Finite Group Databasebeta

← #486×D127 #30649033

Subgroups — #486×D127 #30649033

Order 16256.

528 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C4⋊Q8×D127C211yesmaximal, characteristic, complemented
2#127×D127C211yesmaximal, characteristic
3#22550224C211yesmaximal, characteristic, complemented
4#22550223C211yesmaximal, characteristic, complemented
5#22550222C211yesmaximal, characteristic, complemented
6#22550221C214yesmaximal, complemented
7#102×D127C214yesmaximal, complemented
8C4×C8×D127C211yesmaximal, characteristic, complemented
9#486×C127C211yesmaximal, characteristic, complemented
10C2×Q8×D127—21no—
11C2×D4×D127—21no—
12C2×D4×D127—21no—
13C2×D4×D127C2212yes—
14C2×C8×D127C2212yes—
15#16861209C2211yescharacteristic, complemented
16#16861208—21no—
17#16861208C2212yescomplemented
18#16861207—21no—
19#16861206C2211yescharacteristic
20#16861205—21no—
21#16861205—21no—
22#16861205C2212yes—
23C4⋊C4×D127—21no—
24C4⋊C4×D127C2212yes—
25C42×D127C2211yescharacteristic
26#16861131C2214yescomplemented
27#16861130C2214yescomplemented
28#16861129C2214yescomplemented
29C4×C127⋊C8C2211yescharacteristic, complemented
30C8×Dic127C2212yescomplemented
31C4⋊Q8×C127C2211yescharacteristic, complemented
32#127×C127C2211yescharacteristic
33#102×C127C2214yescomplemented
34C4×C8×C127C2211yescharacteristic, complemented
35C23×D127—22no—
36C2×Dic254—22no—
37C2×Dic254—21no—
38Q8×D127—44no—
39C2×D508—21no—
40C2×D508—21no—
41C2×D508C2312yes—
42C2×C22⋊D127—22no—
43C2×C22⋊D127—22no—
44C2×C22⋊D127—22no—
45D4×D127—28no—
46D4×D127—44no—
47D4×D127—44no—
48C2×C4×D127—22no—
49C2×C4×D127D411yescharacteristic
50C2×C4×D127D411yescharacteristic
51C2×C4×D127C2311yescharacteristic
52C2×C127⋊C8C2312yes—
53C8×D127—24no—
54C4⋊Dic127—21no—
55C4⋊Dic127C2312yes—
56Dic127⋊C4—22no—
57Dic127⋊C4—22no—
58C4×Dic127D411yescharacteristic
59C4×Dic127D411yescharacteristic
60C4×Dic127C2311yescharacteristic
61C2×Q8×C127—21no—
62C2×D4×C127—21no—
63C2×D4×C127—21no—
64C2×D4×C127C2312yes—
65C2×C8×C127C2312yes—
66C4⋊C4×C127—21no—
67C4⋊C4×C127C2312yes—
68C42×C127C2311yescharacteristic
69C22×D127—22no—
70C22×D127—44no—
71C22×D127—44no—
72C22×D127—44no—
73C22×D127C2×D411yescharacteristic
74Dic254—42no—
75Dic254—44no—
76D508—24no—
77D508—42no—
78D508—42no—
79C22⋊D127—44no—
80C22⋊D127—44no—
81C22⋊D127—44no—
82C2×Dic127—22no—
83C2×Dic127C2×D411yescharacteristic
84C2×Dic127C2×D411yescharacteristic
85C2×Dic127C2×D411yescharacteristic
86C4×D127—44no—
87C4×D127D814yes—
88C4×D127SD1614yes—
89C4×D127#4114yes—
90C127⋊C8—22no—
91C23×C127—22no—
92Q8×C127—42no—
93D4×C127—24no—
94D4×C127—42no—
95D4×C127—42no—
96C2×C4×C127—22no—
97C2×C4×C127C2×D411yescharacteristic
98C2×C4×C127C2×D411yescharacteristic
99C2×C4×C127C2411yescharacteristic
100C8×C127—22no—
101C22×C127—42no—
102C22×C127—42no—
103C22×C127—42no—
104C22×C127C22×D411yescharacteristic
105C2×D127—42no—
106C2×D127—42no—
107C2×D127—42no—
108C2×D127—88no—
109C2×D127#12412yes—
110C2×D127C2×D812yes—
111C2×D127C2×SD1612yes—
112C4×C127—42no—
113C4×C127C2×D812yes—
114C4×C127C2×SD1612yes—
115C4×C127C2×#4112yes—
116Dic127—42no—
117Dic127#12412yes—
118Dic127C2×D812yes—
119Dic127C2×SD1612yes—
120C2×C127—82no—
121C2×C127C2×#12411yescharacteristic
122C2×C127C22×D811yescharacteristic
123C2×C127C22×SD1611yescharacteristic
124D127—82no—
125D127#48614yesdirect product factor
126C2×#486—1271nomaximal, Sylow
127C127C2×#48611yescharacteristic, complemented, Sylow
128C2×C4⋊Q8—1271no—
129C2×#127—1271no—
130#486—1271no—
131#486—1271no—
132#486—1274no—
133#486—1271no—
134#486D12711yescharacteristic, direct product factor
135C2×#102—1274no—
136C2×C4×C8—1271no—
137C22×Q8—2541no—
138C22×D4—1272no—
139C22×D4—2541no—
140C22×D4—2541no—
141C22×C8—1272no—
142C4⋊Q8—1272no—
143C4⋊Q8—1271no—
144C4⋊Q8—2541no—
145C4⋊Q8—2541no—
146C4⋊Q8C2×D12711yescharacteristic, complemented
147#127—1272no—
148#127—1271no—
149#127—2541no—
150#127—2541no—
151#127C2×D12711yescharacteristic
152C2×C4⋊C4—1272no—
153C2×C4⋊C4—2541no—
154C2×C42—1271no—
155#102—1274no—
156#102—1274no—
157#102—1274no—
158#102C2×D12714yescomplemented
159C4×C8—1272no—
160C4×C8—1271no—
161C4×C8C2×D12711yescharacteristic, complemented
162C24—2542no—
163C2×Q8—21no—
164C2×Q8—5084no—
165C2×Q8—2542no—
166C2×Q8—2541no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—1272no—
170C2×D4—5084no—
171C2×D4—5084no—
172C2×D4—2548no—
173C2×D4—2542no—
174C2×D4—2542no—
175C2×D4—2542no—
176C2×D4—2541no—
177C2×D4—2541no—
178C2×D4C22×D12712yes—
179C22×C4—1271no—
180C22×C4—1271no—
181C22×C4—1271no—
182C22×C4—2542no—
183C2×C8—1272no—
184C2×C8—2544no—
185C2×C8C22×D12712yes—
186C4⋊C4—21no—
187C4⋊C4—1272no—
188C4⋊C4—2542no—
189C4⋊C4—2542no—
190C4⋊C4—2541no—
191C4⋊C4C22×D12712yes—
192C42—1271no—
193C42—1271no—
194C42—1271no—
195C42C22×D12711yescharacteristic
196C23—22no—
197C23—1271no—
198C23—5084no—
199C23—5084no—
200C23—5084no—
201C23—2542no—
202Q8—42no—
203Q8—5084no—
204Q8—5082no—
205D4—24no—
206D4—42no—
207D4—42no—
208D4—5084no—
209D4—5084no—
210D4—5084no—
211D4—5082no—
212D4—5082no—
213D4—2544no—
214C2×C4—22no—
215C2×C4—1271no—
216C2×C4—1271no—
217C2×C4—1271no—
218C2×C4—1274no—
219C2×C4—1274no—
220C2×C4—1274no—
221C2×C4—5084no—
222C2×C4—2542no—
223C2×C4D4×D12711yescharacteristic
224C2×C4D4×D12711yescharacteristic
225C2×C4C23×D12711yescharacteristic
226C8—22no—
227C8—2542no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—1272no—
232C22—1272no—
233C22—1272no—
234C22—10168no—
235C22—5082no—
236C22—5082no—
237C22—5082no—
238C22C2×D4×D12711yescharacteristic
239C4—42no—
240C4—1272no—
241C4—1272no—
242C4—1272no—
243C4—5082no—
244C4D8×D12712yes—
245C4SD16×D12712yes—
246C4#41×D12712yes—
247C2—82no—
248C2—1274no—
249C2—10162no—
250C2#124×D12711yescharacteristic
251C2C2×D8×D12711yescharacteristic
252C2C2×SD16×D12711yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#486×C127C2
socleC22×C127C22×D4
prime radical#127×D127C2
hypercenter#486D127
hypocenterC127C2×#486
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. C4⋊Q8×D127C2semidirect
3. #22550224C2semidirect
4. #22550223C2semidirect
5. #22550222C2semidirect
6. #22550221C2semidirect
7. #102×D127C2semidirect
8. C4×C8×D127C2semidirect
9. #486×C127C2semidirect
15. #16861209C22semidirect
17. #16861208C22semidirect
26. #16861131C22semidirect
27. #16861130C22semidirect
28. #16861129C22semidirect
29. C4×C127⋊C8C22semidirect
30. C8×Dic127C22semidirect
31. C4⋊Q8×C127C22semidirect
33. #102×C127C22semidirect
34. C4×C8×C127C22semidirect
125. D127#486direct (N × H)
127. C127C2×#486semidirect
134. #486D127direct (N × H)
146. C4⋊Q8C2×D127semidirect
158. #102C2×D127semidirect
161. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1272
#127×D1272
#225502242
#225502232
#225502222
#225502212
#102×D1272
C4×C8×D1272
#486×C1272
C2×#486127