Finite Group Databasebeta

← #72252829

Subgroups — #72252829

Order 5184.

2656 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 2656 conjugacy classes of subgroups — too many to draw. Only its 65 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
2C22×#88907C313yesmaximal, complemented
3#13180074C316yesmaximal, complemented
4#13180071C313yesmaximal, complemented
5#13049240—43nomaximal
6C2×#88907—33no—
7C26×C32C3211yescharacteristic
8C3×C22⋊(C22⋊A4)—33no—
9C3×C22⋊(C22⋊A4)C3212yes—
10C3×C22⋊(C22⋊A4)C3211yescharacteristic
11C22×C22⋊(C22⋊C9)C3219yescomplemented
12C32×C22⋊A4—43no—
13C32×C22⋊A4—123no—
14C32×C22⋊A4—123no—
15#88907A413yescomplemented
16C22×#1094—46no—
17C22×#1094—123no—
18#88881—418no—
19(C22⋊A4)⋊C9—49no—
20A4×3−1+2—163no—
21C25×C32—33no—
22C25×C32—93no—
23C25×C32—93no—
24C2×C22⋊(C22⋊C9)—39no—
25C2×#1094—126no—
26C2×#1094—129no—
27C26×C3—31no—
28C26×C3C3311yescharacteristic
29C22⋊(C22⋊A4)—32no—
30C22⋊(C22⋊A4)3−1+213yescomplemented
31C24×C32—33no—
32C24×C32—33no—
33C24×C32—33no—
34C24×C32—93no—
35C24×C32—99no—
36C24×C32—99no—
37C24×C32—96no—
38C24×C32—93no—
39C24×C32—93no—
40C24×C32—918no—
41C24×C32—99no—
42C24×C32—99no—
43C24×C32C3×A413yes—
44C3×C22⋊A4—43no—
45C3×C22⋊A4—46no—
46C3×C22⋊A4—123no—
47C3×C22⋊A4—126no—
48C3×C22⋊A4—129no—
49C3×C22⋊A4—1227no—
50C3×C22⋊A4—1227no—
51C3×C22⋊A4—126no—
52C3×C22⋊A4—123no—
53C22⋊(C22⋊C9)C3×A419yescomplemented
54C22×C22⋊C9—418no—
55C22×C22⋊C9—129no—
56C32×A4—163no—
57C32×A4—483no—
58C32×A4—483no—
59C22×3−1+2—163no—
60#1094—46no—
61#1094—123no—
62C9⋊A4—1618no—
63C9×A4—169no—
64C25×C3—33no—
65C25×C3—93no—
66C25×C3—93no—
67C25×C3—93no—
68C25×C3—99no—
69C25×C3—99no—
70C3×3−1+2—641noSylow
71C23×C32—39no—
72C23×C32—36no—
73C23×C32—93no—
74C23×C32—93no—
75C23×C32—99no—
76C23×C32—96no—
77C23×C32—99no—
78C23×C32—99no—
79C23×C32—99no—
80C23×C32—93no—
81C23×C32—918no—
82C23×C32—99no—
83C23×C32—99no—
84C23×C32—936no—
85C23×C32—96no—
86C23×C32—918no—
87C23×C32—93no—
88C2×C22⋊C9—1218no—
89C2×C22⋊C9—1227no—
90C26C3×3−1+211yescharacteristic, complemented, Sylow
91C2×3−1+2—483no—
92C24×C3—33no—
93C24×C3—33no—
94C24×C3—39no—
95C24×C3—39no—
96C24×C3—33no—
97C24×C3—33no—
98C24×C3—93no—
99C24×C3—93no—
100C24×C3—96no—
101C24×C3—99no—
102C24×C3—99no—
103C24×C3—99no—
104C24×C3—99no—
105C24×C3—918no—
106C24×C3—927no—
107C24×C3—927no—
108C24×C3—927no—
109C24×C3—93no—
110C24×C3—93no—
111C24×C3—918no—
112C24×C3—99no—
113C24×C3—99no—
114C24×C3—99no—
115C24×C3—954no—
116C24×C3—927no—
117C24×C3C32×A413yes—
118C22⋊A4—49no—
119C22⋊A4—129no—
120C22⋊A4—1218no—
121C22⋊A4—1218no—
122C22⋊A4—126no—
123C22⋊A4—129no—
124C22×C32—33no—
125C22×C32—33no—
126C22×C32—33no—
127C22×C32—93no—
128C22×C32—96no—
129C22×C32—99no—
130C22×C32—99no—
131C22×C32—99no—
132C22×C32—93no—
133C22×C32—918no—
134C22×C32—99no—
135C22×C32—93no—
136C22×C32A4213yes—
137C3×A4—163no—
138C3×A4—166no—
139C3×A4—483no—
140C3×A4—486no—
141C3×A4—483no—
142C3×A4—486no—
143C3×A4—489no—
144C3×A4—4827no—
145C3×A4—4827no—
146C22×C9—169no—
147C22⋊C9—418no—
148C22⋊C9—129no—
149C25—33no—
150C25—93no—
151C25—93no—
152C33—641no—
1533−1+2—163no—
1543−1+2—646no—
155C3×C9—643no—
156C23×C3—39no—
157C23×C3—36no—
158C23×C3—96no—
159C23×C3—99no—
160C23×C3—93no—
161C23×C3—99no—
162C23×C3—93no—
163C23×C3—99no—
164C23×C3—96no—
165C23×C3—99no—
166C23×C3—99no—
167C23×C3—927no—
168C23×C3—918no—
169C23×C3—927no—
170C23×C3—99no—
171C23×C3—99no—
172C23×C3—99no—
173C23×C3—927no—
174C23×C3—927no—
175C23×C3—93no—
176C23×C3—918no—
177C23×C3—99no—
178C23×C3—99no—
179C23×C3—936no—
180C23×C3—96no—
181C23×C3—918no—
182C23×C3—99no—
183C23×C3—954no—
184C23×C3—927no—
185C23×C3—927no—
186C23×C3—9108no—
187C23×C3—918no—
188C23×C3—954no—
189C23×C3—93no—
190C2×C32—33no—
191C2×C32—93no—
192C2×C32—93no—
193C2×C9—489no—
194C24—33no—
195C24—33no—
196C24—33no—
197C24—93no—
198C24—96no—
199C24—99no—
200C24—99no—
201C24—99no—
202C24—93no—
203C24—93no—
204C24—918no—
205C24—99no—
206C24A4×3−1+213yescomplemented
207C22×C3—33no—
208C22×C3—39no—
209C22×C3—39no—
210C22×C3—33no—
211C22×C3—33no—
212C22×C3—33no—
213C22×C3—93no—
214C22×C3—93no—
215C22×C3—96no—
216C22×C3—99no—
217C22×C3—99no—
218C22×C3—99no—
219C22×C3—99no—
220C22×C3—918no—
221C22×C3—927no—
222C22×C3—927no—
223C22×C3—927no—
224C22×C3—99no—
225C22×C3—954no—
226C22×C3—927no—
227C22×C3—99no—
228C22×C3—93no—
229C22×C3—918no—
230C22×C3—99no—
231C22×C3—93no—
232C22×C3C3×A4213yes—
233A4—169no—
234A4—489no—
235A4—489no—
236A4—4818no—
237A4—4818no—
238A4—486no—
239C32—641no—
240C32—642no—
241C32—1923no—
242C32#1059678211yescharacteristic
243C9—169no—
244C23—39no—
245C23—36no—
246C23—93no—
247C23—93no—
248C23—99no—
249C23—96no—
250C23—99no—
251C23—99no—
252C23—99no—
253C23—93no—
254C23—918no—
255C23—99no—
256C23—99no—
257C23—936no—
258C23—96no—
259C23—918no—
260C23—93no—
261C2×C3—33no—
262C2×C3—93no—
263C2×C3—99no—
264C2×C3—99no—
265C2×C3—93no—
266C2×C3—93no—
267C22—33no—
268C22—33no—
269C22—33no—
270C22—93no—
271C22—96no—
272C22—99no—
273C22—99no—
274C22—99no—
275C22—93no—
276C22—918no—
277C22—99no—
278C22—93no—
279C22#1304924013yescomplemented
280C3—31no—
281C3—643no—
282C3—1922no—
283C3C3×#1059678211yescharacteristic
284C2—33no—
285C2—93no—
286C2—93no—
subgroupG/H
center, FrattiniC3C3×#10596782
commutator, socle, cosocleC26×C3C33
FittingC26×C32C32
prime radicalC32×C22⋊(C22⋊A4)C3
hypercenterC32#10596782
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. C22×#88907C3semidirect
3. #13180074C3semidirect
4. #13180071C3semidirect
11. C22×C22⋊(C22⋊C9)C32semidirect
15. #88907A4semidirect
30. C22⋊(C22⋊A4)3−1+2semidirect
53. C22⋊(C22⋊C9)C3×A4semidirect
90. C26C3×3−1+2semidirect
206. C24A4×3−1+2semidirect
279. C22#13049240semidirect

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
C22×#889073
#131800743
#131800713
#130492404