Finite Group Databasebeta

← #72252810

Subgroups — #72252810

Order 5184.

3008 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 3008 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C32×C22⋊(C22⋊A4)C311yesmaximal, characteristic
2#13180116C311yesmaximal, characteristic, complemented
3C22×#88907C311yesmaximal, characteristic, complemented
4C24×#1094C311yesmaximal, characteristic, complemented
5#13180069C316yesmaximal, complemented
6#13180065C313yesmaximal, complemented
73−1+2×C22⋊A4—41nomaximal
8#13049240—45nomaximal
9C2×#88907—31no—
10C23×#1094—35no—
11C26×C32C3211yescharacteristic
12C3×C22⋊(C22⋊A4)—33no—
13C3×C22⋊(C22⋊A4)C3212yes—
14C3×C22⋊(C22⋊A4)C3211yescharacteristic
15C22⋊(C22⋊(C22⋊C9))C3213yescomplemented
16C22×C22⋊(C22⋊C9)C3213yescomplemented
17C24×C22⋊C9C3213yescomplemented
18C32×C22⋊A4—41no—
19C32×C22⋊A4—45no—
20C32×C22⋊A4—125no—
21C24×3−1+2—41no—
22#88907—41no—
23#88907—45no—
24#88907—125no—
25#88907A411yescharacteristic, complemented
26C9⋊(C22⋊A4)—46no—
27C9×C22⋊A4—43no—
28C22×#1094—310no—
29C22×#1094—45no—
30C22×#1094A415yescomplemented
31#88881—430no—
32(C22⋊A4)⋊C9—415no—
33A4×3−1+2—165no—
34A4×3−1+2—161no—
35C25×C32—35no—
36C25×C32—31no—
37C25×C32—95no—
38C2×C22⋊(C22⋊C9)—33no—
39C23×C22⋊C9—315no—
40C23×3−1+2—125no—
41C2×#1094—35no—
42C2×#1094—125no—
43C26×C3—31no—
44C26×C3C3311yescharacteristic
45C22⋊(C22⋊A4)—32no—
46C22⋊(C22⋊A4)3−1+213yescomplemented
47C24×C32—310no—
48C24×C32—35no—
49C24×C32—35no—
50C24×C32—95no—
51C24×C32—95no—
52C24×C32—95no—
53C24×C32—920no—
54C24×C32—930no—
55C24×C32C3×A411yescharacteristic
56C24×C32C3×A415yes—
57C3×C22⋊A4—41no—
58C3×C22⋊A4—42no—
59C3×C22⋊A4—45no—
60C3×C22⋊A4—410no—
61C3×C22⋊A4—125no—
62C3×C22⋊A4—1210no—
63C3×C22⋊A4—123no—
64C3×C22⋊A4—1215no—
65C3×C22⋊A4—1245no—
66C24×C9—43no—
67C22⋊(C22⋊C9)—43no—
68C22⋊(C22⋊C9)—415no—
69C22⋊(C22⋊C9)—1215no—
70C22⋊(C22⋊C9)C3×A413yescomplemented
71C22×C22⋊C9—330no—
72C22×C22⋊C9—415no—
73C22×C22⋊C9C3×A4115yescomplemented
74C32×A4—165no—
75C32×A4—161no—
76C32×A4—485no—
77C22×3−1+2—45no—
78C22×3−1+2—1210no—
79C22×3−1+2—161no—
80#1094—45no—
81#1094—165no—
82#1094—161no—
83#1094—485no—
84#1094C22⋊A411yescharacteristic, complemented
85C9⋊A4—1630no—
86C9⋊A4—166no—
87C9×A4—1615no—
88C9×A4—163no—
89C25×C3—35no—
90C25×C3—31no—
91C25×C3—95no—
92C25×C3—91no—
93C25×C3—95no—
94C25×C3—915no—
95C3×3−1+2—641noSylow
96C23×C32—35no—
97C23×C32—35no—
98C23×C32—35no—
99C23×C32—95no—
100C23×C32—920no—
101C23×C32—910no—
102C23×C32—920no—
103C23×C32—930no—
104C23×C32—930no—
105C23×C32—930no—
106C23×C32—95no—
107C23×C9—1215no—
108C2×C22⋊C9—315no—
109C2×C22⋊C9—1215no—
110C26C3×3−1+211yescharacteristic, complemented, Sylow
111C2×3−1+2—125no—
112C2×3−1+2—481no—
113C24×C3—310no—
114C24×C3—315no—
115C24×C3—35no—
116C24×C3—35no—
117C24×C3—31no—
118C24×C3—35no—
119C24×C3—910no—
120C24×C3—95no—
121C24×C3—95no—
122C24×C3—95no—
123C24×C3—95no—
124C24×C3—930no—
125C24×C3—915no—
126C24×C3—915no—
127C24×C3—915no—
128C24×C3—990no—
129C24×C3—920no—
130C24×C3—960no—
131C24×C3C32×A411yescharacteristic
132C24×C3C32×A415yes—
133C22⋊A4—43no—
134C22⋊A4—415no—
135C22⋊A4—1215no—
136C22⋊A4—1230no—
137C22⋊A4—1210no—
138C22⋊A4—122no—
139C22×C32—35no—
140C22×C32—310no—
141C22×C32—35no—
142C22×C32—95no—
143C22×C32—95no—
144C22×C32—95no—
145C22×C32—930no—
146C22×C32—920no—
147C22×C32A4215yes—
148C22×C32C3×C22⋊A411yescharacteristic
149C3×A4—165no—
150C3×A4—161no—
151C3×A4—1610no—
152C3×A4—162no—
153C3×A4—485no—
154C3×A4—4810no—
155C3×A4—4815no—
156C3×A4—483no—
157C3×A4—4845no—
158C22×C9—415no—
159C22×C9—1230no—
160C22×C9—163no—
161C22⋊C9—415no—
162C22⋊C9—1615no—
163C22⋊C9—163no—
164C22⋊C9—4815no—
165C22⋊C9C3×C22⋊A413yescomplemented
166C25—35no—
167C25—31no—
168C25—95no—
169C33—641no—
1703−1+2—41no—
1713−1+2—161no—
1723−1+2—646no—
1733−1+2—641no—
174C3×C9—643no—
175C23×C3—35no—
176C23×C3—35no—
177C23×C3—35no—
178C23×C3—95no—
179C23×C3—95no—
180C23×C3—95no—
181C23×C3—95no—
182C23×C3—915no—
183C23×C3—910no—
184C23×C3—920no—
185C23×C3—920no—
186C23×C3—960no—
187C23×C3—915no—
188C23×C3—930no—
189C23×C3—930no—
190C23×C3—930no—
191C23×C3—960no—
192C23×C3—990no—
193C23×C3—990no—
194C23×C3—930no—
195C23×C3—990no—
196C23×C3—95no—
197C2×C32—35no—
198C2×C32—31no—
199C2×C32—95no—
200C2×C9—1215no—
201C2×C9—483no—
202C24—310no—
203C24—35no—
204C24—35no—
205C24—95no—
206C24—95no—
207C24—95no—
208C24—930no—
209C24—920no—
210C24A4×3−1+211yescharacteristic, complemented
211C24A4×3−1+215yescomplemented
212C22×C3—310no—
213C22×C3—35no—
214C22×C3—315no—
215C22×C3—35no—
216C22×C3—35no—
217C22×C3—31no—
218C22×C3—910no—
219C22×C3—95no—
220C22×C3—95no—
221C22×C3—95no—
222C22×C3—95no—
223C22×C3—930no—
224C22×C3—915no—
225C22×C3—915no—
226C22×C3—915no—
227C22×C3—990no—
228C22×C3—960no—
229C22×C3—920no—
230C22×C3C3×A4215yes—
231C22×C3C32×C22⋊A411yescharacteristic
232A4—1615no—
233A4—163no—
234A4—4815no—
235A4—4830no—
236A4—4810no—
237A4—482no—
238C32—641no—
239C32—642no—
240C32—1923no—
241C32A4×C22⋊A411yescharacteristic
242C9—43no—
243C9—163no—
244C9—643no—
245C23—35no—
246C23—35no—
247C23—35no—
248C23—95no—
249C23—910no—
250C23—920no—
251C23—920no—
252C23—930no—
253C23—930no—
254C23—930no—
255C23—95no—
256C2×C3—35no—
257C2×C3—31no—
258C2×C3—95no—
259C2×C3—91no—
260C2×C3—915no—
261C2×C3—95no—
262C22—310no—
263C22—35no—
264C22—35no—
265C22—95no—
266C22—95no—
267C22—95no—
268C22—930no—
269C22—920no—
270C22#1304924015yescomplemented
271C223−1+2×C22⋊A411yescharacteristic, complemented
272C3—31no—
273C3—643no—
274C3—1922no—
275C3C3×A4×C22⋊A411yescharacteristic
276C2—35no—
277C2—31no—
278C2—95no—
subgroupG/H
center, FrattiniC3C3×A4×C22⋊A4
commutator, socle, cosocleC26×C3C33
FittingC26×C32C32
prime radicalC32×C22⋊(C22⋊A4)C3
hypercenterC32A4×C22⋊A4
hypocenterC26C3×3−1+2
LCSG ▷ C26×C3 ▷ C26
UCSC1 ◁ C3 ◁ C32

Complements

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

NH ≅ G/Ntype
2. #13180116C3semidirect
3. C22×#88907C3semidirect
4. C24×#1094C3semidirect
5. #13180069C3semidirect
6. #13180065C3semidirect
15. C22⋊(C22⋊(C22⋊C9))C32semidirect
16. C22×C22⋊(C22⋊C9)C32semidirect
17. C24×C22⋊C9C32semidirect
25. #88907A4semidirect
30. C22×#1094A4semidirect
46. C22⋊(C22⋊A4)3−1+2semidirect
70. C22⋊(C22⋊C9)C3×A4semidirect
73. C22×C22⋊C9C3×A4semidirect
84. #1094C22⋊A4semidirect
110. C26C3×3−1+2semidirect
165. C22⋊C9C3×C22⋊A4semidirect
210. C24A4×3−1+2semidirect
211. C24A4×3−1+2semidirect
270. C22#13049240semidirect
271. C223−1+2×C22⋊A4semidirect

Maximal subgroups

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

subgroupindex
C32×C22⋊(C22⋊A4)3
#131801163
C22×#889073
C24×#10943
#131800693
#131800653
3−1+2×C22⋊A44
#130492404