Finite Group Databasebeta

← #30648104

Subgroups — #30648104

Order 16256.

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

normal subgroups only

This group has 736 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
1#22550712C211yesmaximal, characteristic, complemented
2C2×#16861237C212yesmaximal, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550051C214yesmaximal, complemented
5#22550046C214yesmaximal, complemented
6#22549892C212yesmaximal, complemented
7#413×C127C211yesmaximal, characteristic, complemented
8#16861257C2212yescomplemented
9#16861256—22no—
10#16861255—22no—
11#16861254—22no—
12#16861254C2212yescomplemented
13C2×#14523248—22no—
14C2×#14523248C2212yes—
15#41×D127—24no—
16C22×D508C2211yescharacteristic, complemented
17#16861237—24no—
18#16861237C2218yescomplemented
19C2×#14523237—22no—
20C2×D1016—22no—
21#16861205C2212yescomplemented
22#16861198C2212yescomplemented
23#16861183—22no—
24#16861177—22no—
25#16861168C2211yescharacteristic, complemented
26#16861135C2218yescomplemented
27C2×#41×C127C2211yescharacteristic
28C2×M16×C127C2212yescomplemented
29#101×C127C2212yescomplemented
30#100×C127C2212yescomplemented
31C23×D127—21no—
32#14523249—24no—
33#14523248—24no—
34#14523248—24no—
35#14523248—22no—
36#14523248—24no—
37#14523248—22no—
38#14523248D414yescomplemented
39#14523248C2314yes—
40#14523247—28no—
41#14523247—24no—
42C2×Dic254—22no—
43C2×Dic254D412yescomplemented
44Q8×D127—24no—
45C2×D508—22no—
46C2×D508—22no—
47C2×D508—22no—
48C2×D508—21no—
49C2×D508D412yescomplemented
50C2×D508C2312yes—
51C2×C22⋊D127—24no—
52C2×C22⋊D127—22no—
53C2×C22⋊D127—24no—
54D4×D127—28no—
55D4×D127—24no—
56C2×C4×D127—24no—
57C2×C4×D127—22no—
58#14523237—28no—
59D1016—28no—
60C4⋊Dic127—21no—
61C22⋊Dic127—21no—
62(C2×D127)⋊C4—24no—
63C4×Dic127C2312yes—
64#41×C127—22no—
65#41×C127—22no—
66C2×Q8×C127C2311yescharacteristic
67C2×D4×C127D412yes—
68C2×D4×C127C2311yescharacteristic
69C22×C4×C127D412yes—
70C22×C4×C127C2311yescharacteristic
71M16×C127—22no—
72M16×C127C2314yes—
73C2×C8×C127—22no—
74C22×D127—24no—
75C22×D127—22no—
76C22×D127—22no—
77C22×D127—42no—
78C22×D127—82no—
79Dic254—24no—
80Dic254—22no—
81Dic254C2×D414yes—
82D508—24no—
83D508—22no—
84D508—24no—
85D508—41no—
86D508—41no—
87D508C2×D414yes—
88C22⋊D127—28no—
89C22⋊D127—24no—
90C22⋊D127—24no—
91C22⋊D127—84no—
92C2×Dic127—24no—
93C2×Dic127—22no—
94C2×Dic127—22no—
95C4×D127—28no—
96C4×D127—24no—
97C4×D127—24no—
98C23×C127C2×D412yes—
99C23×C127C2×D411yescharacteristic
100Q8×C127—22no—
101D4×C127—24no—
102D4×C127—22no—
103C2×C4×C127—22no—
104C2×C4×C127—22no—
105C2×C4×C127C2×D412yes—
106C2×C4×C127C2×D412yes—
107C2×C4×C127C2×D412yes—
108C2×C4×C127C2×D411yescharacteristic
109C2×C4×C127C2411yescharacteristic
110C8×C127—24no—
111C22×C127—22no—
112C22×C127—41no—
113C22×C127—42no—
114C22×C127#12012yes—
115C22×C127C22×D411yescharacteristic
116C2×D127—24no—
117C2×D127—44no—
118C2×D127—42no—
119C2×D127—82no—
120C2×D127—84no—
121C2×D127—84no—
122C4×C127—22no—
123C4×C127#12011yescharacteristic
124C4×C127#12011yescharacteristic
125C4×C127C22×D412yes—
126Dic127—24no—
127Dic127—42no—
128C2×C127—21no—
129C2×C127—22no—
130C2×C127—42no—
131C2×C127C2×#12011yescharacteristic
132D127—44no—
133D127—82no—
134#3021—1271nomaximal, Sylow
135C127#302111yescharacteristic, complemented, Sylow
136#585—1271no—
137C2×AGL(1,ℤ/8)—1272no—
138#535—1271no—
139#454—1274no—
140#453—1274no—
141#421—1272no—
142#413D12711yescharacteristic, complemented
1432−1+4—1272no—
1442−1+4—2542no—
145H(5,𝔽2)—1272no—
146H(5,𝔽2)—2542no—
147H(5,𝔽2)—2542no—
148C2×#41—1272no—
149C2×#41—2544no—
150C2×#41—2542no—
151C2×#41C2×D12711yescharacteristic
152C22×D4—1271no—
153AGL(1,ℤ/8)—1278no—
154AGL(1,ℤ/8)—2544no—
155C2×SD16—2542no—
156C2×D8—2542no—
157C2×M16C2×D12712yescomplemented
158#127—1272no—
159#124—1272no—
160#121—2542no—
161#120—2542no—
162#117—1271no—
163#104—1278no—
164#101C2×D12712yescomplemented
165#100C2×D12712yescomplemented
166C24—2541no—
167#41—22no—
168#41—22no—
169#41—1274no—
170#41—1274no—
171#41—2544no—
172#41—2544no—
173#41—2544no—
174#41—2548no—
175#41—2544no—
176#41—2542no—
177#41—2542no—
178#41—2544no—
179C2×Q8—1272no—
180C2×Q8—2542no—
181C2×Q8—2544no—
182C2×Q8C22×D12711yescharacteristic
183C2×D4—1272no—
184C2×D4—1272no—
185C2×D4—2542no—
186C2×D4—2544no—
187C2×D4—2548no—
188C2×D4—2544no—
189C2×D4—2542no—
190C2×D4—2544no—
191C2×D4—2542no—
192C2×D4—2542no—
193C2×D4—2541no—
194C2×D4D50812yes—
195C2×D4C22×D12711yescharacteristic
196C22×C4—2544no—
197C22×C4—2542no—
198C22×C4D50812yes—
199C22×C4C22×D12711yescharacteristic
200SD16—2548no—
201D8—2548no—
202M16—22no—
203M16C22×D12714yes—
204C2×C8—22no—
205C4⋊C4—2541no—
206C22⋊C4—2541no—
207C22⋊C4—2544no—
208C42—1272no—
209C23—10162no—
210C23—5082no—
211C23—2544no—
212C23—2542no—
213C23—2542no—
214C23C2×D50812yes—
215C23C2×D50811yescharacteristic
216Q8—22no—
217Q8—1274no—
218Q8—2544no—
219Q8—2542no—
220D4—24no—
221D4—22no—
222D4—1274no—
223D4—10164no—
224D4—5081no—
225D4—5081no—
226D4—2544no—
227D4—2548no—
228D4—2544no—
229D4—2544no—
230D4—2544no—
231D4—2542no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—2548no—
235C2×C4—2542no—
236C2×C4—2544no—
237C2×C4—2542no—
238C2×C4—2544no—
239C2×C4—2544no—
240C2×C4D4×D12712yes—
241C2×C4D4×D12712yes—
242C2×C4C2×D50812yes—
243C2×C4C2×D50811yescharacteristic
244C2×C4C23×D12711yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—10164no—
250C22—10162no—
251C22—10164no—
252C22—5084no—
253C22—5082no—
254C22—2544no—
255C22#1686117912yes—
256C22C22×D50811yescharacteristic
257C4—22no—
258C4—5082no—
259C4—2544no—
260C4#1686117911yescharacteristic
261C4#1686117911yescharacteristic
262C4C2×D4×D12712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—10162no—
267C2—5084no—
268C2C2×#1686117911yescharacteristic
subgroupG/H
centerC2C2×#16861179
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#413×C127C2
socleC2×C127C2×#120
hypercenter#413D127
hypocenterC127#3021
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #22550712C2semidirect
2. C2×#16861237C2semidirect
3. #22550440C2semidirect
4. #22550051C2semidirect
5. #22550046C2semidirect
6. #22549892C2semidirect
7. #413×C127C2semidirect
8. #16861257C22semidirect
12. #16861254C22semidirect
16. C22×D508C22semidirect
18. #16861237C22semidirect
21. #16861205C22semidirect
22. #16861198C22semidirect
25. #16861168C22semidirect
26. #16861135C22semidirect
28. C2×M16×C127C22semidirect
29. #101×C127C22semidirect
30. #100×C127C22semidirect
38. #14523248D4semidirect
43. C2×Dic254D4semidirect
49. C2×D508D4semidirect
135. C127#3021semidirect
142. #413D127semidirect
157. C2×M16C2×D127semidirect
164. #101C2×D127semidirect
165. #100C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225507122
C2×#168612372
#225504402
#225500512
#225500462
#225498922
#413×C1272
#3021127