Finite Group Databasebeta

← #44408036

Subgroups — #44408036

Order 29056.

696 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 169 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×#20300276C211yesmaximal, characteristic, complemented
2C2×#20300268C211yesmaximal, characteristic, complemented
3C2×#20300263C211yesmaximal, characteristic
4C2×C42×D227C211yesmaximal, characteristic, complemented
5#28408937C211yesmaximal, characteristic, complemented
6#28408933C211yesmaximal, characteristic, complemented
7#28408924C214yesmaximal, complemented
8#28408910C214yesmaximal, complemented
9#382×C227C211yesmaximal, characteristic, complemented
10C22×Dic454C2211yescharacteristic
11C22×D908C2212yes—
12C22×D908C2211yescharacteristic
13C22×C4×D227C2212yes—
14C22×C4×D227C2211yescharacteristic
15#20300276—24no—
16#20300268—24no—
17#20300263—24no—
18C2×C4⋊Dic227C2212yes—
19C2×(C2×D227)⋊C4C2212yes—
20C2×(C2×D227)⋊C4C2214yes—
21C2×(C2×D227)⋊C4C2214yes—
22C2×(C2×D227)⋊C4C2212yes—
23C2×C4×Dic227C2212yes—
24C2×C4×Dic227C2211yescharacteristic
25C42×D227—24no—
26C227⋊#95C2214yes—
27C2×C4⋊C4×C227C2211yescharacteristic
28C2×C4⋊C4×C227C2211yescharacteristic
29C2×C42×C227C2211yescharacteristic
30#95×C227C2214yes—
31C23×D227C2312yes—
32C23×D227C2311yescharacteristic
33C2×Dic454—22no—
34C2×Dic454—22no—
35C2×Dic454—22no—
36C2×D908—24no—
37C2×D908—28no—
38C2×D908—22no—
39C2×D908—22no—
40C2×D908—22no—
41C22×Dic227C2312yes—
42C22×Dic227C2311yescharacteristic
43C22×Dic227C2312yes—
44C2×C4×D227—24no—
45C2×C4×D227—24no—
46C2×C4×D227—24no—
47C2×C4×D227—22no—
48C2×C4×D227—22no—
49C2×C4×D227D414yes—
50C4⋊Dic227—24no—
51(C2×D227)⋊C4—24no—
52(C2×D227)⋊C4—24no—
53(C2×D227)⋊C4—28no—
54(C2×D227)⋊C4—28no—
55C4×Dic227—24no—
56C4×Dic227—22no—
57C22×C4×C227C2312yes—
58C22×C4×C227C2312yes—
59C22×C4×C227C2312yes—
60C22×C4×C227C2311yescharacteristic
61C4⋊C4×C227—22no—
62C4⋊C4×C227—22no—
63C42×C227D414yescomplemented
64C22×D227—21no—
65C22×D227—21no—
66C22×D227—22no—
67C22×D227—22no—
68C22×D227—22no—
69C22×D227—24no—
70C22×D227—22no—
71C22×D227—24no—
72C22×D227C2×D412yes—
73C22×D227#4114yes—
74Dic454—44no—
75D908—48no—
76D908—44no—
77C2×Dic227—21no—
78C2×Dic227—22no—
79C2×Dic227—21no—
80C2×Dic227—22no—
81C2×Dic227—22no—
82C2×Dic227—22no—
83C2×Dic227—22no—
84C2×Dic227C2×D412yes—
85C2×Dic227#4114yes—
86C4×D227—216no—
87C4×D227—28no—
88C23×C227C2411yescharacteristic
89C2×C4×C227—22no—
90C2×C4×C227—22no—
91C2×C4×C227—22no—
92C2×C4×C227—22no—
93C2×C4×C227—24no—
94C2×C4×C227—24no—
95C2×C4×C227C2×D414yes—
96C2×C4×C227C2×D412yes—
97C2×C4×C227C2×D412yes—
98C2×C4×C227#4112yes—
99C22×C227C22×D411yescharacteristic
100C22×C227C22×D411yescharacteristic
101C22×C227C2×#4112yes—
102C22×C227C2×#4112yes—
103C22×C227C2×#4111yescharacteristic
104C2×D227—22no—
105C2×D227—24no—
106C2×D227—24no—
107C2×D227—22no—
108C2×D227—42no—
109C2×D227—44no—
110C2×D227—42no—
111C2×D227—44no—
112C2×D227—42no—
113C2×D227#12414yescomplemented
114C4×C227—24no—
115C4×C227—42no—
116C4×C227—42no—
117C4×C227#12114yes—
118Dic227—22no—
119Dic227—24no—
120Dic227—42no—
121C2×C227C2×#12111yescharacteristic
122C2×C227#52512yes—
123C2×C227C2×#12411yescharacteristic
124C2×C227#52912yes—
125C2×C227#53211yescharacteristic
126D227—24no—
127D227—82no—
128C227#285011yescharacteristic, complemented, Sylow
129#2850—2271nomaximal, Sylow
130C2×#124—2271no—
131C2×C22⋊Q8—2271no—
132C2×#121—2271no—
133C22×C42—2271no—
134#391—2271no—
135#390—2271no—
136#388—2274no—
137#386—2274no—
138#382D22711yescharacteristic, complemented
139C22×Q8—2271no—
140C22×D4—2271no—
141C22×D4—2272no—
142C23×C4—2272no—
143C23×C4—2271no—
144#124—4544no—
145C22⋊Q8—4544no—
146#121—4544no—
147C2×C4⋊C4—2272no—
148C2×C4⋊C4C2×D22711yescharacteristic
149C2×C4⋊C4C2×D22711yescharacteristic
150C2×C22⋊C4—2272no—
151C2×C22⋊C4—2272no—
152C2×C22⋊C4—2274no—
153C2×C22⋊C4—2274no—
154C2×C42—4544no—
155C2×C42—2272no—
156C2×C42—2271no—
157C2×C42C2×D22711yescharacteristic
158#95—2274no—
159#95C2×D22714yes—
160C24—2272no—
161C24—2271no—
162C2×Q8—4542no—
163C2×Q8—4542no—
164C2×Q8—4542no—
165C2×D4—4544no—
166C2×D4—4548no—
167C2×D4—4542no—
168C2×D4—4542no—
169C2×D4—4542no—
170C22×C4—4544no—
171C22×C4—4544no—
172C22×C4—4544no—
173C22×C4—4542no—
174C22×C4—4542no—
175C22×C4—2272no—
176C22×C4—2272no—
177C22×C4—2271no—
178C22×C4—2274no—
179C22×C4C22×D22712yes—
180C22×C4C22×D22712yes—
181C22×C4C22×D22712yes—
182C22×C4C22×D22711yescharacteristic
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—4544no—
186C22⋊C4—4544no—
187C22⋊C4—4548no—
188C22⋊C4—4544no—
189C22⋊C4—4548no—
190C42—4544no—
191C42—4542no—
192C42D90814yescomplemented
193C23—4542no—
194C23—4542no—
195C23—4544no—
196C23—4544no—
197C23—4542no—
198C23—4541no—
199C23—4542no—
200C23—4541no—
201C23—2272no—
202C23—2274no—
203C23C23×D22711yescharacteristic
204Q8—9084no—
205D4—9088no—
206D4—9084no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—24no—
213C2×C4—45416no—
214C2×C4—4542no—
215C2×C4—4542no—
216C2×C4—4542no—
217C2×C4—4541no—
218C2×C4—4542no—
219C2×C4—4542no—
220C2×C4—4541no—
221C2×C4—4548no—
222C2×C4—2272no—
223C2×C4—2274no—
224C2×C4D4×D22712yes—
225C2×C4C2×D90814yes—
226C2×C4C2×D90812yes—
227C2×C4#1681388512yes—
228C22—9082no—
229C22—9084no—
230C22—9082no—
231C22—9084no—
232C22—9082no—
233C22—4542no—
234C22—4544no—
235C22—4544no—
236C22—4542no—
237C22—2274no—
238C22C2×D4×D22711yescharacteristic
239C22C22×D90811yescharacteristic
240C22#41×D22712yes—
241C22#41×D22712yes—
242C22C2×#1681388511yescharacteristic
243C4—24no—
244C4—42no—
245C4—42no—
246C4—9082no—
247C4—4542no—
248C4—4544no—
249C4#2030026314yes—
250C2—18162no—
251C2—4544no—
252C2C2×#2030026311yescharacteristic
253C2#2840954412yes—
254C2#124×D22711yescharacteristic
255C2#2840956712yes—
256C2#2840958111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
Fitting#382×C227C2
prime radicalC2×#20300263C2
hypercenter#382D227
hypocenterC227#2850
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
1. C2×#20300276C2semidirect
2. C2×#20300268C2semidirect
4. C2×C42×D227C2semidirect
5. #28408937C2semidirect
6. #28408933C2semidirect
7. #28408924C2semidirect
8. #28408910C2semidirect
9. #382×C227C2semidirect
63. C42×C227D4semidirect
113. C2×D227#124semidirect
128. C227#2850semidirect
138. #382D227semidirect
192. C42D908semidirect

Maximal subgroups

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

subgroupindex
C2×#203002762
C2×#203002682
C2×#203002632
C2×C42×D2272
#284089372
#284089332
#284089242
#284089102
#382×C2272
#2850227