Finite Group Databasebeta

← #27180481

Subgroups — #27180481

Order 12928.

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#20265975C212yesmaximal, complemented
2#20265723C211yesmaximal, characteristic, complemented
3#20265360C212yesmaximal, complemented
4#20265355C212yesmaximal, complemented
5#20265350C212yesmaximal, complemented
6#20265345C212yesmaximal, complemented
7C2×#15692182C212yesmaximal, complemented
8#20265161C211yesmaximal, characteristic, complemented
9#585×C101C211yesmaximal, characteristic, complemented
10C2×#13108074C2211yescharacteristic, complemented
11#15692286—22no—
12#15692285C2214yescomplemented
13#15692281—22no—
14#15692280C2214yescomplemented
15#15692275—24no—
16C2×#13108058C2212yescomplemented
17#15692253C2211yescharacteristic, complemented
18#15692251C2211yescharacteristic, complemented
19#15692244C2211yescharacteristic, complemented
20#15692242C2211yescharacteristic, complemented
21#15692227—22no—
22C4×C22⋊D101—22no—
23C2×C4×Dic101C2211yescharacteristic, complemented
24#15692182C2214yescomplemented
25#15692182C2214yescomplemented
26#15692174C2212yescomplemented
27#15692172C2212yescomplemented
282−1+4×C101—22no—
292−1+4×C101C2212yes—
30H(5,𝔽2)×C101—22no—
31H(5,𝔽2)×C101—22no—
32H(5,𝔽2)×C101C2212yes—
33C2×#41×C101—24no—
34C2×#41×C101—22no—
35C2×#41×C101C2212yes—
36C2×#41×C101C2211yescharacteristic
37#13108074—22no—
38#13108074—22no—
39C2×Dic202C2311yescharacteristic
40C2×D404C2311yescharacteristic
41C2×C22⋊D101—21no—
42C2×C22⋊D101—22no—
43C22×Dic101—21no—
44C2×C4×D101—21no—
45#13108064—24no—
46#13108062—24no—
47#13108061—24no—
48#13108059—24no—
49#13108058—22no—
50#13108058C2314yes—
51C2×#11717579—22no—
52#13108053—22no—
53C22⋊Dic101—22no—
54#13108051—22no—
55C4×Dic101—21no—
56C4×Dic101C2311yescharacteristic
57C4×Dic101C2311yescharacteristic
58#41×C101—24no—
59#41×C101—28no—
60#41×C101—24no—
61#41×C101—24no—
62#41×C101—24no—
63#41×C101—22no—
64#41×C101—22no—
65#41×C101—24no—
66#41×C101—22no—
67#41×C101—22no—
68#41×C101D414yescomplemented
69#41×C101C2314yes—
70C2×Q8×C101—22no—
71C2×Q8×C101—24no—
72C2×Q8×C101D412yescomplemented
73C2×Q8×C101C2311yescharacteristic
74C2×D4×C101—28no—
75C2×D4×C101—22no—
76C2×D4×C101—24no—
77C2×D4×C101—22no—
78C2×D4×C101—24no—
79C2×D4×C101D412yes—
80C2×D4×C101D412yescomplemented
81C2×D4×C101C2311yescharacteristic
82C22×C4×C101—24no—
83C22×C4×C101—22no—
84C22×C4×C101D412yes—
85C22×C4×C101C2311yescharacteristic
86C23×C101—24no—
87C23×C101—22no—
88C23×C101C2×D412yes—
89C23×C101C2×D411yescharacteristic
90C22×D101—21no—
91Q8×C101—24no—
92Q8×C101—22no—
93Q8×C101—22no—
94Q8×C101C2×D414yes—
95D4×C101—28no—
96D4×C101—24no—
97D4×C101—24no—
98D4×C101—24no—
99D4×C101—24no—
100D4×C101—22no—
101D4×C101—22no—
102D4×C101C2×D414yes—
103C2×C4×C101—24no—
104C2×C4×C101—28no—
105C2×C4×C101—22no—
106C2×C4×C101—24no—
107C2×C4×C101—22no—
108C2×C4×C101—22no—
109C2×C4×C101—24no—
110C2×C4×C101C2×D412yes—
111C2×C4×C101C2×D411yescharacteristic
112C2×C4×C101C2×D412yes—
113C2×C4×C101C2×D412yes—
114C2×C4×C101C2411yescharacteristic
115C22⋊D101—44no—
116C22⋊D101—42no—
117C2×Dic101—21no—
118C2×Dic101—21no—
119C2×Dic101—21no—
120C2×Dic101—42no—
121D404—22no—
122C4×D101—41no—
123C4×D101—41no—
124Dic202—22no—
125#11717579—24no—
126C22×C101—22no—
127C22×C101—24no—
128C22×C101—41no—
129C22×C101—42no—
130C22×C101—82no—
131C22×C101—84no—
132C22×C101#12012yes—
133C22×C101C22×D411yescharacteristic
134C2×D101—42no—
135C2×D101—41no—
136C4×C101—24no—
137C4×C101—22no—
138C4×C101#12011yescharacteristic
139C4×C101#12011yescharacteristic
140C4×C101C22×D412yes—
141Dic101—22no—
142Dic101—41no—
143Dic101—41no—
144C2×C101—21no—
145C2×C101—22no—
146C2×C101—42no—
147C2×C101—44no—
148C2×C101C2×#12011yescharacteristic
149D101—81no—
150#3018—1011nomaximal, Sylow
151C101#301811yescharacteristic, complemented, Sylow
152#585D10111yescharacteristic, complemented
153#575—1012no—
154#532—1011no—
155#456—1012no—
156#455—1012no—
157#454—1012no—
158#453—1012no—
159C2×#104—1012no—
160#413—1011no—
1612−1+4—22no—
1622−1+4C2×D10112yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D10112yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—1011no—
169C2×#41C2×D10112yes—
170C2×#41C2×D10111yescharacteristic
171#137—1014no—
172#137—2022no—
173AGL(1,ℤ/8)—1014no—
174AGL(1,ℤ/8)—2022no—
175#135—2024no—
176C2×M16—1012no—
177C4⋊Q8—1011no—
178#127—1011no—
179#124—1011no—
180#124—1011no—
181#121—2022no—
182C4×D4—2022no—
183C2×C42—1011no—
184#104—1014no—
185#104—1014no—
186#101—1012no—
187#100—1012no—
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—2022no—
199#41—2022no—
200#41C22⋊D10114yescomplemented
201#41C22×D10114yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—1011no—
205C2×Q8C22⋊D10112yescomplemented
206C2×Q8C22×D10111yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—1011no—
213C2×D4—2022no—
214C2×D4—2021no—
215C2×D4C22⋊D10112yes—
216C2×D4C22⋊D10112yescomplemented
217C2×D4C22×D10111yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—2021no—
221C22×C4—2021no—
222C22×C4C22⋊D10112yes—
223C22×C4C22×D10111yescharacteristic
224Q16—2024no—
225SD16—2024no—
226SD16—2024no—
227D8—2024no—
228M16—1014no—
229M16—2022no—
230C2×C8—2022no—
231C4⋊C4—2022no—
232C22⋊C4—2022no—
233C22⋊C4—2022no—
234C42—1011no—
235C42—1011no—
236C42—2021no—
237C23—24no—
238C23—22no—
239C23—2021no—
240C23C2×C22⋊D10112yes—
241C23C2×C22⋊D10111yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—2022no—
246Q8C2×C22⋊D10114yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—4042no—
255D4—4044no—
256D4—2022no—
257D4C2×C22⋊D10114yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—22no—
265C2×C4—4042no—
266C2×C4—4041no—
267C2×C4—4041no—
268C2×C4—2021no—
269C2×C4—2021no—
270C2×C4—2021no—
271C2×C4C2×C22⋊D10112yes—
272C2×C4C2×C22⋊D10112yes—
273C2×C4C2×C22⋊D10112yes—
274C2×C4C2×C22⋊D10111yescharacteristic
275C2×C4C23×D10111yescharacteristic
276C8—2024no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—4042no—
284C22—4041no—
285C22#1569222412yes—
286C22C22×C22⋊D10111yescharacteristic
287C4—24no—
288C4—22no—
289C4—4041no—
290C4—4041no—
291C4—2022no—
292C4#1569222411yescharacteristic
293C4#1569222411yescharacteristic
294C4C22×C22⋊D10112yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—8081no—
300C2C2×#1569222411yescharacteristic
subgroupG/H
centerC4#15692224
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#585×C101C2
socleC2×C101C2×#120
hypercenter#585D101
hypocenterC101#3018
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20265975C2semidirect
2. #20265723C2semidirect
3. #20265360C2semidirect
4. #20265355C2semidirect
5. #20265350C2semidirect
6. #20265345C2semidirect
7. C2×#15692182C2semidirect
8. #20265161C2semidirect
9. #585×C101C2semidirect
10. C2×#13108074C22semidirect
12. #15692285C22semidirect
14. #15692280C22semidirect
16. C2×#13108058C22semidirect
17. #15692253C22semidirect
18. #15692251C22semidirect
19. #15692244C22semidirect
20. #15692242C22semidirect
23. C2×C4×Dic101C22semidirect
24. #15692182C22semidirect
25. #15692182C22semidirect
26. #15692174C22semidirect
27. #15692172C22semidirect
68. #41×C101D4semidirect
72. C2×Q8×C101D4semidirect
80. C2×D4×C101D4semidirect
151. C101#3018semidirect
152. #585D101semidirect
200. #41C22⋊D101semidirect
205. C2×Q8C22⋊D101semidirect
216. C2×D4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202659752
#202657232
#202653602
#202653552
#202653502
#202653452
C2×#156921822
#202651612
#585×C1012
#3018101