Finite Group Databasebeta

← #44408030

Subgroups — #44408030

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×#20300264C211yesmaximal, characteristic, complemented
2C2×#20300259C211yesmaximal, characteristic
3C2×C42×D227C211yesmaximal, characteristic, complemented
4#28408939C211yesmaximal, characteristic, complemented
5#28408935C211yesmaximal, characteristic, complemented
6#28408920C214yesmaximal, complemented
7#28408908C214yesmaximal, complemented
8#28408898C211yesmaximal, characteristic, complemented
9C2×#124×C227C211yesmaximal, characteristic, complemented
10C22×C22⋊D227C2212yes—
11C22×C4×D227C2212yes—
12C22×C4×D227C2211yescharacteristic
13#20300264—24no—
14#20300259—24no—
15C2×C4⋊Dic227C2211yescharacteristic
16C2×C4⋊Dic227C2211yescharacteristic
17C2×Dic227⋊C4C2212yes—
18C2×C22⋊Dic227C2214yes—
19C2×C22⋊Dic227C2212yes—
20C2×(C2×D227)⋊C4C2212yes—
21C2×C4×Dic227C2212yes—
22C2×C4×Dic227C2211yescharacteristic
23C42×D227—24no—
24C227⋊#95C2214yes—
25C227⋊#95C2214yes—
26C22×Q8×C227C2211yescharacteristic
27C22×D4×C227C2211yescharacteristic
28#124×C227—24no—
29C2×C22⋊C4×C227C2214yes—
30C2×C42×C227C2211yescharacteristic
31C23×D227C2311yescharacteristic
32C2×C22⋊D227—24no—
33C2×C22⋊D227—28no—
34C22×Dic227C2312yes—
35C22×Dic227C2312yes—
36C22×Dic227C2311yescharacteristic
37C22×Dic227C2312yes—
38C2×C4×D227—24no—
39C2×C4×D227—24no—
40C2×C4×D227—24no—
41C2×C4×D227—22no—
42C2×C4×D227—22no—
43C2×C4×D227D414yes—
44C4⋊Dic227—22no—
45C4⋊Dic227—22no—
46Dic227⋊C4—24no—
47C22⋊Dic227—24no—
48C22⋊Dic227—28no—
49(C2×D227)⋊C4—24no—
50C4×Dic227—24no—
51C4×Dic227—22no—
52C24×C227C2312yes—
53C2×Q8×C227—22no—
54C2×Q8×C227—22no—
55C2×Q8×C227—22no—
56C2×D4×C227—22no—
57C2×D4×C227—22no—
58C2×D4×C227—22no—
59C22×C4×C227C2312yes—
60C22×C4×C227C2312yes—
61C22×C4×C227C2311yescharacteristic
62C22⋊C4×C227—28no—
63C42×C227D414yescomplemented
64C22×D227—21no—
65C22×D227—21no—
66C22×D227—22no—
67C22×D227C2×D412yes—
68C22×D227#4114yes—
69C22⋊D227—48no—
70C2×Dic227—21no—
71C2×Dic227—22no—
72C2×Dic227—21no—
73C2×Dic227—22no—
74C2×Dic227—22no—
75C2×Dic227—22no—
76C2×Dic227—24no—
77C2×Dic227—24no—
78C2×Dic227C2×D412yes—
79C2×Dic227#4114yes—
80C4×D227—216no—
81C4×D227—28no—
82C23×C227—22no—
83C23×C227—22no—
84C23×C227—22no—
85C23×C227—24no—
86C23×C227—24no—
87C23×C227C2411yescharacteristic
88Q8×C227—44no—
89D4×C227—44no—
90C2×C4×C227—22no—
91C2×C4×C227—22no—
92C2×C4×C227—22no—
93C2×C4×C227—22no—
94C2×C4×C227—22no—
95C2×C4×C227C2×D414yes—
96C2×C4×C227C2×D412yes—
97C2×C4×C227C2×D412yes—
98C2×C4×C227#4112yes—
99C22×C227—42no—
100C22×C227—44no—
101C22×C227—42no—
102C22×C227—44no—
103C22×C227—42no—
104C22×C227C22×D411yescharacteristic
105C22×C227C22×D411yescharacteristic
106C22×C227C2×#4112yes—
107C22×C227C2×#4112yes—
108C22×C227C2×#4111yescharacteristic
109C2×D227—22no—
110C2×D227—24no—
111C2×D227—24no—
112C2×D227—22no—
113C2×D227#12414yescomplemented
114C4×C227—24no—
115C4×C227—42no—
116C4×C227#12114yes—
117Dic227—22no—
118Dic227—24no—
119Dic227—42no—
120Dic227—42no—
121C2×C227—82no—
122C2×C227C2×#12111yescharacteristic
123C2×C227#52512yes—
124C2×C227C2×#12411yescharacteristic
125C2×C227#52912yes—
126C2×C227#53211yescharacteristic
127D227—24no—
128C227#285011yescharacteristic, complemented, Sylow
129#2850—2271nomaximal, Sylow
130C2×#124D22711yescharacteristic, complemented
131C2×C22⋊Q8—2271no—
132C2×#121—2271no—
133C22×C42—2271no—
134#391—2271no—
135#390—2271no—
136#388—2274no—
137#386—2274no—
138#382—2271no—
139C22×Q8C2×D22711yescharacteristic
140C22×D4—2272no—
141C22×D4C2×D22711yescharacteristic
142C23×C4—2272no—
143C23×C4—2271no—
144#124—24no—
145C22⋊Q8—4544no—
146#121—4544no—
147C2×C4⋊C4—2271no—
148C2×C4⋊C4—2271no—
149C2×C4⋊C4—2272no—
150C2×C22⋊C4—2272no—
151C2×C22⋊C4—2274no—
152C2×C22⋊C4—2272no—
153C2×C22⋊C4C2×D22714yes—
154C2×C42—4544no—
155C2×C42—2272no—
156C2×C42—2271no—
157C2×C42C2×D22711yescharacteristic
158#95—2274no—
159#95—2274no—
160C24—2271no—
161C24C22×D22712yes—
162C2×Q8—22no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—4544no—
169C2×D4—4548no—
170C22×C4—4544no—
171C22×C4—4544no—
172C22×C4—4544no—
173C22×C4—4542no—
174C22×C4—4542no—
175C22×C4—2272no—
176C22×C4—2272no—
177C22×C4—2272no—
178C22×C4—2271no—
179C22×C4—2274no—
180C22×C4C22×D22712yes—
181C22×C4C22×D22712yes—
182C22×C4C22×D22711yescharacteristic
183C4⋊C4—4544no—
184C4⋊C4—4542no—
185C4⋊C4—4542no—
186C22⋊C4—28no—
187C22⋊C4—4544no—
188C22⋊C4—4548no—
189C22⋊C4—4544no—
190C42—4544no—
191C42—4542no—
192C42C22⋊D22714yescomplemented
193C23—22no—
194C23—22no—
195C23—24no—
196C23—24no—
197C23—22no—
198C23—4541no—
199C23—4542no—
200C23—4541no—
201C23—2272no—
202C23—2274no—
203C23C23×D22711yescharacteristic
204Q8—44no—
205D4—44no—
206D4—9088no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—45416no—
213C2×C4—4542no—
214C2×C4—4542no—
215C2×C4—4544no—
216C2×C4—4544no—
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×C22⋊D22714yes—
226C2×C4C2×C22⋊D22712yes—
227C2×C4#1681388312yes—
228C22—42no—
229C22—44no—
230C22—42no—
231C22—44no—
232C22—42no—
233C22—4542no—
234C22—4544no—
235C22—4544no—
236C22—4542no—
237C22—2274no—
238C22C2×D4×D22711yescharacteristic
239C22C22×C22⋊D22711yescharacteristic
240C22#41×D22712yes—
241C22#41×D22712yes—
242C22C2×#1681388311yescharacteristic
243C4—24no—
244C4—42no—
245C4—9082no—
246C4—9082no—
247C4—4542no—
248C4—4544no—
249C4#2030025914yes—
250C2—82no—
251C2—4544no—
252C2C2×#2030025911yescharacteristic
253C2#2840953912yes—
254C2#124×D22711yescharacteristic
255C2#2840956112yes—
256C2#2840958111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D227
commutator, socle, cosocleC23×C227C24
FittingC2×#124×C227C2
prime radicalC2×#20300259C2
hypercenterC2×#124D227
hypocenterC227#2850
LCSG ▷ C23×C227 ▷ C227
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#20300264C2semidirect
3. C2×C42×D227C2semidirect
4. #28408939C2semidirect
5. #28408935C2semidirect
6. #28408920C2semidirect
7. #28408908C2semidirect
8. #28408898C2semidirect
9. C2×#124×C227C2semidirect
63. C42×C227D4semidirect
113. C2×D227#124semidirect
128. C227#2850semidirect
130. C2×#124D227semidirect
192. C42C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203002642
C2×#203002592
C2×C42×D2272
#284089392
#284089352
#284089202
#284089082
#284088982
C2×#124×C2272
#2850227