Finite Group Databasebeta

← #44409018

Subgroups — #44409018

Order 29056.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20300335C211yesmaximal, characteristic, complemented
2C2×#20300314C211yesmaximal, characteristic, complemented
3C22×#16813872C211yesmaximal, characteristic, complemented
4#28409175C214yesmaximal, complemented
5#28409170C214yesmaximal, complemented
6#28409040C211yesmaximal, characteristic, complemented
7#28409026C211yesmaximal, characteristic, complemented
8#28408999C211yesmaximal, characteristic, complemented
9#535×C227C211yesmaximal, characteristic, complemented
10#20300335—24no—
11#20300335—24no—
12C2×#16813884C2211yescharacteristic
13C2×#16813884C2212yescomplemented
14C2×#16813883—22no—
15C2×#16813883—21no—
16C22×Dic454C2211yescharacteristic
17C22×Dic454C2212yescomplemented
18C2×Q8×D227—21no—
19#20300314—24no—
20C2×#16813872—24no—
21C2×#16813872C2212yes—
22C2×#16813872C2214yes—
23C2×#16813869C2212yes—
24C2×#16813868C2211yescharacteristic
25C22×C227⋊C8C2211yescharacteristic
26#20300211C2212yescomplemented
27#20300211C2214yescomplemented
28#20300208C2212yescomplemented
29C22⋊(C227⋊C8)C2214yescomplemented
30C2×#41×C227C2211yescharacteristic
31C22×D4×C227C2211yescharacteristic
32#127×C227—21no—
33#124×C227—21no—
34#121×C227C2214yes—
35#120×C227—22no—
36#117×C227C2211yescharacteristic
37#16813884—24no—
38#16813884—24no—
39#16813884—22no—
40#16813884D414yescomplemented
41#16813883—216no—
42#16813883—28no—
43C2×Dic454—24no—
44C2×Dic454—24no—
45C2×Dic454—21no—
46C2×Dic454—22no—
47C2×Dic454—22no—
48C2×Dic454D414yescomplemented
49C2×Dic454C2312yes—
50C2×Dic454C2311yescharacteristic
51Q8×D227—28no—
52C2×D908C2311yescharacteristic
53C2×C22⋊D227—21no—
54C2×C22⋊D227—22no—
55C22×Dic227—21no—
56C22×Dic227—22no—
57C2×C4×D227—21no—
58C2×C4×D227—22no—
59#16813872—28no—
60#16813872—24no—
61#16813869—24no—
62#16813868—22no—
63C2×C227⋊C8—22no—
64C2×C227⋊C8C2312yes—
65C2×C227⋊C8C2312yes—
66C24×C227—21no—
67#41×C227—22no—
68#41×C227—22no—
69C2×Q8×C227D411yescharacteristic
70C2×D4×C227—24no—
71C2×D4×C227—22no—
72C2×D4×C227—22no—
73C2×D4×C227—21no—
74C2×D4×C227D412yes—
75C2×D4×C227D411yescharacteristic
76C2×D4×C227C2312yes—
77C22×C4×C227C2312yes—
78C22×C4×C227C2311yescharacteristic
79C4⋊C4×C227C2312yes—
80C22⋊C4×C227—21no—
81C22⋊C4×C227—24no—
82C42×C227—21no—
83C22×D227—21no—
84Dic454—28no—
85Dic454—24no—
86Dic454—21no—
87Dic454—22no—
88Dic454C2×D412yes—
89Dic454C2×D414yes—
90D908—21no—
91D908C2×D412yes—
92C22⋊D227—28no—
93C22⋊D227—24no—
94C2×Dic227—28no—
95C2×Dic227—24no—
96C2×Dic227—21no—
97C2×Dic227—22no—
98C4×D227—28no—
99C4×D227—24no—
100C227⋊C8—22no—
101C227⋊C8—22no—
102C23×C227—22no—
103C23×C227—42no—
104C23×C227—82no—
105C23×C227C2×D412yes—
106C23×C227C2×D411yescharacteristic
107Q8×C227—22no—
108D4×C227—24no—
109D4×C227—22no—
110D4×C227—22no—
111D4×C227—22no—
112D4×C227—42no—
113D4×C227—84no—
114C2×C4×C227—22no—
115C2×C4×C227—22no—
116C2×C4×C227—22no—
117C2×C4×C227C2×D412yes—
118C2×C4×C227C2×D411yescharacteristic
119C2×C4×C227C2×D412yes—
120C2×C4×C227C2×D412yes—
121C2×C4×C227C2411yescharacteristic
122C22×C227—22no—
123C22×C227—22no—
124C22×C227—24no—
125C22×C227—42no—
126C22×C227—44no—
127C22×C227—84no—
128C22×C227#12012yes—
129C22×C227C22×D411yescharacteristic
130C2×D227—22no—
131C2×D227—42no—
132C4×C227—22no—
133C4×C227—42no—
134C4×C227#12012yes—
135C4×C227C22×D411yescharacteristic
136C4×C227C22×D411yescharacteristic
137Dic227—24no—
138Dic227—22no—
139C2×C227—22no—
140C2×C227—42no—
141C2×C227—82no—
142C2×C227C2×#12011yescharacteristic
143C2×C227#57712yes—
144D227—42no—
145C227#301011yescharacteristic, complemented, Sylow
146#3010—2271nomaximal, Sylow
147C2×2−1+4—2271no—
148C2×AGL(1,ℤ/8)—2271no—
149C22×SD16—2271no—
150#535D22711yescharacteristic, complemented
151#449—2274no—
152#448—2274no—
153#419—2271no—
154#416—2271no—
155#408—2271no—
1562−1+4—4544no—
1572−1+4—4544no—
158C2×#41—4542no—
159C2×#41—4541no—
160C2×#41—2272no—
161C2×#41—2271no—
162C2×#41C2×D22711yescharacteristic
163C22×Q8—4541no—
164C22×Q8—2272no—
165C22×Q8—2271no—
166C22×D4C2×D22711yescharacteristic
167AGL(1,ℤ/8)—4544no—
168C2×SD16—4544no—
169C2×SD16—2274no—
170C2×SD16—2272no—
171C2×D8—2272no—
172C2×M16—2271no—
173C22×C8—2271no—
174#127—21no—
175#124—21no—
176#121C2×D22714yes—
177#120—22no—
178#117C2×D22711yescharacteristic
179#103—2272no—
180#103—2274no—
181#102—2272no—
182C22⋊C8—2274no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—4544no—
187#41—4544no—
188#41—45416no—
189#41—4548no—
190#41—4542no—
191#41—2274no—
192C2×Q8—4544no—
193C2×Q8—4544no—
194C2×Q8—4548no—
195C2×Q8—4542no—
196C2×Q8—4542no—
197C2×Q8—4541no—
198C2×Q8—2274no—
199C2×Q8—2271no—
200C2×Q8—2272no—
201C2×Q8C22⋊D22711yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—4542no—
207C2×D4—4541no—
208C2×D4—2271no—
209C2×D4C22⋊D22712yes—
210C2×D4C22⋊D22711yescharacteristic
211C2×D4C22×D22712yes—
212C22×C4—4542no—
213C22×C4—4541no—
214C22×C4—4542no—
215C22×C4—4541no—
216C22×C4C22×D22712yes—
217C22×C4C22×D22711yescharacteristic
218SD16—4548no—
219SD16—4544no—
220D8—4544no—
221M16—4542no—
222C2×C8—4542no—
223C2×C8—2272no—
224C2×C8—2272no—
225C4⋊C4C22×D22712yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—4541no—
233C23C2×C22⋊D22712yes—
234C23C2×C22⋊D22711yescharacteristic
235Q8—22no—
236Q8—4548no—
237Q8—4541no—
238Q8—4542no—
239Q8—4544no—
240Q8—2272no—
241Q8—2274no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—4548no—
249D4—4544no—
250D4—4541no—
251D4—2272no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—4548no—
256C2×C4—4548no—
257C2×C4—4542no—
258C2×C4—4541no—
259C2×C4—4544no—
260C2×C4—4544no—
261C2×C4D4×D22712yes—
262C2×C4D4×D22712yes—
263C2×C4C2×C22⋊D22712yes—
264C2×C4C2×C22⋊D22711yescharacteristic
265C2×C4C23×D22711yescharacteristic
266C8—4542no—
267C8—4542no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—42no—
272C22—44no—
273C22—84no—
274C22—9082no—
275C22—4542no—
276C22#2030025612yes—
277C22C22×C22⋊D22711yescharacteristic
278C4—22no—
279C4—42no—
280C4—4544no—
281C4—4542no—
282C4#2030025612yes—
283C4C2×D4×D22711yescharacteristic
284C4C2×D4×D22711yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—9082no—
289C2C2×#2030025611yescharacteristic
290C2#2840984312yes—
subgroupG/H
centerC22C22×C22⋊D227
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#535×C227C2
socleC22×C227C22×D4
hypercenter#535D227
hypocenterC227#3010
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#20300335C2semidirect
2. C2×#20300314C2semidirect
3. C22×#16813872C2semidirect
4. #28409175C2semidirect
5. #28409170C2semidirect
6. #28409040C2semidirect
7. #28409026C2semidirect
8. #28408999C2semidirect
9. #535×C227C2semidirect
13. C2×#16813884C22semidirect
17. C22×Dic454C22semidirect
26. #20300211C22semidirect
27. #20300211C22semidirect
28. #20300208C22semidirect
29. C22⋊(C227⋊C8)C22semidirect
40. #16813884D4semidirect
48. C2×Dic454D4semidirect
145. C227#3010semidirect
150. #535D227semidirect

Maximal subgroups

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

subgroupindex
C2×#203003352
C2×#203003142
C22×#168138722
#284091752
#284091702
#284090402
#284090262
#284089992
#535×C2272
#3010227