Finite Group Databasebeta

← #30650512

Subgroups — #30650512

Order 16256.

454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#22550690C211yesmaximal, characteristic, complemented
2#22550687C212yesmaximal, complemented
3#22550259C211yesmaximal, characteristic, complemented
4#22550151C211yesmaximal, characteristic, complemented
5#22550143C211yesmaximal, characteristic, complemented
6#22550137C212yesmaximal, complemented
7#22550054C212yesmaximal, complemented
8#22550047C212yesmaximal, complemented
9#22549993C211yesmaximal, characteristic, complemented
10#22549991C211yesmaximal, characteristic, complemented
11#576×C127C211yesmaximal, characteristic, complemented
12#16861256C2212yescomplemented
13#16861239—22no—
14#16861239C2212yes—
15#16861239C2212yescomplemented
16#16861238—22no—
17#16861235—22no—
18#16861235C2212yescomplemented
19#16861229—22no—
20#16861229C2211yescharacteristic
21#16861228—21no—
22C2×Dic508C2211yescharacteristic, complemented
23Q16×D127—21no—
24C2×#14523236—22no—
25C2×#14523236C2212yescomplemented
26#16861216C2212yescomplemented
27#16861215C2211yescharacteristic, complemented
28#16861196C2212yescomplemented
29#16861145C2211yescharacteristic, complemented
30#16861144C2211yescharacteristic, complemented
31#16861136C2211yescharacteristic
32#16861136C2212yescomplemented
33#16861135C2211yescharacteristic, complemented
34#16861127C2212yescomplemented
35#16861126C2212yescomplemented
36C8×Dic127C2211yescharacteristic, complemented
37H(5,𝔽2)×C127C2212yes—
38AGL(1,ℤ/8)×C127—22no—
39AGL(1,ℤ/8)×C127C2212yes—
40#135×C127—21no—
41#135×C127C2211yescharacteristic
42C2×D8×C127—21no—
43C2×D8×C127C2211yescharacteristic
44#131×C127C2211yescharacteristic
45#14523248—22no—
46#14523248C2311yescharacteristic
47#14523247—22no—
48#14523247—42no—
49C2×Dic254—22no—
50C2×Dic254C2312yes—
51Q8×D127—22no—
52Dic508—21no—
53Dic508—21no—
54#14523238—22no—
55#14523238—22no—
56#14523238D411yescharacteristic
57#14523236—22no—
58#14523236—22no—
59#14523236—22no—
60#14523236—22no—
61#14523236—22no—
62#14523236D411yescharacteristic
63#14523235D411yescharacteristic
64#14523233—22no—
65#14523233D411yescharacteristic
66#14523232—22no—
67#14523232C2311yescharacteristic
68#14523232C2312yes—
69#14523231—21no—
70C2×C127⋊C8—22no—
71C2×C127⋊C8C2311yescharacteristic
72C8×D127—21no—
73C22⋊Dic127—42no—
74C4×Dic127C2311yescharacteristic
75#41×C127—22no—
76#41×C127—22no—
77#41×C127C2312yes—
78#41×C127C2311yescharacteristic
79C2×D4×C127—22no—
80C2×D4×C127—22no—
81C2×D4×C127—42no—
82C2×D4×C127C2312yes—
83Q16×C127D411yescharacteristic
84SD16×C127—22no—
85SD16×C127D412yes—
86D8×C127—22no—
87D8×C127—21no—
88D8×C127—21no—
89D8×C127D411yescharacteristic
90M16×C127—21no—
91M16×C127C2311yescharacteristic
92C2×C8×C127—21no—
93C2×C8×C127C2311yescharacteristic
94Dic254—22no—
95Dic254—22no—
96Dic254—42no—
97Dic254C2×D411yescharacteristic
98D508C2×D411yescharacteristic
99C22⋊D127—21no—
100C22⋊D127—42no—
101C2×Dic127—21no—
102C2×Dic127—22no—
103C2×Dic127—42no—
104C4×D127—21no—
105C4×D127—42no—
106C127⋊C8—21no—
107C127⋊C8—22no—
108C127⋊C8C2×D411yescharacteristic
109C127⋊C8C2×D411yescharacteristic
110C23×C127—22no—
111C23×C127—42no—
112Q8×C127C2×D411yescharacteristic
113Q8×C127C2×D412yes—
114D4×C127—21no—
115D4×C127—22no—
116D4×C127—22no—
117D4×C127—22no—
118D4×C127—42no—
119D4×C127—42no—
120D4×C127C2×D411yescharacteristic
121D4×C127C2×D412yes—
122C2×C4×C127—21no—
123C2×C4×C127—22no—
124C2×C4×C127—42no—
125C2×C4×C127C2411yescharacteristic
126C8×C127—21no—
127C8×C127C2×D411yescharacteristic
128C8×C127C2×D411yescharacteristic
129C22×C127—21no—
130C22×C127—22no—
131C22×C127—42no—
132C22×C127—82no—
133C22×C127—162no—
134C22×C127H(5,𝔽2)11yescharacteristic
135C2×D127—21no—
136C4×C127—21no—
137C4×C127—22no—
138C4×C127C22×D411yescharacteristic
139C4×C127C22×D411yescharacteristic
140Dic127—21no—
141Dic127—41no—
142Dic127—42no—
143C2×C127—21no—
144C2×C127—42no—
145C2×C127—41no—
146C2×C127—82no—
147C2×C127D4211yescharacteristic
148D127—41no—
149#3290—1271nomaximal, Sylow
150C127#329011yescharacteristic, complemented, Sylow
151#578—1271no—
152#577—1272no—
153#576D12711yescharacteristic, complemented
154#495—1271no—
155#473—1271no—
156#472—1271no—
157#471—1272no—
158#455—1272no—
159#454—1272no—
160#443—1271no—
161#443—1271no—
1622−1+4—1272no—
163H(5,𝔽2)C2×D12712yes—
164#137—1272no—
165#137—1272no—
166#137—2542no—
167#137—2542no—
168AGL(1,ℤ/8)—22no—
169AGL(1,ℤ/8)—1272no—
170AGL(1,ℤ/8)—2542no—
171AGL(1,ℤ/8)C2×D12712yes—
172#135—21no—
173#135—1271no—
174#135—2542no—
175#135—2541no—
176#135C2×D12711yescharacteristic
177C2×Q16—1271no—
178C2×Q16—2541no—
179C2×SD16—1272no—
180C2×SD16—2542no—
181C2×D8—21no—
182C2×D8C2×D12711yescharacteristic
183#131—1272no—
184#131—1271no—
185#131C2×D12711yescharacteristic
186#124—1272no—
187#108—1271no—
188#108—1271no—
189#104—1272no—
190#104—1271no—
191#104—1271no—
192#101—1272no—
193#100—1272no—
194C4×C8—1271no—
195#41—22no—
196#41—22no—
197#41—1271no—
198#41—5082no—
199#41—2542no—
200#41—2542no—
201#41C22×D12711yescharacteristic
202#41C22×D12712yes—
203C2×Q8—1272no—
204C2×Q8—2542no—
205C2×Q8—2542no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4C22×D12712yes—
210Q16—1271no—
211Q16—2542no—
212Q16—2542no—
213Q16—2541no—
214Q16—2541no—
215Q16C22⋊D12711yescharacteristic
216SD16—22no—
217SD16—1271no—
218SD16—1271no—
219SD16—2542no—
220SD16—2542no—
221SD16—2542no—
222SD16—2542no—
223SD16—2542no—
224SD16C22⋊D12712yes—
225D8—22no—
226D8—21no—
227D8—21no—
228D8—1271no—
229D8—2542no—
230D8C22⋊D12711yescharacteristic
231M16—21no—
232M16—1271no—
233M16—1272no—
234M16—2542no—
235M16—2541no—
236M16C22×D12711yescharacteristic
237C2×C8—21no—
238C2×C8—1271no—
239C2×C8—2542no—
240C2×C8—2541no—
241C2×C8C22×D12711yescharacteristic
242C22⋊C4—5082no—
243C42—1271no—
244C23—22no—
245C23—42no—
246Q8—1271no—
247Q8—5082no—
248Q8—2542no—
249Q8—2542no—
250Q8D4×D12711yescharacteristic
251Q8C2×C22⋊D12712yes—
252D4—21no—
253D4—22no—
254D4—22no—
255D4—22no—
256D4—42no—
257D4—42no—
258D4—1271no—
259D4—5082no—
260D4—2541no—
261D4D4×D12711yescharacteristic
262D4C2×C22⋊D12712yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—5082no—
267C2×C4—5082no—
268C2×C4—2541no—
269C2×C4—2542no—
270C2×C4—2541no—
271C2×C4C23×D12711yescharacteristic
272C8—21no—
273C8—1271no—
274C8—1271no—
275C8—2542no—
276C8—2541no—
277C8C2×C22⋊D12711yescharacteristic
278C8C2×C22⋊D12711yescharacteristic
279C22—22no—
280C22—21no—
281C22—42no—
282C22—82no—
283C22—162no—
284C22—2541no—
285C22#1686125411yescharacteristic
286C4—21no—
287C4—22no—
288C4—5081no—
289C4—5082no—
290C4—2541no—
291C4C2×D4×D12711yescharacteristic
292C4C22×C22⋊D12711yescharacteristic
293C2—21no—
294C2—41no—
295C2—42no—
296C2—82no—
297C2—5081no—
298C2D4×C22⋊D12711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#576×C127C2
socleC2×C127D42
hypercenter#576D127
hypocenterC127#3290
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #22550690C2semidirect
2. #22550687C2semidirect
3. #22550259C2semidirect
4. #22550151C2semidirect
5. #22550143C2semidirect
6. #22550137C2semidirect
7. #22550054C2semidirect
8. #22550047C2semidirect
9. #22549993C2semidirect
10. #22549991C2semidirect
11. #576×C127C2semidirect
12. #16861256C22semidirect
15. #16861239C22semidirect
18. #16861235C22semidirect
22. C2×Dic508C22semidirect
25. C2×#14523236C22semidirect
26. #16861216C22semidirect
27. #16861215C22semidirect
28. #16861196C22semidirect
29. #16861145C22semidirect
30. #16861144C22semidirect
32. #16861136C22semidirect
33. #16861135C22semidirect
34. #16861127C22semidirect
35. #16861126C22semidirect
36. C8×Dic127C22semidirect
150. C127#3290semidirect
153. #576D127semidirect

Maximal subgroups

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

subgroupindex
#225506902
#225506872
#225502592
#225501512
#225501432
#225501372
#225500542
#225500472
#225499932
#225499912
#576×C1272
#3290127