Finite Group Databasebeta

← #27185913

Subgroups — #27185913

Order 12928.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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#20266013C211yesmaximal, characteristic, complemented
2C2×#15692301C211yesmaximal, characteristic, complemented
3#20265989C218yesmaximal, complemented
4#20265982C218yesmaximal, complemented
5#20265977C216yesmaximal, complemented
6C2×#15692283C216yesmaximal, complemented
7#568×C101C211yesmaximal, characteristic, complemented
8#15692303—21no—
9#15692302C2214yescomplemented
10#15692301—24no—
11#15692301C2218yescomplemented
12#15692301C2214yescomplemented
13#15692300—23no—
14C2×#13108075—21no—
15C2×#13108074C2213yescomplemented
16C2×#13108074C2216yescomplemented
17#41×D101—24no—
18C22×D404C2213yescomplemented
19C2×D4×D101—23no—
20#15692288C2214yescomplemented
21#15692288C22112yescomplemented
22#15692283C22112yescomplemented
23#15692283C22124yescomplemented
24#15692283C22112yescomplemented
25#15692277C22124yescomplemented
26C2×#13108063C22112yescomplemented
27C2×D808C22112yescomplemented
28C2×#41×C101C2211yescharacteristic
29#131×C101C2218yescomplemented
30C2×M16×C101C2216yescomplemented
31C23×D101—23no—
32#13108075—24no—
33#13108075—28no—
34#13108074—26no—
35#13108074—26no—
36#13108074—26no—
37#13108074C2314yes—
38#13108074C23112yes—
39#13108074C23112yescomplemented
40#13108073—212no—
41C2×Dic202C2311yescharacteristic
42C2×Dic202C2316yescomplemented
43Q8×D101—24no—
44C2×D404—23no—
45C2×D404—26no—
46C2×D404C2313yes—
47C2×D404C23112yes—
48C2×D404C2316yescomplemented
49C2×C22⋊D101—23no—
50C2×C22⋊D101—26no—
51D4×D101—224no—
52D4×D101—212no—
53C2×C4×D101—23no—
54C2×C4×D101—26no—
55Dic404C2318yescomplemented
56#13108063C23124yescomplemented
57#13108063C2318yescomplemented
58D808C23124yescomplemented
59#41×C101D414yes—
60#41×C101C2314yes—
61C2×Q8×C101D411yescharacteristic
62C2×D4×C101D413yes—
63C22×C4×C101C2313yes—
64M16×C101C2314yes—
65M16×C101C23112yescomplemented
66C2×C8×C101C23112yescomplemented
67C23×C101C2×D413yes—
68C22×D101—23no—
69C22×D101—26no—
70C22×D101—212no—
71C22×D101—86no—
72Q8×C101C2×D414yes—
73D4×C101C2×D4112yes—
74C2×C4×C101C2×D413yes—
75C2×C4×C101C2×D416yes—
76C2×C4×C101C2416yes—
77C2×C4×C101C2411yescharacteristic
78C22⋊D101—212no—
79C22⋊D101—212no—
80C22⋊D101—24no—
81C2×Dic101—21no—
82C2×Dic101—26no—
83D404—212no—
84D404—26no—
85D404—23no—
86D404C2414yes—
87D404C2416yes—
88C4×D101—24no—
89C4×D101—212no—
90C4×D101—212no—
91Dic202—26no—
92Dic202—21no—
93Dic202C2414yes—
94Dic202C2412yes—
95C8×C101C2418yes—
96C22×C101—43no—
97C22×C101C22×D411yescharacteristic
98C22×C101C22×D416yes—
99C2×D101—24no—
100C2×D101—26no—
101C2×D101—46no—
102C2×D101—86no—
103C2×D101—812no—
104C4×C101C22×D411yescharacteristic
105C4×C101C22×D416yes—
106C4×C101C2511yescharacteristic
107Dic101—24no—
108Dic101—22no—
109C2×C101—21no—
110C2×C101—26no—
111C2×C101C23×D411yescharacteristic
112D101—46no—
113D101—44no—
114#3585—1011nomaximal, Sylow
115C101#358511yescharacteristic, complemented, Sylow
116#585—1011no—
117C2×H(5,𝔽2)—1011no—
118#577—1018no—
119#576—1018no—
120#575—1016no—
121C2×AGL(1,ℤ/8)—1016no—
122#568D10111yescharacteristic, complemented
1232−1+4—1014no—
1242−1+4—2021no—
125H(5,𝔽2)—1014no—
126H(5,𝔽2)—1018no—
127H(5,𝔽2)—2024no—
128H(5,𝔽2)—2023no—
129C2×#41—1016no—
130C2×#41—1013no—
131C2×#41—2024no—
132C2×#41—2021no—
133C2×#41C2×D10111yescharacteristic
134C22×D4—1013no—
135C22×D4—2023no—
136#137—1014no—
137#137—10112no—
138AGL(1,ℤ/8)—10112no—
139AGL(1,ℤ/8)—10124no—
140AGL(1,ℤ/8)—10112no—
141#135—10124no—
142C2×SD16—10112no—
143C2×D8—10112no—
144#131C2×D10118yescomplemented
145C2×M16C2×D10116yescomplemented
146C24—2023no—
147#41—10112no—
148#41—1014no—
149#41—10112no—
150#41—2028no—
151#41—2024no—
152#41—20212no—
153#41—2026no—
154#41—2026no—
155#41—2026no—
156#41D40414yes—
157#41C22×D10114yes—
158C2×Q8—1011no—
159C2×Q8—1016no—
160C2×Q8—2024no—
161C2×Q8D40411yescharacteristic
162C2×D4—10112no—
163C2×D4—1013no—
164C2×D4—1016no—
165C2×D4—2023no—
166C2×D4—2026no—
167C2×D4—20224no—
168C2×D4—20212no—
169C2×D4—2023no—
170C2×D4—2026no—
171C2×D4D40413yes—
172C22×C4—2023no—
173C22×C4—2026no—
174C22×C4C22×D10113yes—
175Q16—1018no—
176SD16—10124no—
177SD16—1018no—
178D8—10124no—
179M16C22×D10114yes—
180M16C22×D101112yescomplemented
181C2×C8C22×D101112yescomplemented
182C23—8086no—
183C23—2026no—
184C23—20212no—
185C23—2023no—
186C23C2×D40413yes—
187Q8—1014no—
188Q8—1012no—
189Q8—2026no—
190Q8—2021no—
191Q8C2×D40414yes—
192D4—1014no—
193D4—1016no—
194D4—20212no—
195D4—2024no—
196D4—20212no—
197D4—2023no—
198D4—20212no—
199D4—2026no—
200D4C2×D404112yes—
201C2×C4—2021no—
202C2×C4—2026no—
203C2×C4—2024no—
204C2×C4—20212no—
205C2×C4—20212no—
206C2×C4C2×D40413yes—
207C2×C4C2×D40416yes—
208C2×C4C23×D10116yes—
209C2×C4C23×D10111yescharacteristic
210C8C23×D10118yes—
211C22—43no—
212C22—80812no—
213C22—8086no—
214C22—4046no—
215C22—2024no—
216C22—2026no—
217C22C22×D40411yescharacteristic
218C22C22×D40416yes—
219C4—2022no—
220C4—2024no—
221C4C22×D40411yescharacteristic
222C4C22×D40416yes—
223C4C24×D10111yescharacteristic
224C2—21no—
225C2—26no—
226C2—4046no—
227C2—4044no—
228C2C23×D40411yescharacteristic
subgroupG/H
centerC2C23×D404
commutator, cosocleC4×C101C25
FrattiniC4C24×D101
Fitting#568×C101C2
socleC2×C101C23×D4
hypercenter#568D101
hypocenterC101#3585
LCSG ▷ C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #20266013C2semidirect
2. C2×#15692301C2semidirect
3. #20265989C2semidirect
4. #20265982C2semidirect
5. #20265977C2semidirect
6. C2×#15692283C2semidirect
7. #568×C101C2semidirect
9. #15692302C22semidirect
11. #15692301C22semidirect
12. #15692301C22semidirect
15. C2×#13108074C22semidirect
16. C2×#13108074C22semidirect
18. C22×D404C22semidirect
20. #15692288C22semidirect
21. #15692288C22semidirect
22. #15692283C22semidirect
23. #15692283C22semidirect
24. #15692283C22semidirect
25. #15692277C22semidirect
26. C2×#13108063C22semidirect
27. C2×D808C22semidirect
29. #131×C101C22semidirect
30. C2×M16×C101C22semidirect
39. #13108074C23semidirect
42. C2×Dic202C23semidirect
48. C2×D404C23semidirect
55. Dic404C23semidirect
56. #13108063C23semidirect
57. #13108063C23semidirect
58. D808C23semidirect
65. M16×C101C23semidirect
66. C2×C8×C101C23semidirect
115. C101#3585semidirect
122. #568D101semidirect
144. #131C2×D101semidirect
145. C2×M16C2×D101semidirect
180. M16C22×D101semidirect
181. C2×C8C22×D101semidirect

Maximal subgroups

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

subgroupindex
#202660132
C2×#156923012
#202659892
#202659822
#202659772
C2×#156922832
#568×C1012
#3585101