Finite Group Databasebeta

← #30647998

Subgroups — #30647998

Order 16256.

784 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 165 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×#16861234C211yesmaximal, characteristic, complemented
2C22×#14523233C211yesmaximal, characteristic, complemented
3#22550440C211yesmaximal, characteristic, complemented
4#22550021C214yesmaximal, complemented
5#22550010C214yesmaximal, complemented
6#22549881C211yesmaximal, characteristic, complemented
7#22549872C211yesmaximal, characteristic, complemented
8#22549843C211yesmaximal, characteristic, complemented
9C2×H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
10C22×D508C2211yescharacteristic, complemented
11#16861234—24no—
12C2×#14523235C2212yescomplemented
13C2×#14523233—24no—
14C2×#14523233C2214yescomplemented
15C2×#14523233C2212yescomplemented
16C2×#14523232C2211yescharacteristic, complemented
17C22×C127⋊C8C2211yescharacteristic, complemented
18#16861205—21no—
19#16861198—21no—
20#16861183C2214yescomplemented
21#16861177—22no—
22#16861168C2211yescharacteristic, complemented
23#16861133C2212yescomplemented
24#16861130C2214yescomplemented
25#16861130C2212yescomplemented
26C22⋊(C127⋊C8)C2214yescomplemented
27H(5,𝔽2)×C127—24no—
28H(5,𝔽2)×C127—24no—
29C2×#41×C127—21no—
30C2×#41×C127C2211yescharacteristic
31C2×#41×C127C2211yescharacteristic
32C2×#41×C127C2212yescomplemented
33C22×D4×C127—21no—
34C22×D4×C127—22no—
35C22×D4×C127C2211yescharacteristic
36C22×D4×C127C2212yescomplemented
37C23×D127—21no—
38C2×D508—22no—
39C2×D508—22no—
40C2×D508—21no—
41C2×D508C2312yes—
42C2×C22⋊D127—24no—
43#14523235—24no—
44#14523233—28no—
45#14523233—24no—
46#14523232—22no—
47C2×C127⋊C8—22no—
48C2×C127⋊C8C2312yes—
49C2×C127⋊C8C2312yes—
50C4⋊Dic127C2312yes—
51C22⋊Dic127—21no—
52(C2×D127)⋊C4—24no—
53C4×Dic127—21no—
54C24×C127—21no—
55C24×C127—22no—
56#41×C127—28no—
57#41×C127—24no—
58#41×C127—24no—
59#41×C127—22no—
60#41×C127—22no—
61#41×C127—22no—
62#41×C127D414yescomplemented
63C2×Q8×C127D411yescharacteristic
64C2×Q8×C127C2311yescharacteristic
65C2×D4×C127—216no—
66C2×D4×C127—28no—
67C2×D4×C127—21no—
68C2×D4×C127—22no—
69C2×D4×C127—24no—
70C2×D4×C127—24no—
71C2×D4×C127—21no—
72C2×D4×C127—22no—
73C2×D4×C127—22no—
74C2×D4×C127D411yescharacteristic
75C2×D4×C127D412yes—
76C2×D4×C127D414yescomplemented
77C2×D4×C127C2312yes—
78C2×D4×C127C2311yescharacteristic
79C22×C4×C127—21no—
80C22×C4×C127—22no—
81C22×C4×C127C2312yes—
82C22×C4×C127C2311yescharacteristic
83C22×D127—22no—
84C22×D127—42no—
85C22×D127—82no—
86D508—22no—
87D508—22no—
88D508—42no—
89C22⋊D127—84no—
90C2×Dic127—22no—
91C127⋊C8—22no—
92C127⋊C8—22no—
93C23×C127—28no—
94C23×C127—21no—
95C23×C127—22no—
96C23×C127—24no—
97C23×C127—82no—
98C23×C127—84no—
99C23×C127C2×D412yes—
100C23×C127C2×D411yescharacteristic
101Q8×C127—22no—
102Q8×C127—21no—
103Q8×C127C2×D412yes—
104D4×C127—28no—
105D4×C127—28no—
106D4×C127—24no—
107D4×C127—24no—
108D4×C127—22no—
109D4×C127—24no—
110D4×C127—22no—
111D4×C127—21no—
112D4×C127C2×D412yes—
113D4×C127C2×D414yes—
114C2×C4×C127—28no—
115C2×C4×C127—21no—
116C2×C4×C127—24no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127C2×D412yes—
120C2×C4×C127C2×D411yescharacteristic
121C2×C4×C127C2×D412yes—
122C2×C4×C127C2×D412yes—
123C2×C4×C127C2411yescharacteristic
124C22×C127—22no—
125C22×C127—22no—
126C22×C127—24no—
127C22×C127—22no—
128C22×C127—24no—
129C22×C127—44no—
130C22×C127—42no—
131C22×C127—84no—
132C22×C127—88no—
133C22×C127#12012yes—
134C22×C127C22×D411yescharacteristic
135C2×D127—42no—
136C2×D127—44no—
137C2×D127—84no—
138C4×C127—22no—
139C4×C127—22no—
140C4×C127#12012yes—
141C4×C127C22×D411yescharacteristic
142C4×C127C22×D411yescharacteristic
143Dic127—42no—
144C2×C127—22no—
145C2×C127—42no—
146C2×C127—44no—
147C2×C127—42no—
148C2×C127C2×#12011yescharacteristic
149C2×C127#57612yes—
150D127—82no—
151#3008—1271nomaximal, Sylow
152C127#300811yescharacteristic, complemented, Sylow
153C2×H(5,𝔽2)D12711yescharacteristic, complemented
154C2×AGL(1,ℤ/8)—1271no—
155C22×D8—1271no—
156#535—1271no—
157#449—1274no—
158#447—1274no—
159#418—1271no—
160#416—1271no—
161#408—1271no—
162H(5,𝔽2)—24no—
163H(5,𝔽2)—24no—
164C2×#41—21no—
165C2×#41C2×D12711yescharacteristic
166C2×#41C2×D12711yescharacteristic
167C2×#41C2×D12712yescomplemented
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—1271no—
171C22×D4C2×D12711yescharacteristic
172C22×D4C2×D12712yescomplemented
173AGL(1,ℤ/8)—2544no—
174C2×SD16—1272no—
175C2×D8—1274no—
176C2×D8—1272no—
177C2×D8—2544no—
178C2×M16—1271no—
179C22×C8—1271no—
180#127—2541no—
181#124—2541no—
182#121—1274no—
183#120—2542no—
184#117—1271no—
185#103—1272no—
186#102—1272no—
187#102—1274no—
188C22⋊C8—1274no—
189C24—21no—
190C24—22no—
191C24—2541no—
192#41—28no—
193#41—24no—
194#41—24no—
195#41—22no—
196#41—22no—
197#41—22no—
198#41C22⋊D12714yescomplemented
199C2×Q8C22⋊D12711yescharacteristic
200C2×Q8C22×D12711yescharacteristic
201C2×D4—216no—
202C2×D4—28no—
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—1272no—
211C2×D4—2544no—
212C2×D4—2541no—
213C2×D4—2542no—
214C2×D4—2542no—
215C2×D4C22⋊D12712yes—
216C2×D4C22⋊D12711yescharacteristic
217C2×D4C22⋊D12714yescomplemented
218C2×D4C22×D12712yes—
219C2×D4C22×D12711yescharacteristic
220C22×C4—21no—
221C22×C4—22no—
222C22×C4C22×D12712yes—
223C22×C4C22×D12711yescharacteristic
224SD16—2544no—
225D8—2548no—
226D8—2544no—
227M16—2542no—
228C2×C8—1272no—
229C2×C8—1272no—
230C2×C8—2542no—
231C4⋊C4—1272no—
232C22⋊C4—2541no—
233C22⋊C4—2544no—
234C42—2541no—
235C23—28no—
236C23—21no—
237C23—22no—
238C23—24no—
239C23—82no—
240C23—84no—
241C23—10162no—
242C23—5082no—
243C23—2542no—
244C23C2×C22⋊D12712yes—
245C23C2×C22⋊D12711yescharacteristic
246Q8—21no—
247Q8—22no—
248Q8C2×C22⋊D12712yes—
249D4—28no—
250D4—28no—
251D4—24no—
252D4—24no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—24no—
257D4—10164no—
258D4—5082no—
259D4—2542no—
260D4—2542no—
261D4C2×C22⋊D12712yes—
262D4C2×C22⋊D12714yes—
263C2×C4—28no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—2542no—
269C2×C4C2×C22⋊D12712yes—
270C2×C4C2×C22⋊D12712yes—
271C2×C4C2×C22⋊D12712yes—
272C2×C4C2×C22⋊D12711yescharacteristic
273C2×C4C23×D12711yescharacteristic
274C8—2542no—
275C8—2542no—
276C22—22no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—24no—
281C22—44no—
282C22—42no—
283C22—84no—
284C22—88no—
285C22—10164no—
286C22—5082no—
287C22—5084no—
288C22#1686117812yes—
289C22C22×C22⋊D12711yescharacteristic
290C4—22no—
291C4—22no—
292C4—5082no—
293C4#1686117812yes—
294C4C22×C22⋊D12711yescharacteristic
295C4C22×C22⋊D12711yescharacteristic
296C2—22no—
297C2—42no—
298C2—44no—
299C2—42no—
300C2—10162no—
301C2C2×#1686117811yescharacteristic
302C2#2255067912yes—
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×H(5,𝔽2)×C127C2
socleC22×C127C22×D4
hypercenterC2×H(5,𝔽2)D127
hypocenterC127#3008
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16861234C2semidirect
2. C22×#14523233C2semidirect
3. #22550440C2semidirect
4. #22550021C2semidirect
5. #22550010C2semidirect
6. #22549881C2semidirect
7. #22549872C2semidirect
8. #22549843C2semidirect
9. C2×H(5,𝔽2)×C127C2semidirect
10. C22×D508C22semidirect
12. C2×#14523235C22semidirect
14. C2×#14523233C22semidirect
15. C2×#14523233C22semidirect
16. C2×#14523232C22semidirect
17. C22×C127⋊C8C22semidirect
20. #16861183C22semidirect
22. #16861168C22semidirect
23. #16861133C22semidirect
24. #16861130C22semidirect
25. #16861130C22semidirect
26. C22⋊(C127⋊C8)C22semidirect
32. C2×#41×C127C22semidirect
36. C22×D4×C127C22semidirect
62. #41×C127D4semidirect
76. C2×D4×C127D4semidirect
152. C127#3008semidirect
153. C2×H(5,𝔽2)D127semidirect
167. C2×#41C2×D127semidirect
172. C22×D4C2×D127semidirect
198. #41C22⋊D127semidirect
217. C2×D4C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612342
C22×#145232332
#225504402
#225500212
#225500102
#225498812
#225498722
#225498432
C2×H(5,𝔽2)×C1272
#3008127