Finite Group Databasebeta

← #30648014

Subgroups — #30648014

Order 16256.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16861256C211yesmaximal, characteristic, complemented
2C2×#16861235C211yesmaximal, characteristic, complemented
3C22×#14523236C211yesmaximal, characteristic, complemented
4#22550019C214yesmaximal, complemented
5#22550014C214yesmaximal, complemented
6#22549884C211yesmaximal, characteristic, complemented
7#22549870C211yesmaximal, characteristic, complemented
8#22549843C211yesmaximal, characteristic, complemented
9#535×C127C211yesmaximal, characteristic, complemented
10#16861256—24no—
11#16861256—24no—
12C2×#14523248C2211yescharacteristic
13C2×#14523248C2212yescomplemented
14C2×#14523247—22no—
15C2×#14523247—21no—
16C22×Dic254C2211yescharacteristic
17C22×Dic254C2212yescomplemented
18C2×Q8×D127—21no—
19#16861235—24no—
20C2×#14523236—24no—
21C2×#14523236C2214yes—
22C2×#14523236C2212yes—
23C2×#14523233C2212yes—
24C2×#14523232C2211yescharacteristic
25C22×C127⋊C8C2211yescharacteristic
26#16861132C2214yescomplemented
27#16861132C2212yescomplemented
28#16861129C2212yescomplemented
29C22⋊(C127⋊C8)C2214yescomplemented
30C2×#41×C127C2211yescharacteristic
31C22×D4×C127C2211yescharacteristic
32#127×C127—21no—
33#124×C127—21no—
34#121×C127C2214yes—
35#120×C127—22no—
36#117×C127C2211yescharacteristic
37#14523248—24no—
38#14523248—24no—
39#14523248—22no—
40#14523248D414yescomplemented
41#14523247—216no—
42#14523247—28no—
43C2×Dic254—24no—
44C2×Dic254—24no—
45C2×Dic254—21no—
46C2×Dic254—22no—
47C2×Dic254—22no—
48C2×Dic254D414yescomplemented
49C2×Dic254C2312yes—
50C2×Dic254C2311yescharacteristic
51Q8×D127—28no—
52C2×D508C2311yescharacteristic
53C2×C22⋊D127—21no—
54C2×C22⋊D127—22no—
55C22×Dic127—21no—
56C22×Dic127—22no—
57C2×C4×D127—21no—
58C2×C4×D127—22no—
59#14523236—28no—
60#14523236—24no—
61#14523233—24no—
62#14523232—22no—
63C2×C127⋊C8—22no—
64C2×C127⋊C8C2312yes—
65C2×C127⋊C8C2312yes—
66C24×C127—21no—
67#41×C127—22no—
68#41×C127—22no—
69C2×Q8×C127D411yescharacteristic
70C2×D4×C127—24no—
71C2×D4×C127—22no—
72C2×D4×C127—22no—
73C2×D4×C127—21no—
74C2×D4×C127D412yes—
75C2×D4×C127D411yescharacteristic
76C2×D4×C127C2312yes—
77C22×C4×C127C2312yes—
78C22×C4×C127C2311yescharacteristic
79C4⋊C4×C127C2312yes—
80C22⋊C4×C127—21no—
81C22⋊C4×C127—24no—
82C42×C127—21no—
83C22×D127—21no—
84Dic254—28no—
85Dic254—24no—
86Dic254—21no—
87Dic254—22no—
88Dic254C2×D412yes—
89Dic254C2×D414yes—
90D508—21no—
91D508C2×D412yes—
92C22⋊D127—28no—
93C22⋊D127—24no—
94C2×Dic127—28no—
95C2×Dic127—24no—
96C2×Dic127—21no—
97C2×Dic127—22no—
98C4×D127—28no—
99C4×D127—24no—
100C127⋊C8—22no—
101C127⋊C8—22no—
102C23×C127—22no—
103C23×C127—42no—
104C23×C127—82no—
105C23×C127C2×D412yes—
106C23×C127C2×D411yescharacteristic
107Q8×C127—22no—
108D4×C127—24no—
109D4×C127—22no—
110D4×C127—22no—
111D4×C127—22no—
112D4×C127—42no—
113D4×C127—84no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127C2×D412yes—
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2411yescharacteristic
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—24no—
125C22×C127—42no—
126C22×C127—44no—
127C22×C127—84no—
128C22×C127#12012yes—
129C22×C127C22×D411yescharacteristic
130C2×D127—22no—
131C2×D127—42no—
132C4×C127—22no—
133C4×C127—42no—
134C4×C127#12012yes—
135C4×C127C22×D411yescharacteristic
136C4×C127C22×D411yescharacteristic
137Dic127—24no—
138Dic127—22no—
139C2×C127—22no—
140C2×C127—42no—
141C2×C127—82no—
142C2×C127C2×#12011yescharacteristic
143C2×C127#57712yes—
144D127—42no—
145#3010—1271nomaximal, Sylow
146C127#301011yescharacteristic, complemented, Sylow
147C2×2−1+4—1271no—
148C2×AGL(1,ℤ/8)—1271no—
149C22×SD16—1271no—
150#535D12711yescharacteristic, complemented
151#449—1274no—
152#448—1274no—
153#419—1271no—
154#416—1271no—
155#408—1271no—
1562−1+4—2544no—
1572−1+4—2544no—
158C2×#41—1271no—
159C2×#41—1272no—
160C2×#41—2542no—
161C2×#41—2541no—
162C2×#41C2×D12711yescharacteristic
163C22×Q8—1271no—
164C22×Q8—1272no—
165C22×Q8—2541no—
166C22×D4C2×D12711yescharacteristic
167AGL(1,ℤ/8)—2544no—
168C2×SD16—1274no—
169C2×SD16—1272no—
170C2×SD16—2544no—
171C2×D8—1272no—
172C2×M16—1271no—
173C22×C8—1271no—
174#127—21no—
175#124—21no—
176#121C2×D12714yes—
177#120—22no—
178#117C2×D12711yescharacteristic
179#103—1272no—
180#103—1274no—
181#102—1272no—
182C22⋊C8—1274no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—1274no—
187#41—2544no—
188#41—2544no—
189#41—25416no—
190#41—2548no—
191#41—2542no—
192C2×Q8—1274no—
193C2×Q8—1272no—
194C2×Q8—1271no—
195C2×Q8—2544no—
196C2×Q8—2544no—
197C2×Q8—2548no—
198C2×Q8—2542no—
199C2×Q8—2542no—
200C2×Q8—2541no—
201C2×Q8C22⋊D12711yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—1271no—
207C2×D4—2542no—
208C2×D4—2541no—
209C2×D4C22⋊D12712yes—
210C2×D4C22⋊D12711yescharacteristic
211C2×D4C22×D12712yes—
212C22×C4—2542no—
213C22×C4—2541no—
214C22×C4—2542no—
215C22×C4—2541no—
216C22×C4C22×D12712yes—
217C22×C4C22×D12711yescharacteristic
218SD16—2548no—
219SD16—2544no—
220D8—2544no—
221M16—2542no—
222C2×C8—1272no—
223C2×C8—1272no—
224C2×C8—2542no—
225C4⋊C4C22×D12712yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—2541no—
233C23C2×C22⋊D12712yes—
234C23C2×C22⋊D12711yescharacteristic
235Q8—22no—
236Q8—1274no—
237Q8—1272no—
238Q8—2548no—
239Q8—2542no—
240Q8—2541no—
241Q8—2544no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—1272no—
249D4—2548no—
250D4—2544no—
251D4—2541no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—2548no—
256C2×C4—2548no—
257C2×C4—2542no—
258C2×C4—2541no—
259C2×C4—2544no—
260C2×C4—2544no—
261C2×C4D4×D12712yes—
262C2×C4D4×D12712yes—
263C2×C4C2×C22⋊D12712yes—
264C2×C4C2×C22⋊D12711yescharacteristic
265C2×C4C23×D12711yescharacteristic
266C8—2542no—
267C8—2542no—
268C22—22no—
269C22—22no—
270C22—24no—
271C22—42no—
272C22—44no—
273C22—84no—
274C22—5082no—
275C22—2542no—
276C22#1686117712yes—
277C22C22×C22⋊D12711yescharacteristic
278C4—22no—
279C4—42no—
280C4—2544no—
281C4—2542no—
282C4#1686117712yes—
283C4C2×D4×D12711yescharacteristic
284C4C2×D4×D12711yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—5082no—
289C2C2×#1686117711yescharacteristic
290C2#2255068712yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#535×C127C2
socleC22×C127C22×D4
hypercenter#535D127
hypocenterC127#3010
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16861256C2semidirect
2. C2×#16861235C2semidirect
3. C22×#14523236C2semidirect
4. #22550019C2semidirect
5. #22550014C2semidirect
6. #22549884C2semidirect
7. #22549870C2semidirect
8. #22549843C2semidirect
9. #535×C127C2semidirect
13. C2×#14523248C22semidirect
17. C22×Dic254C22semidirect
26. #16861132C22semidirect
27. #16861132C22semidirect
28. #16861129C22semidirect
29. C22⋊(C127⋊C8)C22semidirect
40. #14523248D4semidirect
48. C2×Dic254D4semidirect
146. C127#3010semidirect
150. #535D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612562
C2×#168612352
C22×#145232362
#225500192
#225500142
#225498842
#225498702
#225498432
#535×C1272
#3010127