Finite Group Databasebeta

← #30649101

Subgroups — #30649101

Order 16256.

484 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 125 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#22550431C211yesmaximal, characteristic, complemented
2#127×D127C211yesmaximal, characteristic
3#22550239C214yesmaximal, complemented
4#22550224C211yesmaximal, characteristic, complemented
5#22550223C211yesmaximal, characteristic, complemented
6#22550222C211yesmaximal, characteristic, complemented
7#22549879C214yesmaximal, complemented
8C4×#14523231C211yesmaximal, characteristic, complemented
9#486×C127C211yesmaximal, characteristic, complemented
10C2×D4×D127—21no—
11C2×D4×D127—21no—
12C2×D4×D127C2212yes—
13C2×#14523231C2212yes—
14#16861209C2211yescharacteristic, complemented
15#16861206C2211yescharacteristic
16#16861205—21no—
17#16861205—21no—
18#16861205C2212yes—
19#16861200C2212yescomplemented
20#16861189—21no—
21#16861185—21no—
22Q8×Dic127—21no—
23#16861174—21no—
24#16861167C2212yes—
25C42×D127C2211yescharacteristic
26#16861131C2214yescomplemented
27#16861130C2214yescomplemented
28#16861129C2214yescomplemented
29#16861119C2212yescomplemented
30C4×C127⋊C8C2211yescharacteristic, complemented
31C4⋊Q8×C127C2211yescharacteristic, complemented
32#127×C127C2211yescharacteristic
33#102×C127C2214yescomplemented
34C4×C8×C127C2211yescharacteristic, complemented
35C23×D127—22no—
36C2×Dic254—21no—
37C2×D508—21no—
38C2×D508—21no—
39C2×D508C2312yes—
40C2×C22⋊D127—22no—
41C2×C22⋊D127—22no—
42C2×C22⋊D127—22no—
43D4×D127—28no—
44D4×D127—44no—
45D4×D127—44no—
46C2×C4×D127D411yescharacteristic
47C2×C4×D127D411yescharacteristic
48C2×C4×D127C2311yescharacteristic
49#14523231—24no—
50C2×C127⋊C8C2312yes—
51C4⋊Dic127—22no—
52C4⋊Dic127—21no—
53C4⋊Dic127C2312yes—
54Dic127⋊C4—22no—
55(C2×D127)⋊C4—22no—
56C4×Dic127—22no—
57C4×Dic127D411yescharacteristic
58C4×Dic127D411yescharacteristic
59C4×Dic127C2311yescharacteristic
60C2×Q8×C127—21no—
61C2×D4×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127C2312yes—
64C2×C8×C127C2312yes—
65C4⋊C4×C127—21no—
66C4⋊C4×C127C2312yes—
67C42×C127C2311yescharacteristic
68C22×D127—22no—
69C22×D127—44no—
70C22×D127—44no—
71C22×D127—44no—
72C22×D127C2×D411yescharacteristic
73Dic254—42no—
74D508—24no—
75D508—42no—
76D508—42no—
77C22⋊D127—44no—
78C22⋊D127—44no—
79C22⋊D127—44no—
80C2×Dic127—22no—
81C2×Dic127C2×D411yescharacteristic
82C2×Dic127C2×D411yescharacteristic
83C2×Dic127C2×D411yescharacteristic
84C4×D127—22no—
85C4×D127—22no—
86C4×D127#4114yes—
87C127⋊C8—22no—
88C23×C127—22no—
89Q8×C127—42no—
90D4×C127—24no—
91D4×C127—42no—
92D4×C127—42no—
93C2×C4×C127—22no—
94C2×C4×C127C2×D411yescharacteristic
95C2×C4×C127C2×D411yescharacteristic
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127—42no—
99C22×C127—42no—
100C22×C127—42no—
101C22×C127C22×D411yescharacteristic
102C2×D127—21no—
103C2×D127—21no—
104C2×D127—42no—
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—88no—
108C2×D127#12412yes—
109C4×C127—42no—
110C4×C127AGL(1,ℤ/8)12yes—
111C4×C127AGL(1,ℤ/8)12yes—
112C4×C127C2×#4112yes—
113Dic127—21no—
114Dic127—21no—
115Dic127—42no—
116Dic127#12412yes—
117C2×C127—82no—
118C2×C127C2×#12411yescharacteristic
119C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
121D127—22no—
122D127—82no—
123#3138—1271nomaximal, Sylow
124C127#313811yescharacteristic, complemented, Sylow
125#533—1271no—
126C2×#127—1271no—
127#490—1274no—
128#486—1271no—
129#486—1271no—
130#486—1271no—
131#486D12711yescharacteristic, complemented
132#418—1274no—
133C4×M16—1271no—
134C22×D4—1272no—
135C22×D4—2541no—
136C22×D4—2541no—
137C2×M16—1272no—
138C4⋊Q8—1271no—
139C4⋊Q8C2×D12711yescharacteristic, complemented
140#127—1272no—
141#127—1271no—
142#127—2541no—
143#127—2541no—
144#127C2×D12711yescharacteristic
145#125—1272no—
146C22⋊Q8—2541no—
147C22⋊Q8—2541no—
148C4×Q8—2541no—
149C4×Q8—2541no—
150#117—1272no—
151C2×C42—1271no—
152#102—1274no—
153#102—1274no—
154#102—1274no—
155#102C2×D12714yescomplemented
156#97—1272no—
157C4×C8—1271no—
158C4×C8C2×D12711yescharacteristic, complemented
159C24—2542no—
160C2×Q8—21no—
161C2×Q8—2541no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—1272no—
165C2×D4—5084no—
166C2×D4—5084no—
167C2×D4—2548no—
168C2×D4—2542no—
169C2×D4—2542no—
170C2×D4—2542no—
171C2×D4—2541no—
172C2×D4—2541no—
173C2×D4C22×D12712yes—
174C22×C4—1271no—
175C22×C4—1271no—
176C22×C4—1271no—
177M16—2544no—
178C2×C8—1272no—
179C2×C8C22×D12712yes—
180C4⋊C4—21no—
181C4⋊C4—1272no—
182C4⋊C4—2542no—
183C4⋊C4—2542no—
184C4⋊C4—2541no—
185C4⋊C4C22×D12712yes—
186C22⋊C4—2542no—
187C42—1271no—
188C42—1271no—
189C42—1271no—
190C42—2542no—
191C42C22×D12711yescharacteristic
192C23—22no—
193C23—1271no—
194C23—5084no—
195C23—5084no—
196C23—5084no—
197C23—2542no—
198Q8—42no—
199Q8—5082no—
200D4—24no—
201D4—42no—
202D4—42no—
203D4—5084no—
204D4—5084no—
205D4—5084no—
206D4—5082no—
207D4—5082no—
208D4—2544no—
209C2×C4—22no—
210C2×C4—1271no—
211C2×C4—1274no—
212C2×C4—1271no—
213C2×C4—1271no—
214C2×C4—2542no—
215C2×C4—2542no—
216C2×C4—2542no—
217C2×C4D4×D12711yescharacteristic
218C2×C4D4×D12711yescharacteristic
219C2×C4C23×D12711yescharacteristic
220C8—22no—
221C8—2542no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—1272no—
226C22—10168no—
227C22—5082no—
228C22—5082no—
229C22—5082no—
230C22—2541no—
231C22—2541no—
232C22C2×D4×D12711yescharacteristic
233C4—42no—
234C4—1272no—
235C4—5082no—
236C4—2541no—
237C4—2541no—
238C4#1686123312yes—
239C4#1686123612yes—
240C4#41×D12712yes—
241C2—82no—
242C2—10162no—
243C2—2542no—
244C2#124×D12711yescharacteristic
245C2C2×#1686123311yescharacteristic
246C2C2×#1686123611yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#486×C127C2
socleC22×C127C22×D4
prime radical#127×D127C2
hypercenter#486D127
hypocenterC127#3138
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #22550431C2semidirect
3. #22550239C2semidirect
4. #22550224C2semidirect
5. #22550223C2semidirect
6. #22550222C2semidirect
7. #22549879C2semidirect
8. C4×#14523231C2semidirect
9. #486×C127C2semidirect
14. #16861209C22semidirect
19. #16861200C22semidirect
26. #16861131C22semidirect
27. #16861130C22semidirect
28. #16861129C22semidirect
29. #16861119C22semidirect
30. C4×C127⋊C8C22semidirect
31. C4⋊Q8×C127C22semidirect
33. #102×C127C22semidirect
34. C4×C8×C127C22semidirect
124. C127#3138semidirect
131. #486D127semidirect
139. C4⋊Q8C2×D127semidirect
155. #102C2×D127semidirect
158. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225504312
#127×D1272
#225502392
#225502242
#225502232
#225502222
#225498792
C4×#145232312
#486×C1272
#3138127