Finite Group Databasebeta

← C22×#16861236 #30653464

Subgroups — C22×#16861236 #30653464

Order 16256.

1720 conjugacy classes of subgroups; only the 525 normal subgroups are shown.

normal subgroups only

This group has 1720 conjugacy classes of subgroups — too many to draw. Only its 525 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
1C22×#14523249C211yesmaximal, characteristic, complemented
2C22×D4×D127C211yesmaximal, characteristic, complemented
3C2×#16861236C2124yesmaximal, direct product factor
4C22×#14523235C211yesmaximal, characteristic, complemented
5C22×D1016C211yesmaximal, characteristic, complemented
6C22×#14523233C211yesmaximal, characteristic, complemented
7C22×#14523231C211yesmaximal, characteristic, complemented
8C22×SD16×C127C211yesmaximal, characteristic, complemented
9C24×D127—21no—
10C2×#14523249—26no—
11C2×#14523249C22112yescomplemented
12C22×D508—21no—
13C22×D508C2211yescharacteristic
14C22×D508C2211yescharacteristic, complemented
15C22×C22⋊D127—21no—
16C2×D4×D127—26no—
17C2×D4×D127C22112yescomplemented
18C22×C4×D127—21no—
19C22×C4×D127C2211yescharacteristic
20#16861236C22164yesdirect product factor
21C2×#14523235C22112yescomplemented
22C2×D1016C22112yescomplemented
23C2×#14523233C22112yescomplemented
24C2×#14523231C22112yescomplemented
25C22×C127⋊C8C2211yescharacteristic, complemented
26C22×Q8×C127C2211yescharacteristic, complemented
27C22×D4×C127C2211yescharacteristic
28C2×SD16×C127C22112yescomplemented
29C22×C8×C127C2211yescharacteristic, complemented
30C23×D127—212no—
31C23×D127—21no—
32C23×D127—21no—
33C23×D127—48no—
34C23×D127D411yescharacteristic
35#14523249—224no—
36#14523249C23116yescomplemented
37C2×D508—212no—
38C2×D508—23no—
39C2×D508—23no—
40C2×D508C2316yes—
41C2×D508C2316yescomplemented
42C2×C22⋊D127—212no—
43D4×D127—224no—
44D4×D127C23116yescomplemented
45C22×Dic127D411yescharacteristic
46C2×C4×D127—212no—
47C2×C4×D127D416yes—
48C2×C4×D127C2316yes—
49#14523235C23116yescomplemented
50D1016C23116yescomplemented
51#14523233C23116yescomplemented
52#14523231C23116yescomplemented
53C2×C127⋊C8C2316yescomplemented
54C24×C127—21no—
55C2×Q8×C127—23no—
56C2×Q8×C127C2316yescomplemented
57C2×D4×C127—23no—
58C2×D4×C127C2316yes—
59C22×C4×C127—21no—
60C22×C4×C127C2311yescharacteristic
61SD16×C127C23116yescomplemented
62C2×C8×C127C2316yescomplemented
63C22×D127—24no—
64C22×D127—26no—
65C22×D127—26no—
66C22×D127—216no—
67C22×D127—448no—
68C22×D127—44no—
69C22×D127—44no—
70C22×D127C2×D416yes—
71D508—216no—
72D508—26no—
73D508—26no—
74D508C2414yes—
75D508C2414yescomplemented
76C22⋊D127—216no—
77C2×Dic127C2×D416yes—
78C4×D127—216no—
79C4×D127C2×D4112yes—
80C4×D127C2414yes—
81C127⋊C8C2414yescomplemented
82C23×C127—26no—
83C23×C127—44no—
84C23×C127C2×D411yescharacteristic
85Q8×C127—26no—
86Q8×C127C2414yescomplemented
87D4×C127—26no—
88D4×C127C2414yes—
89C2×C4×C127—26no—
90C2×C4×C127C2×D413yes—
91C2×C4×C127C2413yes—
92C8×C127C2414yescomplemented
93C22×C127—24no—
94C22×C127—412no—
95C22×C127AGL(1,ℤ/8)14yescomplemented
96C22×C127C22×D413yes—
97C2×D127—212no—
98C2×D127—24no—
99C2×D127—24no—
100C2×D127—432no—
101C2×D127—412no—
102C2×D127—412no—
103C2×D127C22×D414yes—
104C4×C127—24no—
105C4×C127C22×D413yes—
106C4×C127C2511yescharacteristic
107Dic127C22×D414yes—
108C2×C127—44no—
109C2×C127C2×AGL(1,ℤ/8)16yescomplemented
110C2×C127C23×D411yescharacteristic
111D127—24no—
112D127—44no—
113D127—44no—
114C22×AGL(1,ℤ/8)—1271nomaximal, Sylow
115C127C22×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
116C22×#41—1271no—
117C23×D4—1271no—
118C2×AGL(1,ℤ/8)—12724no—
119C22×SD16—1271no—
120C22×SD16D12711yescharacteristic, complemented
121C22×D8—1271no—
122C22×D8—1271no—
123C22×M16—1271no—
124C25—2541no—
125C2×#41—12712no—
126C2×#41—2546no—
127C22×Q8C2×D12711yescharacteristic, complemented
128C22×D4—12712no—
129C22×D4—1271no—
130C22×D4—1271no—
131C22×D4—2541no—
132C22×D4—2546no—
133C22×D4—2541no—
134C22×D4C2×D12711yescharacteristic
135C23×C4—1271no—
136C23×C4—2541no—
137AGL(1,ℤ/8)—12764no—
138C2×SD16—12712no—
139C2×SD16C2×D127112yescomplemented
140C2×D8—12712no—
141C2×D8—12712no—
142C2×M16—12712no—
143C22×C8—1271no—
144C22×C8C2×D12711yescharacteristic, complemented
145C24—21no—
146C24—1271no—
147C24—5088no—
148C24—25412no—
149C24—2541no—
150C24—2541no—
151#41—12716no—
152#41—25424no—
153C2×Q8—23no—
154C2×Q8C22×D12716yescomplemented
155C2×D4—23no—
156C2×D4—12716no—
157C2×D4—1276no—
158C2×D4—1276no—
159C2×D4—25412no—
160C2×D4—25424no—
161C2×D4—25412no—
162C2×D4—2543no—
163C2×D4—2543no—
164C2×D4C22×D12716yes—
165C22×C4—21no—
166C22×C4—1271no—
167C22×C4—1276no—
168C22×C4—1276no—
169C22×C4—25412no—
170C22×C4C22×D12711yescharacteristic
171SD16—12716no—
172SD16C22×D127116yescomplemented
173D8—12716no—
174D8—12716no—
175M16—12716no—
176C2×C8—1276no—
177C2×C8C22×D12716yescomplemented
178C23—26no—
179C23—44no—
180C23—1276no—
181C23—50848no—
182C23—5084no—
183C23—5084no—
184C23—25416no—
185C23—2546no—
186C23—2546no—
187C23—2544no—
188C23D4×D12711yescharacteristic
189Q8—26no—
190Q8C23×D12714yescomplemented
191D4—26no—
192D4—1274no—
193D4—1274no—
194D4—25416no—
195D4—25416no—
196D4—2546no—
197D4—2546no—
198D4C23×D12714yes—
199C2×C4—26no—
200C2×C4—1276no—
201C2×C4—12712no—
202C2×C4—1274no—
203C2×C4—25416no—
204C2×C4D4×D12713yes—
205C2×C4C23×D12713yes—
206C8—1274no—
207C8C23×D12714yescomplemented
208C22—24no—
209C22—412no—
210C22—1274no—
211C22—50832no—
212C22—50812no—
213C22—50812no—
214C22—2544no—
215C22—2544no—
216C22—25412no—
217C22#1686123614yesdirect product factor
218C22C2×D4×D12713yes—
219C4—24no—
220C4—1274no—
221C4C2×D4×D12713yes—
222C4C24×D12711yescharacteristic
223C2—44no—
224C2—5084no—
225C2—5084no—
226C2—2544no—
227C2C2×#1686123616yesdirect product factor
228C2C22×D4×D12711yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC4×C127C25
FrattiniC4C24×D127
FittingC22×SD16×C127C2
socleC23×C127C2×D4
hypercenterC22×SD16D127
hypocenterC127C22×AGL(1,ℤ/8)
LCSG ▷ C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C22×#14523249C2semidirect
2. C22×D4×D127C2semidirect
3. C2×#16861236C2direct (N × H)
4. C22×#14523235C2semidirect
5. C22×D1016C2semidirect
6. C22×#14523233C2semidirect
7. C22×#14523231C2semidirect
8. C22×SD16×C127C2semidirect
11. C2×#14523249C22semidirect
14. C22×D508C22semidirect
17. C2×D4×D127C22semidirect
20. #16861236C22direct (N × H)
21. C2×#14523235C22semidirect
22. C2×D1016C22semidirect
23. C2×#14523233C22semidirect
24. C2×#14523231C22semidirect
25. C22×C127⋊C8C22semidirect
26. C22×Q8×C127C22semidirect
28. C2×SD16×C127C22semidirect
29. C22×C8×C127C22semidirect
36. #14523249C23semidirect
41. C2×D508C23semidirect
44. D4×D127C23semidirect
49. #14523235C23semidirect
50. D1016C23semidirect
51. #14523233C23semidirect
52. #14523231C23semidirect
53. C2×C127⋊C8C23semidirect
56. C2×Q8×C127C23semidirect
61. SD16×C127C23semidirect
62. C2×C8×C127C23semidirect
75. D508C24semidirect
81. C127⋊C8C24semidirect
86. Q8×C127C24semidirect
92. C8×C127C24semidirect
95. C22×C127AGL(1,ℤ/8)semidirect
109. C2×C127C2×AGL(1,ℤ/8)semidirect
115. C127C22×AGL(1,ℤ/8)semidirect
120. C22×SD16D127semidirect
127. C22×Q8C2×D127semidirect
139. C2×SD16C2×D127semidirect
144. C22×C8C2×D127semidirect
154. C2×Q8C22×D127semidirect
172. SD16C22×D127semidirect
177. C2×C8C22×D127semidirect
190. Q8C23×D127semidirect
207. C8C23×D127semidirect
217. C22#16861236direct (N × H)
227. C2C2×#16861236direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232492
C22×D4×D1272
C2×#168612362
C22×#145232352
C22×D10162
C22×#145232332
C22×#145232312
C22×SD16×C1272
C22×AGL(1,ℤ/8)127