Finite Group Databasebeta

← #30649179

Subgroups — #30649179

Order 16256.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16861236C212yesmaximal, complemented
2C2×#16861233C212yesmaximal, complemented
3#22550425C211yesmaximal, characteristic, complemented
4#127×D127C211yesmaximal, characteristic, complemented
5#22550266C211yesmaximal, characteristic, complemented
6#22550265C211yesmaximal, characteristic, complemented
7#22550263C211yesmaximal, characteristic, complemented
8#22550251C212yesmaximal, complemented
9#22550247C212yesmaximal, complemented
10#22549828C211yesmaximal, characteristic, complemented
11#496×C127C211yesmaximal, characteristic, complemented
12C2×#14523249C2211yescharacteristic
13C2×#14523247C2211yescharacteristic
14C2×D4×D127—22no—
15C2×D4×D127C2212yes—
16#16861236—28no—
17#16861233—28no—
18C2×#14523237C2212yescomplemented
19C2×#14523236C2212yescomplemented
20C2×#14523235C2212yescomplemented
21C2×D1016C2212yescomplemented
22C2×#14523233C2212yescomplemented
23C2×#14523233C2212yescomplemented
24C2×#14523231C2212yes—
25#16861207C2211yescharacteristic, complemented
26#16861206C2211yescharacteristic
27#16861205—22no—
28#16861205C2212yes—
29#16861205C2211yescharacteristic, complemented
30#16861197C2211yescharacteristic, complemented
31#16861182—22no—
32(C22⋊D127)⋊C4—22no—
33C42×D127C2211yescharacteristic
34#16861120C2211yescharacteristic, complemented
35C8×Dic127C2212yescomplemented
36C2×SD16×C127C2212yescomplemented
37C2×D8×C127C2212yescomplemented
38#127×C127C2211yescharacteristic
39#124×C127C2211yescharacteristic, complemented
40#97×C127C2211yescharacteristic, complemented
41C23×D127—22no—
42#14523249—24no—
43#14523247—24no—
44C2×Dic254C2311yescharacteristic
45C2×D508—21no—
46C2×D508—22no—
47C2×D508C2311yescharacteristic
48C2×D508C2312yes—
49C2×C22⋊D127—21no—
50C2×C22⋊D127—22no—
51C2×C22⋊D127—24no—
52C2×C22⋊D127—22no—
53D4×D127—24no—
54D4×D127—24no—
55D4×D127—48no—
56C22×Dic127—21no—
57C2×C4×D127—21no—
58C2×C4×D127D412yes—
59C2×C4×D127C2311yescharacteristic
60#14523237—24no—
61#14523236—24no—
62#14523235—24no—
63D1016—24no—
64#14523233—24no—
65#14523233—24no—
66#14523231D418yescomplemented
67C2×C127⋊C8C2312yes—
68Dic127⋊C4—22no—
69(C2×D127)⋊C4—22no—
70C4×Dic127D412yes—
71C4×Dic127C2311yescharacteristic
72C2×Q8×C127C2311yescharacteristic
73C2×D4×C127—22no—
74C2×D4×C127C2312yes—
75C2×D4×C127C2311yescharacteristic
76SD16×C127—24no—
77D8×C127—24no—
78C2×C8×C127C2312yes—
79C22⋊C4×C127—22no—
80C42×C127C2311yescharacteristic
81C22×D127—21no—
82C22×D127—22no—
83C22×D127—48no—
84C22×D127—44no—
85C22×D127C2×D411yescharacteristic
86Dic254—22no—
87D508—22no—
88D508—22no—
89D508—22no—
90D508—44no—
91D508—42no—
92C22⋊D127—42no—
93C22⋊D127—44no—
94C22⋊D127—48no—
95C22⋊D127—44no—
96C2×Dic127—21no—
97C2×Dic127—42no—
98C2×Dic127C2×D411yescharacteristic
99C2×Dic127C2×D412yes—
100C4×D127—24no—
101C4×D127—42no—
102C4×D127C2×D414yes—
103C127⋊C8C2×D414yescomplemented
104C23×C127—21no—
105C23×C127—22no—
106Q8×C127—22no—
107D4×C127—22no—
108D4×C127—22no—
109D4×C127—22no—
110D4×C127—44no—
111C2×C4×C127—21no—
112C2×C4×C127C2×D412yes—
113C2×C4×C127C2411yescharacteristic
114C8×C127C2×D414yescomplemented
115C22×C127—41no—
116C22×C127—42no—
117C22×C127—44no—
118C22×C127—42no—
119C22×C127C22×D411yescharacteristic
120C2×D127—22no—
121C2×D127—41no—
122C2×D127—42no—
123C2×D127—44no—
124C2×D127—42no—
125C2×D127—88no—
126C2×D127#12712yes—
127C4×C127—22no—
128C4×C127—41no—
129C4×C127C22×D412yes—
130Dic127—41no—
131Dic127#12712yes—
132Dic127AGL(1,ℤ/8)14yescomplemented
133C2×C127—82no—
134C2×C127—81no—
135C2×C127C2×#12711yescharacteristic
136C2×C127C2×AGL(1,ℤ/8)12yes—
137D127—22no—
138D127—81no—
139D127—82no—
140#3151—1271nomaximal, Sylow
141C127#315111yescharacteristic, complemented, Sylow
142C2×AGL(1,ℤ/8)—1272no—
143C2×AGL(1,ℤ/8)—1272no—
144#532—1271no—
145C2×#127—1271no—
146#496—1271no—
147#496—1271no—
148#496—1271no—
149#496D12711yescharacteristic, complemented
150#493—1272no—
151#492—1272no—
152C4×M16—1271no—
153C2×#41—1271no—
154C2×#41—1271no—
155C22×D4—1272no—
156C22×D4—2542no—
157AGL(1,ℤ/8)—2548no—
158AGL(1,ℤ/8)—2548no—
159C2×SD16—1272no—
160C2×SD16—1272no—
161C2×SD16—1272no—
162C2×SD16C2×D12712yescomplemented
163C2×D8—1272no—
164C2×D8—1272no—
165C2×D8—1272no—
166C2×D8C2×D12712yescomplemented
167C2×M16—1272no—
168C4⋊Q8—1271no—
169#127—1272no—
170#127—1271no—
171#127—1271no—
172#127—2542no—
173#127C2×D12711yescharacteristic
174#124—1271no—
175#124C2×D12711yescharacteristic, complemented
176#121—2542no—
177C4×D4—2542no—
178C2×C42—1271no—
179#97—1271no—
180#97C2×D12711yescharacteristic, complemented
181C4×C8—1272no—
182C24—2542no—
183#41—2544no—
184#41—2544no—
185C2×Q8—1271no—
186C2×Q8C22×D12711yescharacteristic
187C2×D4—22no—
188C2×D4—1271no—
189C2×D4—1272no—
190C2×D4—5088no—
191C2×D4—2541no—
192C2×D4—2544no—
193C2×D4—2544no—
194C2×D4—2542no—
195C2×D4—2544no—
196C2×D4—2542no—
197C2×D4—2541no—
198C2×D4—2542no—
199C2×D4C22×D12711yescharacteristic
200C2×D4C22×D12712yes—
201C22×C4—1271no—
202C22×C4—1272no—
203C22×C4—2541no—
204C22×C4—2541no—
205SD16—24no—
206SD16—2544no—
207SD16—2544no—
208SD16—2544no—
209D8—24no—
210D8—2544no—
211D8—2544no—
212D8—2544no—
213M16—1278no—
214C2×C8—1272no—
215C2×C8C22×D12712yes—
216C4⋊C4—2542no—
217C22⋊C4—22no—
218C22⋊C4—2542no—
219C42—1272no—
220C42—1271no—
221C42C22×D12711yescharacteristic
222C23—21no—
223C23—22no—
224C23—1271no—
225C23—5084no—
226C23—5088no—
227C23—2541no—
228C23—2542no—
229Q8—22no—
230Q8—2542no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—5084no—
236D4—5082no—
237D4—5088no—
238D4—5084no—
239D4—5084no—
240D4—5082no—
241D4—2542no—
242D4—2542no—
243D4—2542no—
244C2×C4—21no—
245C2×C4—1272no—
246C2×C4—1271no—
247C2×C4—1274no—
248C2×C4—5082no—
249C2×C4—5082no—
250C2×C4—2544no—
251C2×C4—2541no—
252C2×C4D4×D12712yes—
253C2×C4C23×D12711yescharacteristic
254C8—1274no—
255C8D4×D12714yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—1272no—
261C22—10168no—
262C22—5084no—
263C22—5082no—
264C22—5081no—
265C22—5082no—
266C22—2542no—
267C22C2×D4×D12711yescharacteristic
268C4—22no—
269C4—41no—
270C4—1274no—
271C4—1272no—
272C4—5081no—
273C4C2×D4×D12712yes—
274C2—82no—
275C2—81no—
276C2—10162no—
277C2—10161no—
278C2—2542no—
279C2#127×D12711yescharacteristic
280C2AGL(1,ℤ/8)×D12712yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#496×C127C2
socleC22×C127C22×D4
hypercenter#496D127
hypocenterC127#3151
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16861236C2semidirect
2. C2×#16861233C2semidirect
3. #22550425C2semidirect
4. #127×D127C2semidirect
5. #22550266C2semidirect
6. #22550265C2semidirect
7. #22550263C2semidirect
8. #22550251C2semidirect
9. #22550247C2semidirect
10. #22549828C2semidirect
11. #496×C127C2semidirect
18. C2×#14523237C22semidirect
19. C2×#14523236C22semidirect
20. C2×#14523235C22semidirect
21. C2×D1016C22semidirect
22. C2×#14523233C22semidirect
23. C2×#14523233C22semidirect
25. #16861207C22semidirect
29. #16861205C22semidirect
30. #16861197C22semidirect
34. #16861120C22semidirect
35. C8×Dic127C22semidirect
36. C2×SD16×C127C22semidirect
37. C2×D8×C127C22semidirect
39. #124×C127C22semidirect
40. #97×C127C22semidirect
66. #14523231D4semidirect
103. C127⋊C8C2×D4semidirect
114. C8×C127C2×D4semidirect
132. Dic127AGL(1,ℤ/8)semidirect
141. C127#3151semidirect
149. #496D127semidirect
162. C2×SD16C2×D127semidirect
166. C2×D8C2×D127semidirect
175. #124C2×D127semidirect
180. #97C2×D127semidirect
255. C8D4×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612362
C2×#168612332
#225504252
#127×D1272
#225502662
#225502652
#225502632
#225502512
#225502472
#225498282
#496×C1272
#3151127