Finite Group Databasebeta

← #30650346

Subgroups — #30650346

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×#16861235C212yesmaximal, complemented
2#22550547C211yesmaximal, characteristic, complemented
3#22550252C211yesmaximal, characteristic, complemented
4#22550127C212yesmaximal, complemented
5#22550073C211yesmaximal, characteristic, complemented
6#22550071C211yesmaximal, characteristic, complemented
7#22550019C212yesmaximal, complemented
8#22550010C212yesmaximal, complemented
9C4×#14523236C211yesmaximal, characteristic, complemented
10#22549954C211yesmaximal, characteristic, complemented
11D42×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yescomplemented
13#16861235—28no—
14C2×#14523236C2211yescharacteristic
15C2×#14523236C2212yescomplemented
16C2×#14523233C2212yes—
17C2×#14523233C2212yescomplemented
18C2×#14523233C2212yescomplemented
19C2×#14523232C2212yescomplemented
20#16861206C2211yescharacteristic, complemented
21#16861205—21no—
22#16861183C2212yescomplemented
23#16861182—22no—
24C4×Dic254C2211yescharacteristic, complemented
25#16861171—21no—
26#16861140C2211yescharacteristic, complemented
27C4⋊(C127⋊C8)C2211yescharacteristic, complemented
28#16861132C2211yescharacteristic, complemented
29#16861130C2211yescharacteristic
30#16861130C2212yescomplemented
31#16861129C2212yescomplemented
32C22⋊(C127⋊C8)C2212yescomplemented
33C4×C127⋊C8C2211yescharacteristic, complemented
34C22×D4×C127—21no—
35C22×D4×C127C2212yes—
36#127×C127C2211yescharacteristic
37#121×C127—21no—
38#121×C127C2212yes—
39#120×C127—22no—
40C4×D4×C127C2211yescharacteristic
41C4×D4×C127C2211yescharacteristic
42#14523248—24no—
43#14523248—42no—
44C2×Dic254C2311yescharacteristic
45C2×D508—21no—
46C2×D508—21no—
47C2×D508C2312yes—
48C2×C22⋊D127—22no—
49C2×C22⋊D127—22no—
50C2×C4×D127—22no—
51#14523236—24no—
52#14523236D414yescomplemented
53#14523233—24no—
54#14523233—24no—
55#14523233—42no—
56#14523232—24no—
57C2×C127⋊C8C2312yes—
58C2×C127⋊C8C2311yescharacteristic
59C2×C127⋊C8C2311yescharacteristic
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—21no—
62(C2×D127)⋊C4—22no—
63C4×Dic127—21no—
64C24×C127—22no—
65C2×D4×C127—24no—
66C2×D4×C127—22no—
67C2×D4×C127—22no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—42no—
71C2×D4×C127—44no—
72C2×D4×C127D411yescharacteristic
73C2×D4×C127C2312yes—
74C2×D4×C127C2311yescharacteristic
75C22×C4×C127—21no—
76C22×C4×C127C2312yes—
77C4⋊C4×C127D411yescharacteristic
78C4⋊C4×C127C2311yescharacteristic
79C22⋊C4×C127—21no—
80C22⋊C4×C127—22no—
81C22⋊C4×C127D412yes—
82C42×C127C2311yescharacteristic
83C22×D127—22no—
84Dic254—21no—
85Dic254C2×D412yescomplemented
86D508—22no—
87D508—22no—
88D508—42no—
89D508—81no—
90C22⋊D127—44no—
91C22⋊D127—82no—
92C2×Dic127—21no—
93C2×Dic127—21no—
94C4×D127—44no—
95C127⋊C8—22no—
96C127⋊C8—21no—
97C127⋊C8C2×D412yescomplemented
98C23×C127—21no—
99C23×C127—22no—
100C23×C127—44no—
101C23×C127—44no—
102C23×C127—82no—
103C23×C127C2×D412yes—
104D4×C127—22no—
105D4×C127—22no—
106D4×C127—21no—
107D4×C127—22no—
108D4×C127—44no—
109D4×C127—44no—
110D4×C127—42no—
111D4×C127—81no—
112D4×C127C2×D412yes—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—21no—
116C2×C4×C127—42no—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2×D412yes—
119C2×C4×C127C2×D411yescharacteristic
120C2×C4×C127C2411yescharacteristic
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—22no—
124C22×C127—22no—
125C22×C127—41no—
126C22×C127—41no—
127C22×C127—42no—
128C22×C127—42no—
129C22×C127—42no—
130C22×C127—88no—
131C22×C127C22×D411yescharacteristic
132C2×D127—42no—
133C2×D127—42no—
134C2×D127—42no—
135C4×C127—22no—
136C4×C127—21no—
137C4×C127—41no—
138C4×C127AGL(1,ℤ/8)12yescomplemented
139C4×C127C22×D411yescharacteristic
140C4×C127H(5,𝔽2)11yescharacteristic
141Dic127—22no—
142Dic127—41no—
143C2×C127—42no—
144C2×C127—42no—
145C2×C127—82no—
146C2×C127D4211yescharacteristic
147C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C127#57611yescharacteristic
149D127—82no—
150#3274—1271nomaximal, Sylow
151C127#327411yescharacteristic, complemented, Sylow
152C2×AGL(1,ℤ/8)—1272no—
153#550—1271no—
154D42D12711yescharacteristic, complemented
155#493—1271no—
156#469—1272no—
157#460—1271no—
158#459—1271no—
159#449—1272no—
160#447—1272no—
161C4×SD16—1271no—
162#436—1271no—
163C2×#41—1272no—
164C22×D4—21no—
165C22×D4C2×D12712yes—
166AGL(1,ℤ/8)—2548no—
167C2×SD16—1271no—
168C2×SD16—1272no—
169C2×D8—1272no—
170C2×D8—1272no—
171C2×D8—1272no—
172C2×M16—1272no—
173#127—1271no—
174#127—2541no—
175#127C2×D12711yescharacteristic
176#121—21no—
177#121—1272no—
178#121—2542no—
179#121C2×D12712yes—
180#120—22no—
181C4×Q8—1271no—
182C4×D4—2541no—
183C4×D4C2×D12711yescharacteristic
184C4×D4C2×D12711yescharacteristic
185#106—1271no—
186C4⋊C8—1271no—
187#103—1271no—
188#102—1271no—
189#102—1272no—
190#102—1272no—
191C22⋊C8—1272no—
192C4×C8—1271no—
193C24—22no—
194#41—5082no—
195#41—2544no—
196C2×Q8—1271no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—1272no—
205C2×D4—2542no—
206C2×D4—2542no—
207C2×D4—2541no—
208C2×D4—2541no—
209C2×D4C22⋊D12711yescharacteristic
210C2×D4C22×D12712yes—
211C2×D4C22×D12711yescharacteristic
212C22×C4—21no—
213C22×C4—2542no—
214C22×C4C22×D12712yes—
215SD16—1274no—
216SD16—2544no—
217D8—5082no—
218D8—2544no—
219D8—2544no—
220M16—2544no—
221C2×C8—1271no—
222C2×C8—1271no—
223C2×C8—1272no—
224C4⋊C4—1271no—
225C4⋊C4—2541no—
226C4⋊C4C22⋊D12711yescharacteristic
227C4⋊C4C22×D12711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—2542no—
231C22⋊C4C22⋊D12712yes—
232C42—2541no—
233C42C22×D12711yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—2542no—
240C23C2×C22⋊D12712yes—
241Q8—1272no—
242Q8—2541no—
243D4—22no—
244D4—22no—
245D4—21no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—10161no—
252D4—10162no—
253D4—5084no—
254D4—5082no—
255D4—2542no—
256D4—2542no—
257D4D4×D12712yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—5084no—
263C2×C4—2541no—
264C2×C4—2541no—
265C2×C4C2×C22⋊D12711yescharacteristic
266C2×C4C2×C22⋊D12712yes—
267C2×C4C2×C22⋊D12711yescharacteristic
268C2×C4C23×D12711yescharacteristic
269C8—1272no—
270C8—2542no—
271C8—2541no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—5082no—
283C22—5082no—
284C22—5082no—
285C22C22×C22⋊D12711yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—5081no—
290C4—2542no—
291C4#1686123512yescomplemented
292C4C2×D4×D12711yescharacteristic
293C4#1686125411yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—10162no—
298C2D4×C22⋊D12711yescharacteristic
299C2C2×#1686123511yescharacteristic
300C2#2255067911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD42×C127C2
socleC22×C127C22×D4
hypercenterD42D127
hypocenterC127#3274
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#16861235C2semidirect
2. #22550547C2semidirect
3. #22550252C2semidirect
4. #22550127C2semidirect
5. #22550073C2semidirect
6. #22550071C2semidirect
7. #22550019C2semidirect
8. #22550010C2semidirect
9. C4×#14523236C2semidirect
10. #22549954C2semidirect
11. D42×C127C2semidirect
12. C2×#14523248C22semidirect
15. C2×#14523236C22semidirect
17. C2×#14523233C22semidirect
18. C2×#14523233C22semidirect
19. C2×#14523232C22semidirect
20. #16861206C22semidirect
22. #16861183C22semidirect
24. C4×Dic254C22semidirect
26. #16861140C22semidirect
27. C4⋊(C127⋊C8)C22semidirect
28. #16861132C22semidirect
30. #16861130C22semidirect
31. #16861129C22semidirect
32. C22⋊(C127⋊C8)C22semidirect
33. C4×C127⋊C8C22semidirect
52. #14523236D4semidirect
85. Dic254C2×D4semidirect
97. C127⋊C8C2×D4semidirect
138. C4×C127AGL(1,ℤ/8)semidirect
151. C127#3274semidirect
154. D42D127semidirect
291. C4#16861235semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225505472
#225502522
#225501272
#225500732
#225500712
#225500192
#225500102
C4×#145232362
#225499542
D42×C1272
#3274127