C22×C3—96no—
226C22×C3—96no—
227C22×C3—912no—
228C22×C3—93no—
229C22×C3—91no—
230C22×C3—126no—
231C22×C3—366no—
232C22×C3—366no—
233C22×C3C2×C32×A412yes—
234A4—36no—
235A4—46no—
236A4—123no—
237A4—126no—
238A4—124no—
239A4—126no—
240A4—363no—
241C32—46no—
242C32—161no—
243C32—162no—
244C32—163no—
245C32—483no—
246C32—489no—
247C32C2×A4211yescharacteristic, complemented
248C23—31no—
249C23—31no—
250C23—32no—
251C23—32no—
252C23—91no—
253C23—91no—
254C23—92no—
255C23—92no—
256C23—92no—
257C23—91no—
258C23—91no—
259C23—91no—
260C23—94no—
261C23A4×H(3,𝔽3)12yescomplemented
262C2×C3—32no—
263C2×C3—32no—
264C2×C3—31no—
265C2×C3—92no—
266C2×C3—92no—
267C2×C3—93no—
268C2×C3—93no—
269C2×C3—91no—
270C2×C3—91no—
271C2×C3—126no—
272C2×C3—163no—
273C2×C3—366no—
274C2×C3—366no—
275C2×C3—482no—
276C2×C3—483no—
277C2×C3C3×A4211yescharacteristic
278C22—32no—
279C22—32no—
280C22—31no—
281C22—31no—
282C22—91no—
283C22—92no—
284C22—91no—
285C22—92no—
286C22—92no—
287C22—94no—
288C22—91no—
289C22—91no—
290C22—91no—
291C22C2×A4×H(3,𝔽3)12yescomplemented
292C3—31no—
293C3—126no—
294C3—163no—
295C3—482no—
296C3—483no—
297C3C2×C3×A4211yescharacteristic
298C2—32no—
299C2—32no—
300C2—91no—
301C2—91no—
302C2(C22⋊A4)⋊H(3,𝔽3)11yescharacteristic, direct product factor
subgroupG/H
centerC2×C3C3×A42
commutator, cosocleC24×C3C2×C33
FrattiniC3C2×C3×A42
FittingC25×C32C32
socleC25×C3C33
hypercenterC2×C32A42
hypocenterC24C2×C3×H(3,𝔽3)
LCSG ▷ C24×C3 ▷ C24
UCSC1 ◁ C2×C3 ◁ C2×C32

Complements

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

NH ≅ G/Ntype
1. (C22⋊A4)⋊H(3,𝔽3)C2direct (N × H)
2. C2×C32×C22⋊A4C3semidirect
3. C2×C3×A42C3semidirect
4. C2×C32⋊(C22⋊A4)C3semidirect
5. C23×C22⋊H(3,𝔽3)C3semidirect
6. C2×(C3×A4)⋊A4C3semidirect
8. C32×C22⋊A4C2×C3semidirect
9. C3×A42C2×C3semidirect
10. C32⋊(C22⋊A4)C2×C3semidirect
13. C22×C22⋊H(3,𝔽3)C2×C3semidirect
14. (C3×A4)⋊A4C2×C3semidirect
16. C25×C32C32semidirect
19. C2×C3×C22⋊A4C32semidirect
20. C2×C3×C22⋊A4C32semidirect
21. C23×C3×A4C32semidirect
32. C2×C22⋊H(3,𝔽3)A4semidirect
38. C24×C32C2×C32semidirect
41. C3×C22⋊A4C2×C32semidirect
42. C3×C22⋊A4C2×C32semidirect
45. C22×C3×A4C2×C32semidirect
56. C22⋊H(3,𝔽3)C2×A4semidirect
61. C2×C22⋊A4H(3,𝔽3)semidirect
78. C23×C32C3×A4semidirect
90. C2×C3×A4C3×A4semidirect
110. C22⋊A4C2×H(3,𝔽3)semidirect
130. C22×C32C2×C3×A4semidirect
140. C3×A4C2×C3×A4semidirect
141. C25C3×H(3,𝔽3)semidirect
197. C2×C32A42semidirect
202. C24C2×C3×H(3,𝔽3)semidirect
247. C32C2×A42semidirect
261. C23A4×H(3,𝔽3)semidirect
291. C22C2×A4×H(3,𝔽3)semidirect
302. C2(C22⋊A4)⋊H(3,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
(C22⋊A4)⋊H(3,𝔽3)2
C2×C32×C22⋊A43
C2×C3×A423
C2×C32⋊(C22⋊A4)3
C23×C22⋊H(3,𝔽3)3
C2×(C3×A4)⋊A43
C2×A4×H(3,𝔽3)4