Finite Group Databasebeta

← #44408059

Subgroups — #44408059

Order 29056.

582 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C2×#20300272C211yesmaximal, characteristic
2C22×C4×Dic227C211yesmaximal, characteristic, complemented
3#28408922C212yesmaximal, complemented
4#28408921C212yesmaximal, complemented
5#28408919C212yesmaximal, complemented
6#28408918C212yesmaximal, complemented
7#28408911C211yesmaximal, characteristic
8#28408909C211yesmaximal, characteristic
9#28408904C211yesmaximal, characteristic, complemented
10#28408893C211yesmaximal, characteristic, complemented
11#385×C227C211yesmaximal, characteristic, complemented
12C22×C22⋊D227C2211yescharacteristic
13C23×Dic227C2211yescharacteristic
14C23×Dic227C2211yescharacteristic
15#20300272—24no—
16C2×C4⋊Dic227C2211yescharacteristic
17C2×C4⋊Dic227C2211yescharacteristic
18C2×Dic227⋊C4C2212yes—
19C2×C22⋊Dic227C2211yescharacteristic
20C2×(C2×D227)⋊C4C2212yes—
21C2×(C2×D227)⋊C4C2212yes—
22C2×(C2×D227)⋊C4C2211yescharacteristic
23C2×(C2×D227)⋊C4C2211yescharacteristic
24C2×C4×Dic227—24no—
25C2×C4×Dic227C2212yes—
26C2×C4×Dic227C2212yes—
27C227⋊#95C2212yes—
28C227⋊#95C2212yes—
29C227⋊#95C2212yes—
30C227⋊#95C2212yes—
31C227⋊#95C2212yes—
32C23×C4×C227C2211yescharacteristic
33C2×C4⋊C4×C227C2212yes—
34C2×C22⋊C4×C227C2211yescharacteristic
35C2×C22⋊C4×C227C2211yescharacteristic
36#95×C227C2212yes—
37C23×D227C2311yescharacteristic
38C2×C22⋊D227—22no—
39C2×C22⋊D227—22no—
40C2×C22⋊D227—22no—
41C22×Dic227—24no—
42C22×Dic227—22no—
43C22×Dic227—22no—
44C22×Dic227—22no—
45C22×Dic227D414yes—
46C22×Dic227C2311yescharacteristic
47C22×Dic227C2312yes—
48C22×Dic227C2311yescharacteristic
49C22×Dic227C2311yescharacteristic
50C22×Dic227C2312yes—
51C4⋊Dic227—22no—
52C4⋊Dic227—22no—
53Dic227⋊C4—24no—
54C22⋊Dic227—22no—
55(C2×D227)⋊C4—22no—
56(C2×D227)⋊C4—22no—
57(C2×D227)⋊C4—24no—
58(C2×D227)⋊C4—24no—
59C4×Dic227—24no—
60C4×Dic227—24no—
61C24×C227C2311yescharacteristic
62C22×C4×C227—22no—
63C22×C4×C227—24no—
64C22×C4×C227C2311yescharacteristic
65C22×C4×C227C2312yes—
66C22×C4×C227C2311yescharacteristic
67C22×C4×C227C2312yes—
68C4⋊C4×C227—24no—
69C22⋊C4×C227—22no—
70C22⋊C4×C227—22no—
71C22×D227—21no—
72C22×D227—22no—
73C22×D227—22no—
74C22×D227—21no—
75C22×D227—21no—
76C22⋊D227—44no—
77C2×Dic227—28no—
78C2×Dic227—28no—
79C2×Dic227—22no—
80C2×Dic227—21no—
81C2×Dic227—21no—
82C2×Dic227—21no—
83C2×Dic227—21no—
84C2×Dic227—22no—
85C2×Dic227—21no—
86C2×Dic227—22no—
87C2×Dic227—22no—
88C2×Dic227—22no—
89C2×Dic227—21no—
90C2×Dic227—21no—
91C2×Dic227C2×D412yes—
92C2×Dic227C2×D412yes—
93C2×Dic227#4114yes—
94C23×C227—22no—
95C23×C227—22no—
96C23×C227C2×D412yes—
97C23×C227#4112yes—
98C23×C227#4112yes—
99C23×C227C2411yescharacteristic
100C2×C4×C227—28no—
101C2×C4×C227—22no—
102C2×C4×C227—22no—
103C2×C4×C227—21no—
104C2×C4×C227—21no—
105C2×C4×C227—22no—
106C2×C4×C227—22no—
107C2×C4×C227—22no—
108C2×C4×C227—22no—
109C2×C4×C227—22no—
110C2×C4×C227#4114yes—
111C22×C227—22no—
112C22×C227—22no—
113C22×C227—24no—
114C22×C227—24no—
115C22×C227#12314yes—
116C22×C227C22×D411yescharacteristic
117C22×C227C2×#4112yes—
118C22×C227C2×#4112yes—
119C22×C227C2×#4111yescharacteristic
120C22×C227C2×#4111yescharacteristic
121C2×D227—41no—
122C2×D227—41no—
123C2×D227—41no—
124C2×D227—42no—
125C2×D227—42no—
126C4×C227—24no—
127C4×C227—41no—
128C4×C227—41no—
129C4×C227—42no—
130Dic227—22no—
131Dic227—22no—
132Dic227—24no—
133Dic227—41no—
134Dic227—42no—
135C2×C227—24no—
136C2×C227C2×#12311yescharacteristic
137C2×C227#52511yescharacteristic
138C2×C227#52511yescharacteristic
139C2×C227#52912yes—
140C2×C227#52912yes—
141D227—81no—
142C227#285311yescharacteristic, complemented, Sylow
143#2853—2271nomaximal, Sylow
144C2×#123—2271no—
145C22×C42—2271no—
146#388—2272no—
147#388—2272no—
148#387—2272no—
149#387—2272no—
150#386—2271no—
151#386—2271no—
152#385—2271no—
153#385D22711yescharacteristic, complemented
154#381—2271no—
155C22×D4—2271no—
156C23×C4—2271no—
157C23×C4—2271no—
158C23×C4C2×D22711yescharacteristic
159#123—4544no—
160C2×C4⋊C4—2271no—
161C2×C4⋊C4—2271no—
162C2×C4⋊C4—2272no—
163C2×C4⋊C4C2×D22712yes—
164C2×C22⋊C4—2271no—
165C2×C22⋊C4—2271no—
166C2×C22⋊C4—2271no—
167C2×C22⋊C4—2272no—
168C2×C22⋊C4—2272no—
169C2×C22⋊C4C2×D22711yescharacteristic
170C2×C22⋊C4C2×D22711yescharacteristic
171C2×C42—4544no—
172C2×C42—2272no—
173C2×C42—2272no—
174#95—2272no—
175#95—2272no—
176#95—2272no—
177#95—2272no—
178#95—2272no—
179#95C2×D22712yes—
180C24—2271no—
181C24C22×D22711yescharacteristic
182C2×D4—4542no—
183C2×D4—4542no—
184C2×D4—4542no—
185C22×C4—22no—
186C22×C4—24no—
187C22×C4—4542no—
188C22×C4—4542no—
189C22×C4—4542no—
190C22×C4—4544no—
191C22×C4—2272no—
192C22×C4—2271no—
193C22×C4—2274no—
194C22×C4—2272no—
195C22×C4—2271no—
196C22×C4—2271no—
197C22×C4C22×D22712yes—
198C22×C4C22×D22711yescharacteristic
199C22×C4C22×D22711yescharacteristic
200C22×C4C22×D22712yes—
201C4⋊C4—24no—
202C4⋊C4—4544no—
203C4⋊C4—4542no—
204C4⋊C4—4542no—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—4544no—
208C22⋊C4—4542no—
209C22⋊C4—4542no—
210C22⋊C4—4542no—
211C22⋊C4—4544no—
212C42—4544no—
213C42—4544no—
214C23—22no—
215C23—22no—
216C23—4541no—
217C23—4542no—
218C23—4542no—
219C23—4541no—
220C23—4541no—
221C23D4×D22712yes—
222C23#1681388412yes—
223C23#1681388512yes—
224C23C23×D22711yescharacteristic
225D4—9084no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—4541no—
237C2×C4—4542no—
238C2×C4—4542no—
239C2×C4—4541no—
240C2×C4—4542no—
241C2×C4—4541no—
242C2×C4—4548no—
243C2×C4—4542no—
244C2×C4—4542no—
245C2×C4—4541no—
246C2×C4—4541no—
247C2×C4—4541no—
248C2×C4—4541no—
249C2×C4—4548no—
250C2×C4—2272no—
251C2×C4—2272no—
252C2×C4—2274no—
253C2×C4#1681388314yes—
254C22—22no—
255C22—22no—
256C22—24no—
257C22—24no—
258C22—9081no—
259C22—9081no—
260C22—9081no—
261C22—9082no—
262C22—9082no—
263C22#2030027314yes—
264C22C2×D4×D22711yescharacteristic
265C22#41×D22712yes—
266C22C2×#1681388312yes—
267C22C2×#1681388411yescharacteristic
268C22C2×#1681388511yescharacteristic
269C4—24no—
270C4—41no—
271C4—41no—
272C4—42no—
273C4—9081no—
274C4—9082no—
275C4—4542no—
276C4—4542no—
277C4—4544no—
278C2—24no—
279C2—18161no—
280C2C2×#2030027311yescharacteristic
281C2#2840953811yescharacteristic
282C2#2840954011yescharacteristic
283C2#2840956212yes—
284C2#2840956812yes—
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#385×C227C2
prime radicalC22×C22⋊D227C22
hypercenter#385D227
hypocenterC227#2853
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic227C2semidirect
3. #28408922C2semidirect
4. #28408921C2semidirect
5. #28408919C2semidirect
6. #28408918C2semidirect
9. #28408904C2semidirect
10. #28408893C2semidirect
11. #385×C227C2semidirect
142. C227#2853semidirect
153. #385D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002722
C22×C4×Dic2272
#284089222
#284089212
#284089192
#284089182
#284089112
#284089092
#284089042
#284088932
#385×C2272
#2853227