Finite Group Databasebeta

← #44408033

Subgroups — #44408033

Order 29056.

696 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 177 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×#20300275C211yesmaximal, characteristic, complemented
2C2×#20300267C211yesmaximal, characteristic, complemented
3C22×C4×Dic227C211yesmaximal, characteristic, complemented
4#28408937C211yesmaximal, characteristic, complemented
5#28408935C211yesmaximal, characteristic, complemented
6#28408926C214yesmaximal, complemented
7#28408912C214yesmaximal, complemented
8#28408897C211yesmaximal, characteristic, complemented
9C2×#121×C227C211yesmaximal, characteristic
10C22×Dic454C2211yescharacteristic
11C23×Dic227C2212yes—
12#20300275—24no—
13#20300267—24no—
14C2×C4⋊Dic227C2211yescharacteristic
15C2×Dic227⋊C4C2212yes—
16C2×C22⋊Dic227C2212yes—
17C2×C22⋊Dic227C2214yes—
18C2×C22⋊Dic227C2214yes—
19C2×C4×Dic227—24no—
20C2×C4×Dic227C2212yes—
21C2×C4×Dic227C2211yescharacteristic
22C2×C4×Dic227C2211yescharacteristic
23C227⋊#95C2214yes—
24C227⋊#95C2214yes—
25C22×D4×C227C2212yes—
26C22×D4×C227C2211yescharacteristic
27C23×C4×C227C2211yescharacteristic
28#121×C227—24no—
29C2×C4⋊C4×C227C2211yescharacteristic
30C2×C22⋊C4×C227C2212yes—
31C2×Dic454—22no—
32C2×Dic454—22no—
33C2×Dic454—22no—
34C22×Dic227—24no—
35C22×Dic227—24no—
36C22×Dic227—24no—
37C22×Dic227C2312yes—
38C22×Dic227C2312yes—
39C22×Dic227C2312yes—
40C22×Dic227C2312yes—
41C4⋊Dic227—22no—
42Dic227⋊C4—24no—
43C22⋊Dic227—28no—
44C22⋊Dic227—28no—
45C22⋊Dic227—24no—
46C4×Dic227—24no—
47C4×Dic227—22no—
48C4×Dic227D414yescomplemented
49C24×C227C2312yes—
50C24×C227C2311yescharacteristic
51C2×D4×C227—24no—
52C2×D4×C227—28no—
53C2×D4×C227—22no—
54C2×D4×C227—22no—
55C2×D4×C227—22no—
56C22×C4×C227—22no—
57C22×C4×C227—22no—
58C22×C4×C227D414yes—
59C22×C4×C227C2312yes—
60C22×C4×C227C2311yescharacteristic
61C22×C4×C227C2311yescharacteristic
62C4⋊C4×C227—22no—
63C22⋊C4×C227—24no—
64Dic454—44no—
65C2×Dic227—216no—
66C2×Dic227—22no—
67C2×Dic227—22no—
68C2×Dic227—22no—
69C2×Dic227—22no—
70C2×Dic227—22no—
71C2×Dic227—22no—
72C2×Dic227—22no—
73C2×Dic227—24no—
74C2×Dic227—22no—
75C2×Dic227C2×D414yes—
76C2×Dic227#4114yes—
77C23×C227—21no—
78C23×C227—21no—
79C23×C227—22no—
80C23×C227—22no—
81C23×C227—22no—
82C23×C227—22no—
83C23×C227—24no—
84C23×C227—24no—
85C23×C227C2×D412yes—
86C23×C227#4114yes—
87C23×C227C2411yescharacteristic
88D4×C227—48no—
89D4×C227—44no—
90C2×C4×C227—21no—
91C2×C4×C227—21no—
92C2×C4×C227—28no—
93C2×C4×C227—22no—
94C2×C4×C227—24no—
95C2×C4×C227C2×D412yes—
96C2×C4×C227C2×D412yes—
97C2×C4×C227C2×D412yes—
98C2×C4×C227#4112yes—
99C22×C227—22no—
100C22×C227—24no—
101C22×C227—24no—
102C22×C227—22no—
103C22×C227—42no—
104C22×C227—44no—
105C22×C227—42no—
106C22×C227—44no—
107C22×C227—42no—
108C22×C227#12414yescomplemented
109C22×C227C22×D411yescharacteristic
110C22×C227C22×D411yescharacteristic
111C22×C227C2×#4112yes—
112C22×C227C2×#4111yescharacteristic
113C22×C227C2×#4112yes—
114C4×C227—22no—
115C4×C227—42no—
116C4×C227#12114yes—
117Dic227—24no—
118Dic227—24no—
119Dic227—42no—
120Dic227—42no—
121C2×C227—24no—
122C2×C227—82no—
123C2×C227C2×#12111yescharacteristic
124C2×C227#52512yes—
125C2×C227C2×#12411yescharacteristic
126C2×C227#52912yes—
127C2×C227#53211yescharacteristic
128C227#285011yescharacteristic, complemented, Sylow
129#2850—2271nomaximal, Sylow
130C2×#124—2271no—
131C2×C22⋊Q8—2271no—
132C2×#121D22711yescharacteristic
133C22×C42—2271no—
134#391—2271no—
135#390—2271no—
136#388—2274no—
137#386—2274no—
138#382—2271no—
139C22×Q8—2271no—
140C22×D4C2×D22711yescharacteristic
141C22×D4C2×D22712yes—
142C23×C4—2272no—
143C23×C4C2×D22711yescharacteristic
144#124—4544no—
145C22⋊Q8—4544no—
146#121—24no—
147C2×C4⋊C4—2271no—
148C2×C4⋊C4—2272no—
149C2×C4⋊C4C2×D22711yescharacteristic
150C2×C22⋊C4—2272no—
151C2×C22⋊C4—2274no—
152C2×C22⋊C4—2274no—
153C2×C22⋊C4C2×D22712yes—
154C2×C42—4544no—
155C2×C42—2272no—
156C2×C42—2271no—
157C2×C42—2271no—
158#95—2274no—
159#95—2274no—
160C24C22×D22712yes—
161C24C22×D22711yescharacteristic
162C2×Q8—4542no—
163C2×Q8—4542no—
164C2×Q8—4542no—
165C2×D4—24no—
166C2×D4—28no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—4544no—
173C22×C4—4544no—
174C22×C4—4544no—
175C22×C4—2272no—
176C22×C4—2272no—
177C22×C4—2272no—
178C22×C4—2272no—
179C22×C4C22⋊D22714yes—
180C22×C4C22×D22712yes—
181C22×C4C22×D22711yescharacteristic
182C22×C4C22×D22711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—4544no—
185C4⋊C4—4542no—
186C22⋊C4—24no—
187C22⋊C4—4544no—
188C22⋊C4—4548no—
189C22⋊C4—4548no—
190C42—4544no—
191C42—4542no—
192C42—2274no—
193C23—21no—
194C23—22no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23—24no—
199C23—22no—
200C23—24no—
201C23C2×C22⋊D22712yes—
202C23#1681388314yes—
203C23C23×D22711yescharacteristic
204Q8—9084no—
205D4—48no—
206D4—44no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—28no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—4542no—
213C2×C4—4542no—
214C2×C4—4542no—
215C2×C4—4544no—
216C2×C4—4542no—
217C2×C4—45416no—
218C2×C4—4542no—
219C2×C4—4542no—
220C2×C4—4542no—
221C2×C4—4542no—
222C2×C4—2274no—
223C2×C4—2274no—
224C2×C4D4×D22712yes—
225C2×C4C2×C22⋊D22712yes—
226C2×C4C2×C22⋊D22712yes—
227C2×C4#1681388312yes—
228C22—22no—
229C22—24no—
230C22—24no—
231C22—22no—
232C22—42no—
233C22—44no—
234C22—42no—
235C22—44no—
236C22—42no—
237C22#2030027514yescomplemented
238C22C2×D4×D22711yescharacteristic
239C22C22×C22⋊D22711yescharacteristic
240C22#41×D22712yes—
241C22C2×#1681388311yescharacteristic
242C22C2×#1681388312yes—
243C4—22no—
244C4—42no—
245C4—9082no—
246C4—9082no—
247C4—4544no—
248C4—4544no—
249C4#2030026014yes—
250C2—24no—
251C2—82no—
252C2C2×#2030026011yescharacteristic
253C2#2840954212yes—
254C2C2×#2030027511yescharacteristic
255C2#2840956312yes—
256C2#2840957811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting, prime radicalC2×#121×C227C2
hypercenterC2×#121D227
hypocenterC227#2850
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#20300275C2semidirect
2. C2×#20300267C2semidirect
3. C22×C4×Dic227C2semidirect
4. #28408937C2semidirect
5. #28408935C2semidirect
6. #28408926C2semidirect
7. #28408912C2semidirect
8. #28408897C2semidirect
48. C4×Dic227D4semidirect
108. C22×C227#124semidirect
128. C227#2850semidirect
237. C22#20300275semidirect

Maximal subgroups

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

subgroupindex
C2×#203002752
C2×#203002672
C22×C4×Dic2272
#284089372
#284089352
#284089262
#284089122
#284088972
C2×#121×C2272
#2850227