Finite Group Databasebeta

← A4×#88907 #72252825

Subgroups — A4×#88907 #72252825

Order 5184.

3608 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 3608 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C24×C32×A4C311yesmaximal, characteristic
2#13180116C312yesmaximal, complemented
3C22×#88907C311yesmaximal, characteristic, complemented
4#13180067C316yesmaximal, complemented
5A4×C22⋊(C22⋊C9)C313yesmaximal, complemented
6C3×#88907—41nomaximal
7A4×#1094—45nomaximal
8C23×C32×A4—35no—
9C2×#88907—31no—
10C26×C32C3211yescharacteristic
11C24×C3×A4—33no—
12C24×C3×A4C3212yes—
13C24×C3×A4C3211yescharacteristic
14C22⋊(C22⋊(C22⋊C9))C3216yescomplemented
15C22×C22⋊(C22⋊C9)C3213yescomplemented
16C24×C33—41no—
17C22×C32×A4—310no—
18C22×C32×A4A415yes—
19#88907—46no—
20#88907—42no—
21#88907—410no—
22#88907—1210no—
23#88907A411yescharacteristic, direct product factor
24C3×C22⋊(C22⋊C9)—43no—
25C22×#1094—45no—
26#88880—430no—
27A4×C22⋊C9—415no—
28C3×#1094—165no—
29A4×3−1+2—161no—
30C25×C32—35no—
31C25×C32—31no—
32C25×C32—95no—
33C23×C3×A4—345no—
34C23×C3×A4—310no—
35C23×C3×A4—35no—
36C2×C22⋊(C22⋊C9)—33no—
37C23×C33—125no—
38C2×C32×A4—35no—
39C2×#1094—125no—
40C26×C3—31no—
41C26×C3C3311yescharacteristic
42C24×A4—32no—
43C24×A43−1+213yescomplemented
44C24×C32—310no—
45C24×C32—310no—
46C24×C32—41no—
47C24×C32—42no—
48C24×C32—95no—
49C24×C32—95no—
50C24×C32—95no—
51C24×C32—920no—
52C24×C32—930no—
53C24×C32—123no—
54C24×C32C3×A411yescharacteristic
55C24×C32C3×A415yes—
56C22×C3×A4—390no—
57C22×C3×A4—320no—
58C22×C3×A4—310no—
59C22×C3×A4—315no—
60C22×C3×A4C3×A4110yes—
61C22×C3×A4C3×A415yes—
62C22⋊(C22⋊C9)—46no—
63C22⋊(C22⋊C9)—430no—
64C22⋊(C22⋊C9)—1230no—
65C22⋊(C22⋊C9)C3×A413yescomplemented
66C22×C22⋊C9—415no—
67C22×C33—45no—
68C22×C33—1210no—
69C32×A4C22⋊A411yescharacteristic
70C22×3−1+2—161no—
71#1094—45no—
72#1094—1630no—
73#1094—1610no—
74#1094—162no—
75#1094—4810no—
76C3×C22⋊C9—1615no—
77C9⋊A4—166no—
78C9×A4—163no—
79C25×C3—315no—
80C25×C3—35no—
81C25×C3—31no—
82C25×C3—91no—
83C25×C3—95no—
84C25×C3—915no—
85C23×A4—330no—
86C23×A4—315no—
87C3×3−1+2—641noSylow
88C23×C32—35no—
89C23×C32—35no—
90C23×C32—35no—
91C23×C32—95no—
92C23×C32—910no—
93C23×C32—940no—
94C23×C32—930no—
95C23×C32—930no—
96C23×C32—930no—
97C23×C32—95no—
98C23×C32—125no—
99C23×C32—1210no—
100C23×C32—1245no—
101C2×C3×A4—345no—
102C2×C3×A4—310no—
103C2×C3×A4—35no—
104C2×C22⋊C9—1215no—
105C26C3×3−1+211yescharacteristic, complemented, Sylow
106C2×C33—125no—
107C2×3−1+2—481no—
108C24×C3—330no—
109C24×C3—310no—
110C24×C3—310no—
111C24×C3—31no—
112C24×C3—35no—
113C24×C3—43no—
114C24×C3—910no—
115C24×C3—95no—
116C24×C3—95no—
117C24×C3—95no—
118C24×C3—915no—
119C24×C3—915no—
120C24×C3—915no—
121C24×C3—930no—
122C24×C3—990no—
123C24×C3—920no—
124C24×C3—960no—
125C24×C3—122no—
126C24×C3C32×A411yescharacteristic
127C24×C3C32×A415yes—
128C22×A4—360no—
129C22×A4—330no—
130C22×A4—310no—
131C22×A4#1094115yescomplemented
132C22×C32—310no—
133C22×C32—310no—
134C22×C32—45no—
135C22×C32—410no—
136C22×C32—95no—
137C22×C32—95no—
138C22×C32—95no—
139C22×C32—930no—
140C22×C32—920no—
141C22×C32—1215no—
142C22×C32—1210no—
143C22×C32—1220no—
144C22×C32—1290no—
145C22×C32A4215yes—
146C22×C32C3×C22⋊A411yescharacteristic
147C3×A4—33no—
148C3×A4C3×C22⋊A412yes—
149C3×A4C3×C22⋊A411yescharacteristic
150C22×C9—163no—
151C22⋊C9—415no—
152C22⋊C9—1630no—
153C22⋊C9—166no—
154C22⋊C9—4830no—
155C25—35no—
156C25—31no—
157C25—95no—
158C33—41no—
1593−1+2—161no—
1603−1+2—646no—
1613−1+2—642no—
162C3×C9—643no—
163C23×C3—315no—
164C23×C3—35no—
165C23×C3—315no—
166C23×C3—35no—
167C23×C3—35no—
168C23×C3—95no—
169C23×C3—95no—
170C23×C3—910no—
171C23×C3—930no—
172C23×C3—940no—
173C23×C3—930no—
174C23×C3—990no—
175C23×C3—915no—
176C23×C3—915no—
177C23×C3—930no—
178C23×C3—990no—
179C23×C3—9120no—
180C23×C3—930no—
181C23×C3—990no—
182C23×C3—95no—
183C23×C3—1215no—
184C23×C3—1230no—
185C2×A4—330no—
186C2×A4—315no—
187C2×C32—35no—
188C2×C32—31no—
189C2×C32—95no—
190C2×C32—125no—
191C2×C32—1210no—
192C2×C32—1245no—
193C2×C9—483no—
194C24—310no—
195C24—310no—
196C24—95no—
197C24—95no—
198C24—95no—
199C24—930no—
200C24—920no—
201C24A4×3−1+211yescharacteristic, complemented
202C24C3×#109415yescomplemented
203C22×C3—330no—
204C22×C3—310no—
205C22×C3—310no—
206C22×C3—35no—
207C22×C3—31no—
208C22×C3—415no—
209C22×C3—910no—
210C22×C3—95no—
211C22×C3—95no—
212C22×C3—95no—
213C22×C3—915no—
214C22×C3—915no—
215C22×C3—990no—
216C22×C3—930no—
217C22×C3—915no—
218C22×C3—960no—
219C22×C3—920no—
220C22×C3—1230no—
221C22×C3—1260no—
222C22×C3—1210no—
223C22×C3C3×A4215yes—
224C22×C3C32×C22⋊A411yescharacteristic
225A4—32no—
226A4#8890713yesdirect product factor
227C32—41no—
228C32—42no—
229C32—123no—
230C32A4×C22⋊A411yescharacteristic
231C9—163no—
232C9—646no—
233C23—35no—
234C23—35no—
235C23—35no—
236C23—95no—
237C23—910no—
238C23—940no—
239C23—930no—
240C23—930no—
241C23—930no—
242C23—95no—
243C2×C3—315no—
244C2×C3—35no—
245C2×C3—31no—
246C2×C3—91no—
247C2×C3—915no—
248C2×C3—95no—
249C2×C3—1215no—
250C2×C3—1230no—
251C22—310no—
252C22—310no—
253C22—95no—
254C22—95no—
255C22—95no—
256C22—930no—
257C22—920no—
258C22A4×#109415yescomplemented
259C22C3×#8890711yescharacteristic, complemented
260C3—31no—
261C3—43no—
262C3—122no—
263C3C3×A4×C22⋊A411yescharacteristic
264C2—35no—
265C2—31no—
266C2—95no—
subgroupG/H
center, FrattiniC3C3×A4×C22⋊A4
commutator, socle, cosocleC26×C3C33
FittingC26×C32C32
prime radicalC24×C32×A4C3
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. #13180067C3semidirect
5. A4×C22⋊(C22⋊C9)C3semidirect
14. C22⋊(C22⋊(C22⋊C9))C32semidirect
15. C22×C22⋊(C22⋊C9)C32semidirect
23. #88907A4direct (N × H)
43. C24×A43−1+2semidirect
65. C22⋊(C22⋊C9)C3×A4semidirect
105. C26C3×3−1+2semidirect
131. C22×A4#1094semidirect
201. C24A4×3−1+2semidirect
202. C24C3×#1094semidirect
226. A4#88907direct (N × H)
258. C22A4×#1094semidirect
259. C22C3×#88907semidirect

Maximal subgroups

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

subgroupindex
C24×C32×A43
#131801163
C22×#889073
#131800673
A4×C22⋊(C22⋊C9)3
C3×#889074
A4×#10944