Finite Group Databasebeta

← #44408055

Subgroups — #44408055

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×#20300269C211yesmaximal, characteristic
2C22×C4×Dic227C211yesmaximal, characteristic, complemented
3#28408926C212yesmaximal, complemented
4#28408925C212yesmaximal, complemented
5#28408915C212yesmaximal, complemented
6#28408914C212yesmaximal, complemented
7#28408912C211yesmaximal, characteristic
8#28408907C211yesmaximal, characteristic, complemented
9#28408905C211yesmaximal, characteristic, complemented
10#28408893C211yesmaximal, characteristic, complemented
11#386×C227C211yesmaximal, characteristic
12C23×Dic227C2211yescharacteristic
13C23×Dic227C2211yescharacteristic
14#20300269—24no—
15C2×C4⋊Dic227C2212yes—
16C2×Dic227⋊C4C2212yes—
17C2×Dic227⋊C4C2211yescharacteristic
18C2×Dic227⋊C4C2211yescharacteristic
19C2×C22⋊Dic227C2211yescharacteristic
20C2×C22⋊Dic227C2211yescharacteristic
21C2×C22⋊Dic227C2212yes—
22C2×C22⋊Dic227C2211yescharacteristic
23C2×C22⋊Dic227C2211yescharacteristic
24C2×C4×Dic227—24no—
25C2×C4×Dic227C2212yes—
26C2×C4×Dic227C2212yes—
27C227⋊#95C2212yes—
28C227⋊#95C2212yes—
29C227⋊#95C2212yes—
30C227⋊#95C2212yes—
31C227⋊#95C2212yes—
32C22×D4×C227C2211yescharacteristic
33C23×C4×C227C2211yescharacteristic
34C2×C22⋊C4×C227C2211yescharacteristic
35C2×C22⋊C4×C227C2212yes—
36#95×C227C2212yes—
37C22×Dic227—24no—
38C22×Dic227—22no—
39C22×Dic227—22no—
40C22×Dic227—22no—
41C22×Dic227D414yes—
42C22×Dic227C2312yes—
43C22×Dic227C2311yescharacteristic
44C22×Dic227C2311yescharacteristic
45C22×Dic227C2311yescharacteristic
46C22×Dic227C2311yescharacteristic
47C22×Dic227C2312yes—
48C4⋊Dic227—24no—
49Dic227⋊C4—22no—
50Dic227⋊C4—22no—
51Dic227⋊C4—24no—
52C22⋊Dic227—22no—
53C22⋊Dic227—22no—
54C22⋊Dic227—24no—
55C22⋊Dic227—22no—
56C22⋊Dic227—22no—
57C4×Dic227—24no—
58C4×Dic227—24no—
59C24×C227C2311yescharacteristic
60C24×C227C2311yescharacteristic
61C2×D4×C227—22no—
62C2×D4×C227—22no—
63C2×D4×C227—22no—
64C22×C4×C227—22no—
65C22×C4×C227—24no—
66C22×C4×C227C2311yescharacteristic
67C22×C4×C227C2312yes—
68C22×C4×C227C2312yes—
69C22⋊C4×C227—22no—
70C22⋊C4×C227—24no—
71C2×Dic227—28no—
72C2×Dic227—28no—
73C2×Dic227—22no—
74C2×Dic227—21no—
75C2×Dic227—21no—
76C2×Dic227—21no—
77C2×Dic227—21no—
78C2×Dic227—22no—
79C2×Dic227—21no—
80C2×Dic227—21no—
81C2×Dic227—22no—
82C2×Dic227—22no—
83C2×Dic227—22no—
84C2×Dic227—22no—
85C2×Dic227—22no—
86C2×Dic227C2×D412yes—
87C2×Dic227C2×D412yes—
88C2×Dic227#4114yes—
89C23×C227—22no—
90C23×C227—22no—
91C23×C227—21no—
92C23×C227—22no—
93C23×C227—21no—
94C23×C227—21no—
95C23×C227—22no—
96C23×C227C2×D412yes—
97C23×C227#4112yes—
98C23×C227#4112yes—
99C23×C227C2411yescharacteristic
100D4×C227—44no—
101C2×C4×C227—28no—
102C2×C4×C227—22no—
103C2×C4×C227—22no—
104C2×C4×C227—21no—
105C2×C4×C227—22no—
106C2×C4×C227—22no—
107C2×C4×C227—21no—
108C2×C4×C227—21no—
109C2×C4×C227—22no—
110C2×C4×C227#4114yes—
111C22×C227—22no—
112C22×C227—22no—
113C22×C227—24no—
114C22×C227—24no—
115C22×C227—41no—
116C22×C227—41no—
117C22×C227—41no—
118C22×C227—42no—
119C22×C227—42no—
120C22×C227#12314yes—
121C22×C227C22×D411yescharacteristic
122C22×C227C2×#4112yes—
123C22×C227C2×#4111yescharacteristic
124C22×C227C2×#4112yes—
125C22×C227C2×#4111yescharacteristic
126C4×C227—24no—
127C4×C227—41no—
128C4×C227—42no—
129Dic227—22no—
130Dic227—22no—
131Dic227—24no—
132Dic227—41no—
133Dic227—41no—
134Dic227—42no—
135C2×C227—24no—
136C2×C227—81no—
137C2×C227C2×#12311yescharacteristic
138C2×C227#52511yescharacteristic
139C2×C227#52511yescharacteristic
140C2×C227#52912yes—
141C2×C227#52912yes—
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#386D22711yescharacteristic
152#385—2271no—
153#385—2271no—
154#381—2271no—
155C22×D4C2×D22711yescharacteristic
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⋊C4—2272no—
164C2×C22⋊C4—2271no—
165C2×C22⋊C4—2271no—
166C2×C22⋊C4—2271no—
167C2×C22⋊C4—2271no—
168C2×C22⋊C4—2272no—
169C2×C22⋊C4C2×D22711yescharacteristic
170C2×C22⋊C4C2×D22712yes—
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—
180C24C22×D22711yescharacteristic
181C24C22×D22711yescharacteristic
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—22no—
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—2271no—
194C22×C4—2274no—
195C22×C4—2272no—
196C22×C4—2271no—
197C22×C4—2271no—
198C22×C4C22×D22712yes—
199C22×C4C22×D22711yescharacteristic
200C22×C4C22×D22712yes—
201C4⋊C4—4542no—
202C4⋊C4—4542no—
203C4⋊C4—4544no—
204C4⋊C4—4544no—
205C22⋊C4—22no—
206C22⋊C4—24no—
207C22⋊C4—4542no—
208C22⋊C4—4542no—
209C22⋊C4—4544no—
210C22⋊C4—4542no—
211C22⋊C4—4542no—
212C42—4544no—
213C42—4544no—
214C23—22no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—22no—
219C23—21no—
220C23—21no—
221C23D4×D22712yes—
222C23#1681388312yes—
223C23#1681388412yes—
224C23C23×D22711yescharacteristic
225D4—44no—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—4541no—
236C2×C4—4541no—
237C2×C4—4542no—
238C2×C4—4542no—
239C2×C4—4542no—
240C2×C4—4542no—
241C2×C4—4542no—
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—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—42no—
263C22#2030027014yes—
264C22C2×D4×D22711yescharacteristic
265C22#41×D22712yes—
266C22C2×#1681388311yescharacteristic
267C22C2×#1681388312yes—
268C22C2×#1681388411yescharacteristic
269C4—24no—
270C4—41no—
271C4—42no—
272C4—9081no—
273C4—9081no—
274C4—9082no—
275C4—4542no—
276C4—4542no—
277C4—4544no—
278C2—24no—
279C2—81no—
280C2C2×#2030027011yescharacteristic
281C2#2840953811yescharacteristic
282C2#2840954011yescharacteristic
283C2#2840956212yes—
284C2#2840956312yes—
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#386×C227C2
prime radicalC22×D4×C227C22
hypercenter#386D227
hypocenterC227#2853
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic227C2semidirect
3. #28408926C2semidirect
4. #28408925C2semidirect
5. #28408915C2semidirect
6. #28408914C2semidirect
8. #28408907C2semidirect
9. #28408905C2semidirect
10. #28408893C2semidirect
142. C227#2853semidirect

Maximal subgroups

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

subgroupindex
C2×#203002692
C22×C4×Dic2272
#284089262
#284089252
#284089152
#284089142
#284089122
#284089072
#284089052
#284088932
#386×C2272
#2853227