Finite Group Databasebeta

← #30650326

Subgroups — #30650326

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×#16861235C212yesmaximal, complemented
2#22550531C211yesmaximal, characteristic, complemented
3#22550248C211yesmaximal, characteristic, complemented
4#22550120C212yesmaximal, complemented
5#22550081C211yesmaximal, characteristic, complemented
6#22550068C211yesmaximal, characteristic, complemented
7#22550027C212yesmaximal, complemented
8#22550019C212yesmaximal, complemented
9C4×#14523233C211yesmaximal, characteristic, complemented
10#22549954C211yesmaximal, characteristic, complemented
11D42×C127C211yesmaximal, characteristic, complemented
12C2×#14523248C2212yescomplemented
13#16861235—28no—
14C2×#14523236C2212yes—
15C2×#14523236C2212yescomplemented
16C2×#14523236C2212yescomplemented
17C2×#14523233C2211yescharacteristic
18C2×#14523233C2212yescomplemented
19C2×#14523232C2212yescomplemented
20#16861209C2211yescharacteristic, complemented
21#16861191—22no—
22#16861188C2212yescomplemented
23#16861180—21no—
24C4×D508C2211yescharacteristic, complemented
25C4⋊C4×D127—21no—
26#16861142C2211yescharacteristic, complemented
27C4⋊(C127⋊C8)C2211yescharacteristic, complemented
28#16861132C2212yescomplemented
29#16861130C2211yescharacteristic
30#16861130C2212yescomplemented
31#16861129C2211yescharacteristic, complemented
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×Dic254C2312yes—
45C2×D508C2311yescharacteristic
46C2×C22⋊D127—22no—
47C2×C4×D127—22no—
48C2×C4×D127—21no—
49#14523236—24no—
50#14523236—24no—
51#14523236—42no—
52#14523233—24no—
53#14523233D414yescomplemented
54#14523232—24no—
55C2×C127⋊C8C2312yes—
56C2×C127⋊C8C2311yescharacteristic
57C2×C127⋊C8C2311yescharacteristic
58C4⋊Dic127—21no—
59C4⋊Dic127—21no—
60C4⋊Dic127C2311yescharacteristic
61Dic127⋊C4—22no—
62C22⋊Dic127—22no—
63(C2×D127)⋊C4—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—21no—
84Dic254—22no—
85Dic254—22no—
86D508—21no—
87D508C2×D412yescomplemented
88C22⋊D127—44no—
89C2×Dic127—21no—
90C2×Dic127—22no—
91C4×D127—44no—
92C4×D127—42no—
93C127⋊C8—22no—
94C127⋊C8—21no—
95C127⋊C8C2×D412yescomplemented
96C23×C127—21no—
97C23×C127—22no—
98C23×C127—44no—
99C23×C127—44no—
100C23×C127—82no—
101C23×C127C2×D412yes—
102D4×C127—22no—
103D4×C127—22no—
104D4×C127—21no—
105D4×C127—22no—
106D4×C127—44no—
107D4×C127—44no—
108D4×C127—42no—
109D4×C127—81no—
110D4×C127C2×D412yes—
111C2×C4×C127—22no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—42no—
115C2×C4×C127C2×D411yescharacteristic
116C2×C4×C127C2×D412yes—
117C2×C4×C127C2×D411yescharacteristic
118C2×C4×C127C2411yescharacteristic
119C22×C127—22no—
120C22×C127—22no—
121C22×C127—22no—
122C22×C127—22no—
123C22×C127—41no—
124C22×C127—41no—
125C22×C127—42no—
126C22×C127—42no—
127C22×C127—42no—
128C22×C127—88no—
129C22×C127C22×D411yescharacteristic
130C2×D127—22no—
131C2×D127—41no—
132C2×D127—41no—
133C4×C127—22no—
134C4×C127—21no—
135C4×C127—41no—
136C4×C127AGL(1,ℤ/8)12yescomplemented
137C4×C127C22×D411yescharacteristic
138C4×C127H(5,𝔽2)11yescharacteristic
139Dic127—42no—
140Dic127—41no—
141C2×C127—42no—
142C2×C127—42no—
143C2×C127—82no—
144C2×C127D4211yescharacteristic
145C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C127#57711yescharacteristic
147D127—42no—
148#3272—1271nomaximal, Sylow
149C127#327211yescharacteristic, complemented, Sylow
150C2×AGL(1,ℤ/8)—1272no—
151#547—1271no—
152D42D12711yescharacteristic, complemented
153#492—1271no—
154#468—1272no—
155#461—1271no—
156#459—1271no—
157#450—1272no—
158#449—1272no—
159C4×D8—1271no—
160#436—1271no—
161C2×#41—1272no—
162C22×D4—21no—
163C22×D4C2×D12712yes—
164AGL(1,ℤ/8)—2548no—
165C2×SD16—1272no—
166C2×SD16—1272no—
167C2×SD16—1272no—
168C2×D8—1271no—
169C2×D8—1272no—
170C2×M16—1272no—
171C4⋊Q8—1271no—
172#127C2×D12711yescharacteristic
173#123—2542no—
174C22⋊Q8—1272no—
175#121—21no—
176#121—2541no—
177#121C2×D12712yes—
178#120—22no—
179C4×D4—1271no—
180C4×D4C2×D12711yescharacteristic
181C4×D4C2×D12711yescharacteristic
182C2×C4⋊C4—2541no—
183#107—1271no—
184C4⋊C8—1271no—
185#103—1272no—
186#102—1271no—
187#102—1272no—
188#102—1271no—
189C22⋊C8—1272no—
190C4×C8—1271no—
191C24—22no—
192#41—5082no—
193#41—2544no—
194C2×Q8—1272no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—1271no—
203C2×D4—2542no—
204C2×D4C22⋊D12711yescharacteristic
205C2×D4C22×D12712yes—
206C2×D4C22×D12711yescharacteristic
207C22×C4—21no—
208C22×C4—2541no—
209C22×C4—2542no—
210C22×C4C22×D12712yes—
211SD16—5082no—
212SD16—2544no—
213SD16—2544no—
214D8—1274no—
215D8—2544no—
216M16—2544no—
217C2×C8—1271no—
218C2×C8—1271no—
219C2×C8—1272no—
220C4⋊C4—1271no—
221C4⋊C4—2542no—
222C4⋊C4—2541no—
223C4⋊C4—2541no—
224C4⋊C4C22⋊D12711yescharacteristic
225C4⋊C4C22×D12711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—2542no—
229C22⋊C4—2541no—
230C22⋊C4C22⋊D12712yes—
231C42C22×D12711yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—2541no—
238C23C2×C22⋊D12712yes—
239Q8—2542no—
240Q8—2542no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—1272no—
250D4—5084no—
251D4—2541no—
252D4D4×D12712yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—5084no—
258C2×C4—5082no—
259C2×C4—2541no—
260C2×C4—2542no—
261C2×C4C2×C22⋊D12711yescharacteristic
262C2×C4C2×C22⋊D12712yes—
263C2×C4C2×C22⋊D12711yescharacteristic
264C2×C4C23×D12711yescharacteristic
265C8—1272no—
266C8—2542no—
267C8—2541no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—5081no—
279C22—5081no—
280C22—2542no—
281C22C22×C22⋊D12711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—5081no—
286C4—5082no—
287C4#1686123512yescomplemented
288C4C2×D4×D12711yescharacteristic
289C4#1686125411yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—5082no—
294C2D4×C22⋊D12711yescharacteristic
295C2C2×#1686123511yescharacteristic
296C2#2255068711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingD42×C127C2
socleC22×C127C22×D4
hypercenterD42D127
hypocenterC127#3272
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. #22550531C2semidirect
3. #22550248C2semidirect
4. #22550120C2semidirect
5. #22550081C2semidirect
6. #22550068C2semidirect
7. #22550027C2semidirect
8. #22550019C2semidirect
9. C4×#14523233C2semidirect
10. #22549954C2semidirect
11. D42×C127C2semidirect
12. C2×#14523248C22semidirect
15. C2×#14523236C22semidirect
16. C2×#14523236C22semidirect
18. C2×#14523233C22semidirect
19. C2×#14523232C22semidirect
20. #16861209C22semidirect
22. #16861188C22semidirect
24. C4×D508C22semidirect
26. #16861142C22semidirect
27. C4⋊(C127⋊C8)C22semidirect
28. #16861132C22semidirect
30. #16861130C22semidirect
31. #16861129C22semidirect
32. C22⋊(C127⋊C8)C22semidirect
33. C4×C127⋊C8C22semidirect
53. #14523233D4semidirect
87. D508C2×D4semidirect
95. C127⋊C8C2×D4semidirect
136. C4×C127AGL(1,ℤ/8)semidirect
149. C127#3272semidirect
152. D42D127semidirect
287. C4#16861235semidirect

Maximal subgroups

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

subgroupindex
C2×#168612352
#225505312
#225502482
#225501202
#225500812
#225500682
#225500272
#225500192
C4×#145232332
#225499542
D42×C1272
#3272127