Finite Group Databasebeta

← #30653511

Subgroups — #30653511

Order 16256.

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#22550712C211yesmaximal, characteristic, complemented
2C2×#16861255C211yesmaximal, characteristic, complemented
3#22550688C218yesmaximal, complemented
4#22550681C218yesmaximal, complemented
5#22550676C216yesmaximal, complemented
6C2×#16861237C216yesmaximal, complemented
7#568×C127C211yesmaximal, characteristic, complemented
8#16861257—21no—
9#16861256C2214yescomplemented
10#16861255—24no—
11#16861255C2218yescomplemented
12#16861255C2214yescomplemented
13#16861254—23no—
14C2×#14523249—21no—
15C2×#14523248C2213yescomplemented
16C2×#14523248C2216yescomplemented
17#41×D127—24no—
18C22×D508C2213yescomplemented
19C2×D4×D127—23no—
20#16861242C2214yescomplemented
21#16861242C22112yescomplemented
22#16861237C22112yescomplemented
23#16861237C22124yescomplemented
24#16861237C22112yescomplemented
25#16861231C22124yescomplemented
26C2×#14523237C22112yescomplemented
27C2×D1016C22112yescomplemented
28C2×#41×C127C2211yescharacteristic
29#131×C127C2218yescomplemented
30C2×M16×C127C2216yescomplemented
31C23×D127—23no—
32#14523249—24no—
33#14523249—28no—
34#14523248—26no—
35#14523248—26no—
36#14523248—26no—
37#14523248C23112yes—
38#14523248C2314yes—
39#14523248C23112yescomplemented
40#14523247—212no—
41C2×Dic254C2311yescharacteristic
42C2×Dic254C2316yescomplemented
43Q8×D127—24no—
44C2×D508—23no—
45C2×D508—26no—
46C2×D508C23112yes—
47C2×D508C2313yes—
48C2×D508C2316yescomplemented
49C2×C22⋊D127—23no—
50C2×C22⋊D127—26no—
51D4×D127—224no—
52D4×D127—212no—
53C2×C4×D127—23no—
54C2×C4×D127—26no—
55Dic508C2318yescomplemented
56#14523237C23124yescomplemented
57#14523237C2318yescomplemented
58D1016C23124yescomplemented
59#41×C127D414yes—
60#41×C127C2314yes—
61C2×Q8×C127D411yescharacteristic
62C2×D4×C127D413yes—
63C22×C4×C127C2313yes—
64M16×C127C2314yes—
65M16×C127C23112yescomplemented
66C2×C8×C127C23112yescomplemented
67C22×D127—23no—
68C22×D127—26no—
69C22×D127—212no—
70C22×D127—86no—
71Dic254—26no—
72Dic254—21no—
73Dic254C2414yes—
74Dic254C2412yes—
75D508—212no—
76D508—26no—
77D508—23no—
78D508C2414yes—
79D508C2416yes—
80C22⋊D127—212no—
81C22⋊D127—212no—
82C22⋊D127—24no—
83C2×Dic127—21no—
84C2×Dic127—26no—
85C4×D127—24no—
86C4×D127—212no—
87C4×D127—212no—
88C23×C127C2×D413yes—
89Q8×C127C2×D414yes—
90D4×C127C2×D4112yes—
91C2×C4×C127C2×D413yes—
92C2×C4×C127C2×D416yes—
93C2×C4×C127C2416yes—
94C2×C4×C127C2411yescharacteristic
95C8×C127C2418yes—
96C22×C127—43no—
97C22×C127C22×D411yescharacteristic
98C22×C127C22×D416yes—
99C2×D127—24no—
100C2×D127—26no—
101C2×D127—46no—
102C2×D127—86no—
103C2×D127—812no—
104C4×C127C22×D411yescharacteristic
105C4×C127C22×D416yes—
106C4×C127C2511yescharacteristic
107Dic127—22no—
108Dic127—24no—
109C2×C127—21no—
110C2×C127—26no—
111C2×C127C23×D411yescharacteristic
112D127—46no—
113D127—44no—
114#3585—1271nomaximal, Sylow
115C127#358511yescharacteristic, complemented, Sylow
116#585—1271no—
117C2×H(5,𝔽2)—1271no—
118#577—1278no—
119#576—1278no—
120#575—1276no—
121C2×AGL(1,ℤ/8)—1276no—
122#568D12711yescharacteristic, complemented
1232−1+4—1274no—
1242−1+4—2541no—
125H(5,𝔽2)—1274no—
126H(5,𝔽2)—1278no—
127H(5,𝔽2)—2544no—
128H(5,𝔽2)—2543no—
129C2×#41—1276no—
130C2×#41—1273no—
131C2×#41—2544no—
132C2×#41—2541no—
133C2×#41C2×D12711yescharacteristic
134C22×D4—1273no—
135C22×D4—2543no—
136#137—1274no—
137#137—12712no—
138AGL(1,ℤ/8)—12712no—
139AGL(1,ℤ/8)—12724no—
140AGL(1,ℤ/8)—12712no—
141#135—12724no—
142C2×SD16—12712no—
143C2×D8—12712no—
144#131C2×D12718yescomplemented
145C2×M16C2×D12716yescomplemented
146C24—2543no—
147#41—12712no—
148#41—1274no—
149#41—12712no—
150#41—2548no—
151#41—2544no—
152#41—25412no—
153#41—2546no—
154#41—2546no—
155#41—2546no—
156#41D50814yes—
157#41C22×D12714yes—
158C2×Q8—1271no—
159C2×Q8—1276no—
160C2×Q8—2544no—
161C2×Q8D50811yescharacteristic
162C2×D4—12712no—
163C2×D4—1273no—
164C2×D4—1276no—
165C2×D4—2543no—
166C2×D4—2546no—
167C2×D4—25424no—
168C2×D4—25412no—
169C2×D4—2543no—
170C2×D4—2546no—
171C2×D4D50813yes—
172C22×C4—2543no—
173C22×C4—2546no—
174C22×C4C22×D12713yes—
175Q16—1278no—
176SD16—12724no—
177SD16—1278no—
178D8—12724no—
179M16C22×D12714yes—
180M16C22×D127112yescomplemented
181C2×C8C22×D127112yescomplemented
182C23—10166no—
183C23—2546no—
184C23—25412no—
185C23—2543no—
186C23C2×D50813yes—
187Q8—1274no—
188Q8—1272no—
189Q8—2546no—
190Q8—2541no—
191Q8C2×D50814yes—
192D4—1274no—
193D4—1276no—
194D4—25412no—
195D4—2544no—
196D4—25412no—
197D4—2543no—
198D4—25412no—
199D4—2546no—
200D4C2×D508112yes—
201C2×C4—2541no—
202C2×C4—2546no—
203C2×C4—2544no—
204C2×C4—25412no—
205C2×C4—25412no—
206C2×C4C2×D50813yes—
207C2×C4C2×D50816yes—
208C2×C4C23×D12716yes—
209C2×C4C23×D12711yescharacteristic
210C8C23×D12718yes—
211C22—43no—
212C22—101612no—
213C22—10166no—
214C22—5086no—
215C22—2544no—
216C22—2546no—
217C22C22×D50811yescharacteristic
218C22C22×D50816yes—
219C4—2542no—
220C4—2544no—
221C4C22×D50811yescharacteristic
222C4C22×D50816yes—
223C4C24×D12711yescharacteristic
224C2—21no—
225C2—26no—
226C2—5086no—
227C2—5084no—
228C2C23×D50811yescharacteristic
subgroupG/H
centerC2C23×D508
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
Fitting#568×C127C2
socleC2×C127C23×D4
hypercenter#568D127
hypocenterC127#3585
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. C2×#16861255C2semidirect
3. #22550688C2semidirect
4. #22550681C2semidirect
5. #22550676C2semidirect
6. C2×#16861237C2semidirect
7. #568×C127C2semidirect
9. #16861256C22semidirect
11. #16861255C22semidirect
12. #16861255C22semidirect
15. C2×#14523248C22semidirect
16. C2×#14523248C22semidirect
18. C22×D508C22semidirect
20. #16861242C22semidirect
21. #16861242C22semidirect
22. #16861237C22semidirect
23. #16861237C22semidirect
24. #16861237C22semidirect
25. #16861231C22semidirect
26. C2×#14523237C22semidirect
27. C2×D1016C22semidirect
29. #131×C127C22semidirect
30. C2×M16×C127C22semidirect
39. #14523248C23semidirect
42. C2×Dic254C23semidirect
48. C2×D508C23semidirect
55. Dic508C23semidirect
56. #14523237C23semidirect
57. #14523237C23semidirect
58. D1016C23semidirect
65. M16×C127C23semidirect
66. C2×C8×C127C23semidirect
115. C127#3585semidirect
122. #568D127semidirect
144. #131C2×D127semidirect
145. C2×M16C2×D127semidirect
180. M16C22×D127semidirect
181. C2×C8C22×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
C2×#168612552
#225506882
#225506812
#225506762
C2×#168612372
#568×C1272
#3585127