Finite Group Databasebeta

← #30650352

Subgroups — #30650352

Order 16256.

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

normal subgroups only

This group has 556 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#22550547C211yesmaximal, characteristic, complemented
3D4×D508C211yesmaximal, characteristic, complemented
4#22550253C211yesmaximal, characteristic, complemented
5#22550129C212yesmaximal, complemented
6#22550077C211yesmaximal, characteristic, complemented
7#22550072C211yesmaximal, characteristic, complemented
8#22550023C212yesmaximal, complemented
9#22550012C212yesmaximal, complemented
10C4×#14523237C211yesmaximal, characteristic, complemented
11#436×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yescomplemented
13C22×D508C2212yes—
14C2×D4×D127—21no—
15#16861237—28no—
16C2×#14523237C2211yescharacteristic
17C2×#14523237C2212yescomplemented
18C2×D1016C2212yes—
19C2×D1016C2212yescomplemented
20C2×D1016C2212yescomplemented
21#16861206C2211yescharacteristic
22#16861206C2211yescharacteristic, complemented
23#16861205—21no—
24#16861184—21no—
25#16861183C2212yes—
26#16861183C2212yescomplemented
27#16861182—22no—
28#16861179—22no—
29C4×Dic254C2211yescharacteristic, complemented
30C4×D508C2211yescharacteristic
31#16861171—21no—
32#16861141C2211yescharacteristic, complemented
33#16861134C2211yescharacteristic, complemented
34#16861131C2211yescharacteristic
35#16861131C2212yescomplemented
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×Dic254C2311yescharacteristic
46C2×D508—24no—
47C2×D508—21no—
48C2×D508—22no—
49C2×D508—21no—
50C2×D508—21no—
51C2×D508—21no—
52C2×D508—42no—
53C2×D508C2312yes—
54C2×D508C2312yes—
55C2×D508C2311yescharacteristic
56C2×C22⋊D127—22no—
57C2×C22⋊D127—22no—
58C2×C22⋊D127—22no—
59D4×D127—44no—
60C2×C4×D127—21no—
61C2×C4×D127—22no—
62#14523237—24no—
63#14523237D414yescomplemented
64D1016—24no—
65D1016—24no—
66D1016—42no—
67C4⋊Dic127C2311yescharacteristic
68C4⋊Dic127C2311yescharacteristic
69Dic127⋊C4—21no—
70(C2×D127)⋊C4—22no—
71(C2×D127)⋊C4—22no—
72(C2×D127)⋊C4—21no—
73C4×Dic127—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—22no—
85C22×D127—22no—
86C22×D127—44no—
87C22×D127—44no—
88C22×D127—82no—
89Dic254—21no—
90Dic254C2×D412yescomplemented
91D508—22no—
92D508—22no—
93D508—21no—
94D508—22no—
95D508—22no—
96D508—44no—
97D508—42no—
98D508—42no—
99D508—81no—
100D508—81no—
101D508C2×D412yes—
102C22⋊D127—44no—
103C22⋊D127—44no—
104C22⋊D127—82no—
105C2×Dic127—21no—
106C2×Dic127—21no—
107C2×Dic127—21no—
108C4×D127—44no—
109C4×D127—42no—
110C23×C127C2×D412yes—
111D4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127C2×D411yescharacteristic
115C2×C4×C127C2×D412yes—
116C2×C4×C127C2×D411yescharacteristic
117C2×C4×C127C2411yescharacteristic
118C8×C127—22no—
119C8×C127—21no—
120C8×C127C2×D412yescomplemented
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127C22×D411yescharacteristic
125C2×D127—22no—
126C2×D127—41no—
127C2×D127—41no—
128C2×D127—42no—
129C2×D127—42no—
130C2×D127—42no—
131C2×D127—42no—
132C2×D127—42no—
133C2×D127—42no—
134C2×D127—88no—
135C4×C127—22no—
136C4×C127—21no—
137C4×C127AGL(1,ℤ/8)12yescomplemented
138C4×C127C22×D411yescharacteristic
139C4×C127H(5,𝔽2)11yescharacteristic
140Dic127—22no—
141Dic127—41no—
142Dic127—41no—
143C2×C127—42no—
144C2×C127D4211yescharacteristic
145C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C127#57611yescharacteristic
147D127—42no—
148D127—82no—
149D127—82no—
150#3274—1271nomaximal, Sylow
151C127#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1272no—
153#550—1271no—
154D42—1271no—
155#493—1271no—
156#469—1272no—
157#460—1271no—
158#459—1271no—
159#449—1272no—
160#447—1272no—
161C4×SD16—1271no—
162#436D12711yescharacteristic, complemented
163C2×#41—1272no—
164C22×D4—1272no—
165C22×D4—2541no—
166AGL(1,ℤ/8)—2548no—
167C2×SD16—1271no—
168C2×SD16—1272no—
169C2×D8—1272no—
170C2×D8—1272no—
171C2×D8—1272no—
172C2×M16C2×D12712yescomplemented
173#127—1271no—
174#127—1271no—
175#127—2541no—
176#121—1272no—
177#121—1272no—
178#121—2541no—
179#121—2542no—
180#120—2542no—
181C4×Q8—1271no—
182C4×D4—1271no—
183C4×D4—2541no—
184C4×D4C2×D12711yescharacteristic
185#106—1271no—
186C4⋊C8C2×D12711yescharacteristic, complemented
187#103—1271no—
188#102—1271no—
189#102—1272no—
190#102—1272no—
191C22⋊C8C2×D12712yescomplemented
192C4×C8C2×D12711yescharacteristic, complemented
193C24—2542no—
194#41—5082no—
195#41—2544no—
196C2×Q8—1271no—
197C2×D4—1271no—
198C2×D4—1272no—
199C2×D4—1272no—
200C2×D4—5082no—
201C2×D4—5084no—
202C2×D4—2544no—
203C2×D4—2542no—
204C2×D4—2542no—
205C2×D4—2542no—
206C2×D4—2541no—
207C2×D4—2541no—
208C2×D4—2541no—
209C2×D4—2542no—
210C2×D4—2541no—
211C2×D4D50811yescharacteristic
212C22×C4—2541no—
213C22×C4—2542no—
214C22×C4C22×D12712yes—
215SD16—1274no—
216SD16—2544no—
217D8—5082no—
218D8—2544no—
219D8—2544no—
220M16—24no—
221C2×C8C22×D12711yescharacteristic
222C2×C8C22×D12712yes—
223C2×C8C22×D12711yescharacteristic
224C4⋊C4—1271no—
225C4⋊C4—1271no—
226C4⋊C4—2541no—
227C4⋊C4D50811yescharacteristic
228C22⋊C4—2541no—
229C22⋊C4—2542no—
230C22⋊C4—2542no—
231C22⋊C4D50812yes—
232C42—2541no—
233C42C22×D12711yescharacteristic
234C23—10162no—
235C23—5084no—
236C23—5084no—
237C23—2542no—
238C23—2542no—
239C23—2541no—
240C23C2×D50812yes—
241Q8—1272no—
242Q8—2541no—
243D4—22no—
244D4—1272no—
245D4—10161no—
246D4—10161no—
247D4—10162no—
248D4—5084no—
249D4—5084no—
250D4—5084no—
251D4—5082no—
252D4—5082no—
253D4—2542no—
254D4—2542no—
255D4—2542no—
256D4—2542no—
257D4—2541no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—5084no—
261C2×C4—5082no—
262C2×C4—2541no—
263C2×C4—2541no—
264C2×C4—2541no—
265C2×C4C2×D50811yescharacteristic
266C2×C4C2×D50812yes—
267C2×C4C2×D50811yescharacteristic
268C2×C4C23×D12711yescharacteristic
269C8—22no—
270C8—21no—
271C8D4×D12712yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—10168no—
276C22—5082no—
277C22—5082no—
278C22—5082no—
279C22—5082no—
280C22—5082no—
281C22—5082no—
282C22—5081no—
283C22—5081no—
284C22—2542no—
285C22C22×D50811yescharacteristic
286C4—21no—
287C4—22no—
288C4—5081no—
289C4—5081no—
290C4—2542no—
291C4#1686123712yescomplemented
292C4C2×D4×D12711yescharacteristic
293C4#1686125511yescharacteristic
294C2—42no—
295C2—10162no—
296C2—10162no—
297C2—5082no—
298C2D4×D50811yescharacteristic
299C2C2×#1686123711yescharacteristic
300C2#2255068111yescharacteristic
subgroupG/H
centerC22C22×D508
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#436×C127C2
socleC22×C127C22×D4
hypercenter#436D127
hypocenterC127#3274
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. #22550547C2semidirect
3. D4×D508C2semidirect
4. #22550253C2semidirect
5. #22550129C2semidirect
6. #22550077C2semidirect
7. #22550072C2semidirect
8. #22550023C2semidirect
9. #22550012C2semidirect
10. C4×#14523237C2semidirect
11. #436×C127C2semidirect
12. C2×#14523248C22semidirect
17. C2×#14523237C22semidirect
19. C2×D1016C22semidirect
20. C2×D1016C22semidirect
22. #16861206C22semidirect
26. #16861183C22semidirect
29. C4×Dic254C22semidirect
32. #16861141C22semidirect
33. #16861134C22semidirect
35. #16861131C22semidirect
36. #16861131C22semidirect
37. C2×M16×C127C22semidirect
39. C4⋊C8×C127C22semidirect
40. C22⋊C8×C127C22semidirect
41. C4×C8×C127C22semidirect
63. #14523237D4semidirect
90. Dic254C2×D4semidirect
120. C8×C127C2×D4semidirect
137. C4×C127AGL(1,ℤ/8)semidirect
151. C127#3274semidirect
162. #436D127semidirect
172. C2×M16C2×D127semidirect
186. C4⋊C8C2×D127semidirect
191. C22⋊C8C2×D127semidirect
192. C4×C8C2×D127semidirect
271. C8D4×D127semidirect
291. C4#16861237semidirect

Maximal subgroups

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

subgroupindex
C2×#168612372
#225505472
D4×D5082
#225502532
#225501292
#225500772
#225500722
#225500232
#225500122
C4×#145232372
#436×C1272
#3274127