Finite Group Databasebeta

← #30650332

Subgroups — #30650332

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×#16861237C212yesmaximal, complemented
2#22550531C211yesmaximal, characteristic, complemented
3D4×D508C211yesmaximal, characteristic, complemented
4#22550249C211yesmaximal, characteristic, complemented
5#22550122C212yesmaximal, complemented
6#22550082C211yesmaximal, characteristic, complemented
7#22550072C211yesmaximal, characteristic, complemented
8#22550029C212yesmaximal, complemented
9#22550023C212yesmaximal, complemented
10C4×D1016C211yesmaximal, characteristic, complemented
11#436×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yescomplemented
13C22×D508C2212yes—
14C2×D4×D127—21no—
15#16861237—28no—
16C2×#14523237C2212yes—
17C2×#14523237C2212yescomplemented
18C2×#14523237C2212yescomplemented
19C2×D1016C2211yescharacteristic
20C2×D1016C2212yescomplemented
21#16861209C2211yescharacteristic, complemented
22#16861206C2211yescharacteristic
23#16861191—22no—
24#16861188C2212yescomplemented
25#16861184—21no—
26#16861183C2212yes—
27#16861180—21no—
28#16861179—22no—
29C4×D508C2211yescharacteristic
30C4×D508C2211yescharacteristic, complemented
31C4⋊C4×D127—21no—
32#16861143C2211yescharacteristic, complemented
33#16861134C2212yescomplemented
34#16861131C2211yescharacteristic
35#16861131C2211yescharacteristic, complemented
36#16861131C2212yescomplemented
37C2×M16×C127C2212yescomplemented
38C4×D4×C127C2211yescharacteristic
39C4⋊C8×C127C2211yescharacteristic, complemented
40C22⋊C8×C127C2212yescomplemented
41C4×C8×C127C2211yescharacteristic, complemented
42C23×D127—22no—
43#14523248—24no—
44#14523248—42no—
45C2×Dic254C2312yes—
46C2×D508—24no—
47C2×D508—21no—
48C2×D508—22no—
49C2×D508—21no—
50C2×D508—42no—
51C2×D508C2312yes—
52C2×D508C2311yescharacteristic
53C2×D508C2311yescharacteristic
54C2×C22⋊D127—22no—
55C2×C22⋊D127—22no—
56D4×D127—44no—
57C2×C4×D127—21no—
58C2×C4×D127—22no—
59C2×C4×D127—21no—
60#14523237—24no—
61#14523237—24no—
62#14523237—42no—
63D1016—24no—
64D1016D414yescomplemented
65C4⋊Dic127—21no—
66C4⋊Dic127—21no—
67C4⋊Dic127C2311yescharacteristic
68C4⋊Dic127C2311yescharacteristic
69Dic127⋊C4—22no—
70C22⋊Dic127—22no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—21no—
73(C2×D127)⋊C4—21no—
74C2×D4×C127D411yescharacteristic
75C22×C4×C127C2312yes—
76M16×C127—24no—
77C2×C8×C127C2312yes—
78C2×C8×C127C2311yescharacteristic
79C2×C8×C127C2311yescharacteristic
80C4⋊C4×C127D411yescharacteristic
81C22⋊C4×C127D412yes—
82C42×C127C2311yescharacteristic
83C22×D127—21no—
84C22×D127—21no—
85C22×D127—22no—
86C22×D127—44no—
87C22×D127—44no—
88C22×D127—82no—
89Dic254—22no—
90Dic254—22no—
91D508—22no—
92D508—21no—
93D508—21no—
94D508—22no—
95D508—44no—
96D508—42no—
97D508—81no—
98D508C2×D412yes—
99D508C2×D412yescomplemented
100C22⋊D127—44no—
101C22⋊D127—44no—
102C2×Dic127—21no—
103C2×Dic127—21no—
104C2×Dic127—22no—
105C4×D127—44no—
106C4×D127—42no—
107C4×D127—42no—
108C23×C127C2×D412yes—
109D4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127C2×D411yescharacteristic
113C2×C4×C127C2×D412yes—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2411yescharacteristic
116C8×C127—22no—
117C8×C127—21no—
118C8×C127C2×D412yescomplemented
119C22×C127—22no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127C22×D411yescharacteristic
123C2×D127—22no—
124C2×D127—22no—
125C2×D127—41no—
126C2×D127—41no—
127C2×D127—41no—
128C2×D127—41no—
129C2×D127—42no—
130C2×D127—42no—
131C2×D127—42no—
132C2×D127—88no—
133C4×C127—22no—
134C4×C127—21no—
135C4×C127AGL(1,ℤ/8)12yescomplemented
136C4×C127C22×D411yescharacteristic
137C4×C127H(5,𝔽2)11yescharacteristic
138Dic127—42no—
139Dic127—41no—
140Dic127—41no—
141C2×C127—42no—
142C2×C127D4211yescharacteristic
143C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
144C2×C127#57711yescharacteristic
145D127—42no—
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×D8—1271no—
160#436D12711yescharacteristic, complemented
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—1271no—
169C2×D8—1272no—
170C2×M16C2×D12712yescomplemented
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×D4—1271no—
181C4×D4C2×D12711yescharacteristic
182C2×C4⋊C4—2541no—
183#107—1271no—
184C4⋊C8C2×D12711yescharacteristic, complemented
185#103—1272no—
186#102—1271no—
187#102—1271no—
188#102—1272no—
189C22⋊C8C2×D12712yescomplemented
190C4×C8C2×D12711yescharacteristic, complemented
191C24—2542no—
192#41—5082no—
193#41—2544no—
194C2×Q8—1272no—
195C2×D4—1271no—
196C2×D4—1271no—
197C2×D4—1272no—
198C2×D4—5082no—
199C2×D4—5084no—
200C2×D4—2544no—
201C2×D4—2542no—
202C2×D4—2542no—
203C2×D4—2541no—
204C2×D4—2542no—
205C2×D4—2541no—
206C2×D4D50811yescharacteristic
207C22×C4—2541no—
208C22×C4—2542no—
209C22×C4—2541no—
210C22×C4C22×D12712yes—
211SD16—5082no—
212SD16—2544no—
213SD16—2544no—
214D8—1274no—
215D8—2544no—
216M16—24no—
217C2×C8C22×D12711yescharacteristic
218C2×C8C22×D12712yes—
219C2×C8C22×D12711yescharacteristic
220C4⋊C4—1271no—
221C4⋊C4—1271no—
222C4⋊C4—2542no—
223C4⋊C4—2541no—
224C4⋊C4—2541no—
225C4⋊C4D50811yescharacteristic
226C22⋊C4—2542no—
227C22⋊C4—2541no—
228C22⋊C4—2541no—
229C22⋊C4—2542no—
230C22⋊C4D50812yes—
231C42C22×D12711yescharacteristic
232C23—10162no—
233C23—5084no—
234C23—5084no—
235C23—2542no—
236C23—2541no—
237C23—2541no—
238C23C2×D50812yes—
239Q8—2542no—
240Q8—2542no—
241D4—22no—
242D4—1272no—
243D4—1272no—
244D4—10161no—
245D4—5084no—
246D4—5084no—
247D4—5084no—
248D4—5082no—
249D4—2542no—
250D4—2542no—
251D4—2541no—
252D4—2541no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—5084no—
256C2×C4—5082no—
257C2×C4—5082no—
258C2×C4—2541no—
259C2×C4—2541no—
260C2×C4—2542no—
261C2×C4C2×D50811yescharacteristic
262C2×C4C2×D50812yes—
263C2×C4C2×D50811yescharacteristic
264C2×C4C23×D12711yescharacteristic
265C8—22no—
266C8—21no—
267C8D4×D12712yescomplemented
268C22—22no—
269C22—22no—
270C22—22no—
271C22—10168no—
272C22—5082no—
273C22—5082no—
274C22—5082no—
275C22—5081no—
276C22—5081no—
277C22—5081no—
278C22—5081no—
279C22—2542no—
280C22—2542no—
281C22C22×D50811yescharacteristic
282C4—21no—
283C4—22no—
284C4—5081no—
285C4—5081no—
286C4—5082no—
287C4#1686123712yescomplemented
288C4C2×D4×D12711yescharacteristic
289C4#1686125511yescharacteristic
290C2—42no—
291C2—10162no—
292C2—5082no—
293C2—5082no—
294C2D4×D50811yescharacteristic
295C2C2×#1686123711yescharacteristic
296C2#2255068811yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#436×C127C2
socleC22×C127C22×D4
hypercenter#436D127
hypocenterC127#3272
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ #436

Complements

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

NH ≅ G/Ntype
1. C2×#16861237C2semidirect
2. #22550531C2semidirect
3. D4×D508C2semidirect
4. #22550249C2semidirect
5. #22550122C2semidirect
6. #22550082C2semidirect
7. #22550072C2semidirect
8. #22550029C2semidirect
9. #22550023C2semidirect
10. C4×D1016C2semidirect
11. #436×C127C2semidirect
12. C2×#14523248C22semidirect
17. C2×#14523237C22semidirect
18. C2×#14523237C22semidirect
20. C2×D1016C22semidirect
21. #16861209C22semidirect
24. #16861188C22semidirect
30. C4×D508C22semidirect
32. #16861143C22semidirect
33. #16861134C22semidirect
35. #16861131C22semidirect
36. #16861131C22semidirect
37. C2×M16×C127C22semidirect
39. C4⋊C8×C127C22semidirect
40. C22⋊C8×C127C22semidirect
41. C4×C8×C127C22semidirect
64. D1016D4semidirect
99. D508C2×D4semidirect
118. C8×C127C2×D4semidirect
135. C4×C127AGL(1,ℤ/8)semidirect
149. C127#3272semidirect
160. #436D127semidirect
170. C2×M16C2×D127semidirect
184. C4⋊C8C2×D127semidirect
189. C22⋊C8C2×D127semidirect
190. C4×C8C2×D127semidirect
267. C8D4×D127semidirect
287. C4#16861237semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
#225505312
D4×D5082
#225502492
#225501222
#225500822
#225500722
#225500292
#225500232
C4×D10162
#436×C1272
#3272127