Finite Group Databasebeta

← #44409083

Subgroups — #44409083

Order 29056.

724 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 161 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
1#28409830C212yesmaximal, complemented
2#28409578C211yesmaximal, characteristic, complemented
3#28409215C212yesmaximal, complemented
4#28409210C212yesmaximal, complemented
5#28409205C212yesmaximal, complemented
6#28409200C212yesmaximal, complemented
7C2×#20300215C212yesmaximal, complemented
8#28409016C211yesmaximal, characteristic, complemented
9#585×C227C211yesmaximal, characteristic, complemented
10C2×#16813884C2211yescharacteristic, complemented
11#20300319—22no—
12#20300318C2214yescomplemented
13#20300314—22no—
14#20300313C2214yescomplemented
15#20300308—24no—
16C2×#16813868C2212yescomplemented
17#20300286C2211yescharacteristic, complemented
18#20300284C2211yescharacteristic, complemented
19#20300277C2211yescharacteristic, complemented
20#20300275C2211yescharacteristic, complemented
21#20300260—22no—
22C4×C22⋊D227—22no—
23C2×C4×Dic227C2211yescharacteristic, complemented
24#20300215C2214yescomplemented
25#20300215C2214yescomplemented
26#20300207C2212yescomplemented
27#20300205C2212yescomplemented
282−1+4×C227—22no—
292−1+4×C227C2212yes—
30H(5,𝔽2)×C227—22no—
31H(5,𝔽2)×C227—22no—
32H(5,𝔽2)×C227C2212yes—
33C2×#41×C227—24no—
34C2×#41×C227—22no—
35C2×#41×C227C2212yes—
36C2×#41×C227C2211yescharacteristic
37#16813884—22no—
38#16813884—22no—
39C2×Dic454C2311yescharacteristic
40C2×D908C2311yescharacteristic
41C2×C22⋊D227—21no—
42C2×C22⋊D227—22no—
43C22×Dic227—21no—
44C2×C4×D227—21no—
45#16813874—24no—
46#16813872—24no—
47#16813871—24no—
48#16813869—24no—
49#16813868—22no—
50#16813868C2314yes—
51C2×C227⋊C8—22no—
52Dic227⋊C4—22no—
53C22⋊Dic227—22no—
54(C2×D227)⋊C4—22no—
55C4×Dic227—21no—
56C4×Dic227C2311yescharacteristic
57C4×Dic227C2311yescharacteristic
58#41×C227—24no—
59#41×C227—28no—
60#41×C227—24no—
61#41×C227—24no—
62#41×C227—24no—
63#41×C227—22no—
64#41×C227—22no—
65#41×C227—24no—
66#41×C227—22no—
67#41×C227—22no—
68#41×C227D414yescomplemented
69#41×C227C2314yes—
70C2×Q8×C227—22no—
71C2×Q8×C227—24no—
72C2×Q8×C227D412yescomplemented
73C2×Q8×C227C2311yescharacteristic
74C2×D4×C227—28no—
75C2×D4×C227—22no—
76C2×D4×C227—24no—
77C2×D4×C227—22no—
78C2×D4×C227—24no—
79C2×D4×C227D412yes—
80C2×D4×C227D412yescomplemented
81C2×D4×C227C2311yescharacteristic
82C22×C4×C227—24no—
83C22×C4×C227—22no—
84C22×C4×C227D412yes—
85C22×C4×C227C2311yescharacteristic
86C22×D227—21no—
87Dic454—22no—
88D908—22no—
89C22⋊D227—44no—
90C22⋊D227—42no—
91C2×Dic227—21no—
92C2×Dic227—21no—
93C2×Dic227—21no—
94C2×Dic227—42no—
95C4×D227—41no—
96C4×D227—41no—
97C227⋊C8—24no—
98C23×C227—24no—
99C23×C227—22no—
100C23×C227C2×D412yes—
101C23×C227C2×D411yescharacteristic
102Q8×C227—24no—
103Q8×C227—22no—
104Q8×C227—22no—
105Q8×C227C2×D414yes—
106D4×C227—28no—
107D4×C227—24no—
108D4×C227—24no—
109D4×C227—24no—
110D4×C227—24no—
111D4×C227—22no—
112D4×C227—22no—
113D4×C227C2×D414yes—
114C2×C4×C227—24no—
115C2×C4×C227—28no—
116C2×C4×C227—22no—
117C2×C4×C227—24no—
118C2×C4×C227—22no—
119C2×C4×C227—22no—
120C2×C4×C227—24no—
121C2×C4×C227C2×D412yes—
122C2×C4×C227C2×D411yescharacteristic
123C2×C4×C227C2×D412yes—
124C2×C4×C227C2×D412yes—
125C2×C4×C227C2411yescharacteristic
126C22×C227—22no—
127C22×C227—24no—
128C22×C227—41no—
129C22×C227—42no—
130C22×C227—82no—
131C22×C227—84no—
132C22×C227#12012yes—
133C22×C227C22×D411yescharacteristic
134C2×D227—42no—
135C2×D227—41no—
136C4×C227—24no—
137C4×C227—22no—
138C4×C227#12011yescharacteristic
139C4×C227#12011yescharacteristic
140C4×C227C22×D412yes—
141Dic227—22no—
142Dic227—41no—
143Dic227—41no—
144C2×C227—21no—
145C2×C227—22no—
146C2×C227—42no—
147C2×C227—44no—
148C2×C227C2×#12011yescharacteristic
149D227—81no—
150C227#301811yescharacteristic, complemented, Sylow
151#3018—2271nomaximal, Sylow
152#585D22711yescharacteristic, complemented
153#575—2272no—
154#532—2271no—
155#456—2272no—
156#455—2272no—
157#454—2272no—
158#453—2272no—
159C2×#104—2272no—
160#413—2271no—
1612−1+4—22no—
1622−1+4C2×D22712yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D22712yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—2271no—
169C2×#41C2×D22712yes—
170C2×#41C2×D22711yescharacteristic
171#137—4542no—
172#137—2274no—
173AGL(1,ℤ/8)—4542no—
174AGL(1,ℤ/8)—2274no—
175#135—4544no—
176C2×M16—2272no—
177C4⋊Q8—2271no—
178#127—2271no—
179#124—2271no—
180#124—2271no—
181#121—4542no—
182C4×D4—4542no—
183C2×C42—2271no—
184#104—2274no—
185#104—2274no—
186#101—2272no—
187#100—2272no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—4542no—
199#41—4542no—
200#41C22⋊D22714yescomplemented
201#41C22×D22714yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—2271no—
205C2×Q8C22⋊D22712yescomplemented
206C2×Q8C22×D22711yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—4542no—
213C2×D4—4541no—
214C2×D4—2271no—
215C2×D4C22⋊D22712yes—
216C2×D4C22⋊D22712yescomplemented
217C2×D4C22×D22711yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—4541no—
221C22×C4—4541no—
222C22×C4C22⋊D22712yes—
223C22×C4C22×D22711yescharacteristic
224Q16—4544no—
225SD16—4544no—
226SD16—4544no—
227D8—4544no—
228M16—4542no—
229M16—2274no—
230C2×C8—4542no—
231C4⋊C4—4542no—
232C22⋊C4—4542no—
233C22⋊C4—4542no—
234C42—4541no—
235C42—2271no—
236C42—2271no—
237C23—24no—
238C23—22no—
239C23—4541no—
240C23C2×C22⋊D22712yes—
241C23C2×C22⋊D22711yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—4542no—
246Q8C2×C22⋊D22714yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—9082no—
255D4—9084no—
256D4—4542no—
257D4C2×C22⋊D22714yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—9082no—
266C2×C4—9081no—
267C2×C4—9081no—
268C2×C4—4541no—
269C2×C4—4541no—
270C2×C4—4541no—
271C2×C4C2×C22⋊D22712yes—
272C2×C4C2×C22⋊D22712yes—
273C2×C4C2×C22⋊D22712yes—
274C2×C4C2×C22⋊D22711yescharacteristic
275C2×C4C23×D22711yescharacteristic
276C8—4544no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—9082no—
284C22—9081no—
285C22#2030025712yes—
286C22C22×C22⋊D22711yescharacteristic
287C4—24no—
288C4—22no—
289C4—9081no—
290C4—9081no—
291C4—4542no—
292C4#2030025711yescharacteristic
293C4#2030025711yescharacteristic
294C4C22×C22⋊D22712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—18161no—
300C2C2×#2030025711yescharacteristic
subgroupG/H
centerC4#20300257
commutator, cosocleC2×C4×C227C24
FrattiniC2×C4C23×D227
Fitting#585×C227C2
socleC2×C227C2×#120
hypercenter#585D227
hypocenterC227#3018
LCSG ▷ C2×C4×C227 ▷ C2×C227 ▷ C227
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #28409830C2semidirect
2. #28409578C2semidirect
3. #28409215C2semidirect
4. #28409210C2semidirect
5. #28409205C2semidirect
6. #28409200C2semidirect
7. C2×#20300215C2semidirect
8. #28409016C2semidirect
9. #585×C227C2semidirect
10. C2×#16813884C22semidirect
12. #20300318C22semidirect
14. #20300313C22semidirect
16. C2×#16813868C22semidirect
17. #20300286C22semidirect
18. #20300284C22semidirect
19. #20300277C22semidirect
20. #20300275C22semidirect
23. C2×C4×Dic227C22semidirect
24. #20300215C22semidirect
25. #20300215C22semidirect
26. #20300207C22semidirect
27. #20300205C22semidirect
68. #41×C227D4semidirect
72. C2×Q8×C227D4semidirect
80. C2×D4×C227D4semidirect
150. C227#3018semidirect
152. #585D227semidirect
200. #41C22⋊D227semidirect
205. C2×Q8C22⋊D227semidirect
216. C2×D4C22⋊D227semidirect

Maximal subgroups

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

subgroupindex
#284098302
#284095782
#284092152
#284092102
#284092052
#284092002
C2×#203002152
#284090162
#585×C2272
#3018227