Finite Group Databasebeta

← #44408022

Subgroups — #44408022

Order 29056.

1068 conjugacy classes of subgroups; only the 201 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 201 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
1C23×C4×D227C211yesmaximal, characteristic
2C2×#20300268C212yesmaximal, complemented
3C2×#20300267C211yesmaximal, characteristic, complemented
4#28408907C212yesmaximal, complemented
5#28408906C214yesmaximal, complemented
6#28408889C212yesmaximal, complemented
7#28408887C212yesmaximal
8#380×C227C211yesmaximal, characteristic, complemented
9C24×D227C2211yescharacteristic
10C22×Dic454C2211yescharacteristic, complemented
11C23×Dic227C2211yescharacteristic
12C22×C4×D227—28no—
13C22×C4×D227—24no—
14C22×C4×D227C2212yes—
15C22×C4×D227C2212yes—
16#20300268—28no—
17#20300267—24no—
18C2×C4⋊Dic227C2211yescharacteristic, complemented
19C2×C4⋊Dic227C2214yescomplemented
20C2×Dic227⋊C4C2212yescomplemented
21C2×C22⋊Dic227C2212yes—
22C2×(C2×D227)⋊C4C2214yes—
23C2×(C2×D227)⋊C4C2214yes—
24C227⋊#95C2214yescomplemented
25C23×C4×C227C2211yescharacteristic
26C2×C4⋊C4×C227C2212yescomplemented
27C2×C22⋊C4×C227C2212yes—
28#95×C227C2212yescomplemented
29C23×D227—24no—
30C23×D227—24no—
31C23×D227—22no—
32C23×D227—22no—
33C23×D227Q814yescomplemented
34C23×D227C2312yes—
35C2×Dic454—22no—
36C2×Dic454—24no—
37C22×Dic227—24no—
38C22×Dic227—22no—
39C22×Dic227C2312yes—
40C22×Dic227C2312yes—
41C22×Dic227C2312yes—
42C2×C4×D227—232no—
43C2×C4×D227—24no—
44C2×C4×D227—24no—
45C2×C4×D227—24no—
46C2×C4×D227—24no—
47C2×C4×D227—24no—
48C2×C4×D227D418yes—
49C4⋊Dic227—28no—
50C4⋊Dic227—22no—
51Dic227⋊C4—24no—
52C22⋊Dic227—24no—
53(C2×D227)⋊C4—28no—
54(C2×D227)⋊C4—28no—
55C24×C227C2311yescharacteristic
56C22×C4×C227—24no—
57C22×C4×C227D414yes—
58C22×C4×C227C2314yes—
59C22×C4×C227C2311yescharacteristic
60C22×C4×C227C2311yescharacteristic
61C4⋊C4×C227—24no—
62C22⋊C4×C227—24no—
63C22×D227—28no—
64C22×D227—216no—
65C22×D227—28no—
66C22×D227—28no—
67C22×D227—22no—
68C22×D227—22no—
69C22×D227—22no—
70C22×D227—22no—
71C22×D227—216no—
72C22×D227C2×D414yes—
73C22×D227C2×Q814yes—
74C22×D227#4114yes—
75Dic454—44no—
76C2×Dic227—28no—
77C2×Dic227—22no—
78C2×Dic227—22no—
79C2×Dic227—24no—
80C2×Dic227—22no—
81C2×Dic227—22no—
82C2×Dic227—24no—
83C2×Dic227C2×D414yes—
84C4×D227—216no—
85C4×D227—216no—
86C23×C227—22no—
87C23×C227—22no—
88C23×C227C2×D412yes—
89C23×C227C2×Q812yes—
90C23×C227#4112yes—
91C23×C227C2411yescharacteristic
92C2×C4×C227—22no—
93C2×C4×C227—28no—
94C2×C4×C227—24no—
95C2×C4×C227—24no—
96C2×C4×C227—24no—
97C2×C4×C227C2×D412yes—
98C2×C4×C227C2×D414yes—
99C2×C4×C227C2×D412yes—
100C22×C227—22no—
101C22×C227—24no—
102C22×C227—22no—
103C22×C227—24no—
104C22×C227C22⋊Q814yescomplemented
105C22×C227C22×D412yes—
106C22×C227C22×D411yescharacteristic
107C22×C227C22×Q811yescharacteristic
108C22×C227C2×#4112yes—
109C22×C227C2×#4111yescharacteristic
110C2×D227—232no—
111C2×D227—24no—
112C2×D227—24no—
113C2×D227—24no—
114C2×D227—24no—
115C2×D227—24no—
116C2×D227—24no—
117C2×D227C22⋊Q818yescomplemented
118C4×C227—22no—
119C4×C227—44no—
120C4×C227#12014yes—
121Dic227—24no—
122Dic227—42no—
123Dic227—42no—
124C2×C227—24no—
125C2×C227C2×#12011yescharacteristic
126C2×C227C2×C22⋊Q812yes—
127C2×C227C2×C22⋊Q811yescharacteristic
128C2×C227#52512yes—
129C2×C227#52511yescharacteristic
130D227—28no—
131C227#284811yescharacteristic, complemented, Sylow
132#2848—2271nomaximal, Sylow
133C24×C4—2271no—
134C2×C22⋊Q8—2271no—
135C2×C22⋊Q8—2272no—
136#385—2272no—
137#385—2274no—
138#380—2272no—
139#380D22711yescharacteristic, complemented
140#379—2272no—
141C25—2271no—
142C22×Q8—2271no—
143C23×C4—4548no—
144C23×C4—4544no—
145C23×C4—2271no—
146C23×C4—2272no—
147C23×C4—2272no—
148C23×C4C2×D22711yescharacteristic
149C22⋊Q8—4548no—
150C22⋊Q8—4544no—
151C2×C4⋊C4—2271no—
152C2×C4⋊C4—2272no—
153C2×C4⋊C4—2274no—
154C2×C4⋊C4C2×D22712yescomplemented
155C2×C22⋊C4—2272no—
156C2×C22⋊C4—2274no—
157C2×C22⋊C4—2274no—
158C2×C22⋊C4C2×D22712yes—
159#95—2274no—
160#95C2×D22712yescomplemented
161C24—4544no—
162C24—4544no—
163C24—4542no—
164C24—4542no—
165C24—2274no—
166C24—2272no—
167C24C22×D22711yescharacteristic
168C2×Q8—4544no—
169C2×Q8—4542no—
170C22×C4—24no—
171C22×C4—45432no—
172C22×C4—4544no—
173C22×C4—4544no—
174C22×C4—4542no—
175C22×C4—4544no—
176C22×C4—4544no—
177C22×C4—4544no—
178C22×C4—4544no—
179C22×C4—2272no—
180C22×C4—2272no—
181C22×C4—2272no—
182C22×C4—2278no—
183C22×C4D90814yes—
184C22×C4C22×D22714yes—
185C22×C4C22×D22711yescharacteristic
186C22×C4C22×D22711yescharacteristic
187C4⋊C4—24no—
188C4⋊C4—4548no—
189C4⋊C4—4544no—
190C4⋊C4—4542no—
191C22⋊C4—24no—
192C22⋊C4—4548no—
193C22⋊C4—4548no—
194C22⋊C4—4544no—
195C23—22no—
196C23—22no—
197C23—4548no—
198C23—45416no—
199C23—4548no—
200C23—45416no—
201C23—4542no—
202C23—4542no—
203C23—4542no—
204C23—4542no—
205C23—4548no—
206C23—2274no—
207C23—2274no—
208C23—2274no—
209C23C2×D90812yes—
210C23Q8×D22712yes—
211C23#1681388312yes—
212C23C23×D22711yescharacteristic
213Q8—9084no—
214C2×C4—22no—
215C2×C4—28no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—45416no—
220C2×C4—4544no—
221C2×C4—4542no—
222C2×C4—4542no—
223C2×C4—4544no—
224C2×C4—4548no—
225C2×C4—4542no—
226C2×C4—4542no—
227C2×C4—45416no—
228C2×C4—2274no—
229C2×C4D4×D22714yes—
230C2×C4C2×D90812yes—
231C2×C4C2×D90812yes—
232C22—22no—
233C22—24no—
234C22—24no—
235C22—22no—
236C22—45432no—
237C22—4544no—
238C22—4544no—
239C22—4544no—
240C22—4544no—
241C22—4544no—
242C22—4544no—
243C22—2278no—
244C22#2030026814yescomplemented
245C22C2×D4×D22712yes—
246C22C22×D90811yescharacteristic
247C22C2×Q8×D22711yescharacteristic
248C22#41×D22712yes—
249C22C2×#1681388311yescharacteristic
250C4—22no—
251C4—44no—
252C4—9082no—
253C4—9082no—
254C4—4544no—
255C4#2030025814yes—
256C2—24no—
257C2—4548no—
258C2C2×#2030025811yescharacteristic
259C2C22⋊Q8×D22712yes—
260C2C2×#2030026811yescharacteristic
261C2#2840954212yes—
262C2#2840954411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#380×C227C2
prime radicalC24×D227C22
hypercenter#380D227
hypocenterC227#2848
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#20300268C2semidirect
3. C2×#20300267C2semidirect
4. #28408907C2semidirect
5. #28408906C2semidirect
6. #28408889C2semidirect
8. #380×C227C2semidirect
10. C22×Dic454C22semidirect
18. C2×C4⋊Dic227C22semidirect
19. C2×C4⋊Dic227C22semidirect
20. C2×Dic227⋊C4C22semidirect
24. C227⋊#95C22semidirect
26. C2×C4⋊C4×C227C22semidirect
28. #95×C227C22semidirect
33. C23×D227Q8semidirect
104. C22×C227C22⋊Q8semidirect
117. C2×D227C22⋊Q8semidirect
131. C227#2848semidirect
139. #380D227semidirect
154. C2×C4⋊C4C2×D227semidirect
160. #95C2×D227semidirect
244. C22#20300268semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D2272
C2×#203002682
C2×#203002672
#284089072
#284089062
#284088892
#284088872
#380×C2272
#2848227