Finite Group Databasebeta

← #30650331

Subgroups — #30650331

Order 16256.

546 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16861233C212yesmaximal, complemented
2#22550530C211yesmaximal, characteristic, complemented
3D4×D508C211yesmaximal, characteristic, complemented
4#22550249C211yesmaximal, characteristic, complemented
5#22550123C212yesmaximal, complemented
6#22550081C211yesmaximal, characteristic, complemented
7#22550071C211yesmaximal, characteristic, complemented
8#22550028C212yesmaximal, complemented
9#22550022C212yesmaximal, complemented
10#22549955C211yesmaximal, characteristic, complemented
11C4×D8×C127C211yesmaximal, characteristic, complemented
12C2×#14523247C2212yescomplemented
13C22×D508—21no—
14C2×D4×D127C2212yes—
15#16861233—28no—
16C2×#14523237C2212yescomplemented
17C2×#14523237C2212yescomplemented
18C2×#14523236C2212yes—
19C2×#14523233C2212yescomplemented
20C2×#14523231C2212yescomplemented
21#16861209C2211yescharacteristic, complemented
22#16861206C2211yescharacteristic
23#16861193—22no—
24#16861189C2212yescomplemented
25#16861184C2212yes—
26#16861183—21no—
27#16861183—21no—
28#16861179—22no—
29C4×D508C2211yescharacteristic
30C2×C4⋊Dic127—21no—
31C4⋊(C127⋊C8)C2211yescharacteristic, complemented
32#16861132C2212yescomplemented
33#16861129C2212yescomplemented
34(C2×D127)⋊C8C2212yescomplemented
35C2×D8×C127C2211yescharacteristic
36C4×D4×C127C2211yescharacteristic
37C4×D4×C127C2211yescharacteristic, complemented
38#107×C127C2211yescharacteristic, complemented
39#102×C127C2211yescharacteristic
40#102×C127C2211yescharacteristic, complemented
41C4×C8×C127C2211yescharacteristic, complemented
42C23×D127—22no—
43#14523247—24no—
44#14523247—42no—
45C2×Dic254C2312yes—
46C2×D508—22no—
47C2×D508—21no—
48C2×D508—21no—
49C2×D508—44no—
50C2×D508D411yescharacteristic
51C2×D508C2312yes—
52C2×C22⋊D127—22no—
53C2×C22⋊D127—22no—
54D4×D127—24no—
55D4×D127—42no—
56C22×Dic127—22no—
57C2×C4×D127C2312yes—
58#14523237—24no—
59#14523237—42no—
60#14523236—24no—
61#14523233—24no—
62#14523231—24no—
63C2×C127⋊C8C2312yes—
64C4⋊Dic127—22no—
65C4⋊Dic127—21no—
66C4⋊Dic127—21no—
67C4⋊Dic127D411yescharacteristic
68(C2×D127)⋊C4—22no—
69(C2×D127)⋊C4—22no—
70(C2×D127)⋊C4D412yes—
71C2×D4×C127C2311yescharacteristic
72C2×D4×C127C2311yescharacteristic
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75D8×C127D414yescomplemented
76C2×C8×C127C2311yescharacteristic
77C2×C8×C127C2311yescharacteristic
78C4⋊C4×C127C2311yescharacteristic
79C4⋊C4×C127C2311yescharacteristic
80C22⋊C4×C127—21no—
81C22⋊C4×C127—21no—
82C42×C127C2311yescharacteristic
83C22×D127—22no—
84C22×D127—44no—
85C22×D127—44no—
86C22×D127—82no—
87C22×D127C2×D412yes—
88Dic254—22no—
89Dic254—22no—
90D508—22no—
91D508—22no—
92D508—22no—
93D508—44no—
94D508—42no—
95D508—81no—
96C22⋊D127—44no—
97C22⋊D127—44no—
98C2×Dic127—22no—
99C2×Dic127—44no—
100C2×Dic127C2×D412yes—
101C4×D127—22no—
102C4×D127—22no—
103C127⋊C8—22no—
104C23×C127—21no—
105C23×C127—21no—
106D4×C127—21no—
107D4×C127—21no—
108D4×C127C2×D412yes—
109D4×C127C2×D412yescomplemented
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2411yescharacteristic
117C8×C127—21no—
118C8×C127C2×D412yescomplemented
119C22×C127—22no—
120C22×C127—22no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—41no—
125C22×C127C22×D411yescharacteristic
126C2×D127—22no—
127C2×D127—22no—
128C2×D127—22no—
129C2×D127—42no—
130C2×D127—42no—
131C2×D127—42no—
132C2×D127—88no—
133C4×C127—21no—
134C4×C127—41no—
135C4×C127—41no—
136C4×C127AGL(1,ℤ/8)12yescomplemented
137C4×C127C22×D411yescharacteristic
138C4×C127H(5,𝔽2)11yescharacteristic
139Dic127—22no—
140Dic127—42no—
141C2×C127—42no—
142C2×C127—42no—
143C2×C127D4211yescharacteristic
144C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
145C2×C127#57711yescharacteristic
146D127—42no—
147D127—82no—
148#3272—1271nomaximal, Sylow
149C127#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1272no—
151#547—1271no—
152D42—1271no—
153#492—1271no—
154#468—1272no—
155#461—1271no—
156#459—1271no—
157#450—1272no—
158#449—1272no—
159C4×D8D12711yescharacteristic, complemented
160#436—1271no—
161C2×#41—1272no—
162C22×D4—1272no—
163C22×D4—2541no—
164AGL(1,ℤ/8)—2548no—
165C2×SD16—1272no—
166C2×SD16—1272no—
167C2×SD16—1272no—
168C2×D8—1272no—
169C2×D8C2×D12711yescharacteristic
170C2×M16—1272no—
171C4⋊Q8—1271no—
172#127—1271no—
173#123—2542no—
174C22⋊Q8—1272no—
175#121—1272no—
176#121—2541no—
177#121—2541no—
178#120—2542no—
179C4×D4—1271no—
180C4×D4C2×D12711yescharacteristic
181C4×D4C2×D12711yescharacteristic, complemented
182C2×C4⋊C4—2541no—
183#107C2×D12711yescharacteristic, complemented
184C4⋊C8—1271no—
185#103—1272no—
186#102—1272no—
187#102C2×D12711yescharacteristic
188#102C2×D12711yescharacteristic, complemented
189C22⋊C8—1272no—
190C4×C8C2×D12711yescharacteristic, complemented
191C24—2542no—
192#41—5082no—
193#41—2544no—
194C2×Q8—1272no—
195C2×D4—1271no—
196C2×D4—1272no—
197C2×D4—5084no—
198C2×D4—5082no—
199C2×D4—2542no—
200C2×D4—2542no—
201C2×D4—2542no—
202C2×D4—2544no—
203C2×D4—2541no—
204C2×D4—2541no—
205C2×D4C22×D12711yescharacteristic
206C2×D4C22×D12711yescharacteristic
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—1272no—
210C22×C4—2542no—
211SD16—5082no—
212SD16—2544no—
213SD16—2544no—
214D8—2544no—
215D8D50814yescomplemented
216M16—2544no—
217C2×C8—1272no—
218C2×C8C22×D12711yescharacteristic
219C2×C8C22×D12711yescharacteristic
220C4⋊C4—1271no—
221C4⋊C4—2542no—
222C4⋊C4—2541no—
223C4⋊C4—2541no—
224C4⋊C4C22×D12711yescharacteristic
225C4⋊C4C22×D12711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—21no—
228C22⋊C4—1272no—
229C22⋊C4—2542no—
230C22⋊C4—2542no—
231C42C22×D12711yescharacteristic
232C23—21no—
233C23—21no—
234C23—1272no—
235C23—10162no—
236C23—5084no—
237C23—5084no—
238C23—2542no—
239Q8—2542no—
240Q8—2542no—
241D4—21no—
242D4—21no—
243D4—10161no—
244D4—5084no—
245D4—5084no—
246D4—5084no—
247D4—5082no—
248D4—2542no—
249D4—2542no—
250D4—2542no—
251D4C2×D50812yes—
252D4C2×D50812yescomplemented
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—1272no—
258C2×C4—5084no—
259C2×C4—2542no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4D4×D12711yescharacteristic
263C2×C4D4×D12711yescharacteristic
264C2×C4C23×D12711yescharacteristic
265C8—21no—
266C8—2542no—
267C8C2×D50812yescomplemented
268C22—22no—
269C22—22no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—10168no—
275C22—5082no—
276C22—5082no—
277C22—5082no—
278C22—2542no—
279C22—2542no—
280C22—2542no—
281C22C2×D4×D12711yescharacteristic
282C4—21no—
283C4—41no—
284C4—41no—
285C4—5082no—
286C4—2542no—
287C4#1686123312yescomplemented
288C4C22×D50811yescharacteristic
289C4#1686125511yescharacteristic
290C2—42no—
291C2—42no—
292C2—10162no—
293C2—5082no—
294C2D4×D50811yescharacteristic
295C2C2×#1686123311yescharacteristic
296C2#2255068311yescharacteristic
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC4×D8×C127C2
socleC22×C127C22×D4
hypercenterC4×D8D127
hypocenterC127#3272
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ C4×D8

Complements

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

NH ≅ G/Ntype
1. C2×#16861233C2semidirect
2. #22550530C2semidirect
3. D4×D508C2semidirect
4. #22550249C2semidirect
5. #22550123C2semidirect
6. #22550081C2semidirect
7. #22550071C2semidirect
8. #22550028C2semidirect
9. #22550022C2semidirect
10. #22549955C2semidirect
11. C4×D8×C127C2semidirect
12. C2×#14523247C22semidirect
16. C2×#14523237C22semidirect
17. C2×#14523237C22semidirect
19. C2×#14523233C22semidirect
20. C2×#14523231C22semidirect
21. #16861209C22semidirect
24. #16861189C22semidirect
31. C4⋊(C127⋊C8)C22semidirect
32. #16861132C22semidirect
33. #16861129C22semidirect
34. (C2×D127)⋊C8C22semidirect
37. C4×D4×C127C22semidirect
38. #107×C127C22semidirect
40. #102×C127C22semidirect
41. C4×C8×C127C22semidirect
75. D8×C127D4semidirect
109. D4×C127C2×D4semidirect
118. C8×C127C2×D4semidirect
136. C4×C127AGL(1,ℤ/8)semidirect
149. C127#3272semidirect
159. C4×D8D127semidirect
181. C4×D4C2×D127semidirect
183. #107C2×D127semidirect
188. #102C2×D127semidirect
190. C4×C8C2×D127semidirect
215. D8D508semidirect
252. D4C2×D508semidirect
267. C8C2×D508semidirect
287. C4#16861233semidirect

Maximal subgroups

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

subgroupindex
C2×#168612332
#225505302
D4×D5082
#225502492
#225501232
#225500812
#225500712
#225500282
#225500222
#225499552
C4×D8×C1272
#3272127