Finite Group Databasebeta

← #30650515

Subgroups — #30650515

Order 16256.

454 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 117 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#22550689C211yesmaximal, characteristic, complemented
2#22550682C212yesmaximal, complemented
3#22550677C211yesmaximal, characteristic, complemented
4#22550148C211yesmaximal, characteristic, complemented
5#22550143C211yesmaximal, characteristic, complemented
6#22550136C212yesmaximal, complemented
7#22550053C212yesmaximal, complemented
8#22550048C212yesmaximal, complemented
9#22549994C211yesmaximal, characteristic, complemented
10#22549994C211yesmaximal, characteristic, complemented
11#495×C127C211yesmaximal, characteristic, complemented
12#16861257C2212yescomplemented
13#16861254C2212yes—
14#16861241—22no—
15#16861240C2212yes—
16#16861240C2212yescomplemented
17#16861238—22no—
18#16861236—22no—
19#16861235C2212yes—
20#16861235C2212yescomplemented
21#16861233—22no—
22#16861232—21no—
23#16861231C2211yescharacteristic
24#16861231C2211yescharacteristic
25#16861230—22no—
26#16861228—21no—
27Q16×D127—21no—
28SD16×D127—22no—
29D8×D127—21no—
30#16861215C2211yescharacteristic
31#16861215C2212yescomplemented
32#16861215C2211yescharacteristic, complemented
33#16861145C2211yescharacteristic, complemented
34#16861145C2211yescharacteristic, complemented
35#16861137C2211yescharacteristic
36#16861137C2211yescharacteristic, complemented
37#16861137C2212yescomplemented
38#16861128C2212yescomplemented
39#16861126C2212yescomplemented
40C2×Q16×C127C2211yescharacteristic, complemented
41C2×SD16×C127C2212yescomplemented
42C2×D8×C127C2211yescharacteristic
43#124×C127C2212yescomplemented
44C4×C8×C127C2211yescharacteristic, complemented
45#14523249—22no—
46#14523249—22no—
47#14523248—42no—
48#14523248C2311yescharacteristic
49#14523248C2312yes—
50#14523248C2311yescharacteristic
51#14523247—22no—
52#14523247—22no—
53Q8×D127—22no—
54Q8×D127—22no—
55C2×C22⋊D127—42no—
56D4×D127—22no—
57D4×D127—22no—
58Dic508D411yescharacteristic
59Dic508D411yescharacteristic
60#14523238—22no—
61#14523238—22no—
62#14523237D411yescharacteristic
63#14523237D411yescharacteristic
64#14523237D412yes—
65#14523236—22no—
66#14523236—22no—
67#14523235—22no—
68#14523235—22no—
69D1016D411yescharacteristic
70D1016D411yescharacteristic
71#14523233—22no—
72#14523233—22no—
73#14523232C2311yescharacteristic
74#14523232C2312yes—
75#14523232C2311yescharacteristic
76#14523231—21no—
77#14523231—22no—
78#14523231—21no—
79C8×D127—21no—
80C8×D127—22no—
81C8×D127—21no—
82C2×Q8×C127C2312yes—
83C2×D4×C127C2312yes—
84Q16×C127—21no—
85Q16×C127—21no—
86SD16×C127—22no—
87SD16×C127—22no—
88D8×C127—21no—
89D8×C127—21no—
90C2×C8×C127C2311yescharacteristic
91C2×C8×C127C2311yescharacteristic
92C22⋊C4×C127—42no—
93C42×C127C2311yescharacteristic
94C22×D127—42no—
95Dic254—42no—
96Dic254C2×D411yescharacteristic
97Dic254C2×D412yes—
98Dic254C2×D411yescharacteristic
99D508—42no—
100D508C2×D411yescharacteristic
101D508C2×D412yes—
102D508C2×D411yescharacteristic
103C22⋊D127—21no—
104C22⋊D127—22no—
105C22⋊D127—21no—
106C22⋊D127—42no—
107C22⋊D127—42no—
108C2×Dic127—42no—
109C4×D127—21no—
110C4×D127—22no—
111C4×D127—21no—
112C4×D127—42no—
113C4×D127—42no—
114C127⋊C8—21no—
115C127⋊C8—22no—
116C127⋊C8—21no—
117C23×C127—22no—
118Q8×C127—22no—
119Q8×C127—22no—
120D4×C127—22no—
121D4×C127—22no—
122C2×C4×C127—21no—
123C2×C4×C127—22no—
124C2×C4×C127C2411yescharacteristic
125C8×C127C2×D411yescharacteristic
126C8×C127C2×D411yescharacteristic
127C8×C127C2×D411yescharacteristic
128C8×C127C2×D411yescharacteristic
129C22×C127—42no—
130C22×C127—82no—
131C22×C127H(5,𝔽2)11yescharacteristic
132C2×D127—21no—
133C2×D127—22no—
134C2×D127—21no—
135C2×D127—162no—
136C4×C127—41no—
137C4×C127—42no—
138C4×C127C22×D411yescharacteristic
139C4×C127C22×D411yescharacteristic
140Dic127—21no—
141Dic127—22no—
142Dic127—21no—
143C2×C127—21no—
144C2×C127—82no—
145C2×C127D4211yescharacteristic
146D127—41no—
147D127—42no—
148D127—41no—
149#3290—1271nomaximal, Sylow
150C127#329011yescharacteristic, complemented, Sylow
151#578—1271no—
152#577—1272no—
153#576—1271no—
154#495D12711yescharacteristic, complemented
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)—1272no—
164#137—1272no—
165#137—1272no—
166#137—2542no—
167#137—2542no—
168AGL(1,ℤ/8)—1272no—
169AGL(1,ℤ/8)—1272no—
170AGL(1,ℤ/8)—2542no—
171AGL(1,ℤ/8)—2542no—
172#135—1271no—
173#135—1271no—
174#135—2541no—
175#135—2541no—
176#135—2542no—
177C2×Q16—2541no—
178C2×Q16C2×D12711yescharacteristic, complemented
179C2×SD16—2542no—
180C2×SD16C2×D12712yescomplemented
181C2×D8—2541no—
182C2×D8C2×D12711yescharacteristic
183#131—1271no—
184#131—1272no—
185#131—1271no—
186#124C2×D12712yescomplemented
187#108—1271no—
188#108—1271no—
189#104—1271no—
190#104—1272no—
191#104—1271no—
192#101—1272no—
193#100—1272no—
194C4×C8C2×D12711yescharacteristic, complemented
195#41—1271no—
196#41—1272no—
197#41—1271no—
198#41—5082no—
199#41—2542no—
200#41—2542no—
201#41—2542no—
202#41—2542no—
203C2×Q8—2542no—
204C2×Q8—2542no—
205C2×Q8C22×D12712yes—
206C2×D4—5082no—
207C2×D4—2542no—
208C2×D4—2542no—
209C2×D4C22×D12712yes—
210Q16—21no—
211Q16—21no—
212Q16—1271no—
213Q16—1271no—
214Q16—2542no—
215Q16—2542no—
216SD16—22no—
217SD16—22no—
218SD16—1271no—
219SD16—1271no—
220SD16—1272no—
221SD16—2542no—
222SD16—2542no—
223SD16—2542no—
224SD16—2542no—
225D8—21no—
226D8—21no—
227D8—1271no—
228D8—1271no—
229D8—2542no—
230D8—2542no—
231M16—1271no—
232M16—1272no—
233M16—1271no—
234M16—2541no—
235M16—2542no—
236M16—2541no—
237C2×C8—2541no—
238C2×C8—2542no—
239C2×C8—2541no—
240C2×C8C22×D12711yescharacteristic
241C2×C8C22×D12711yescharacteristic
242C22⋊C4—42no—
243C42C22×D12711yescharacteristic
244C23—22no—
245C23—5082no—
246Q8—22no—
247Q8—22no—
248Q8—1271no—
249Q8—1272no—
250Q8—1271no—
251Q8—5082no—
252D4—22no—
253D4—22no—
254D4—1271no—
255D4—1272no—
256D4—1271no—
257D4—5082no—
258D4—5082no—
259D4—5082no—
260D4—2541no—
261D4—2542no—
262D4—2541no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—5082no—
266C2×C4—5082no—
267C2×C4—5082no—
268C2×C4—2541no—
269C2×C4—2542no—
270C2×C4—2541no—
271C2×C4C23×D12711yescharacteristic
272C8—2541no—
273C8—2542no—
274C8—2541no—
275C8D4×D12711yescharacteristic
276C8D4×D12711yescharacteristic
277C8D4×D12711yescharacteristic
278C8D4×D12711yescharacteristic
279C22—42no—
280C22—82no—
281C22—20322no—
282C22—2541no—
283C22—2542no—
284C22—2541no—
285C22#1686125411yescharacteristic
286C4—41no—
287C4—42no—
288C4—2541no—
289C4—2542no—
290C4—2541no—
291C4C2×D4×D12711yescharacteristic
292C4C2×D4×D12711yescharacteristic
293C2—21no—
294C2—82no—
295C2—5081no—
296C2—5082no—
297C2—5081no—
298C2#2255051111yescharacteristic
subgroupG/H
centerC2#22550511
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#495×C127C2
socleC2×C127D42
hypercenter#495D127
hypocenterC127#3290
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #22550689C2semidirect
2. #22550682C2semidirect
3. #22550677C2semidirect
4. #22550148C2semidirect
5. #22550143C2semidirect
6. #22550136C2semidirect
7. #22550053C2semidirect
8. #22550048C2semidirect
9. #22549994C2semidirect
10. #22549994C2semidirect
11. #495×C127C2semidirect
12. #16861257C22semidirect
16. #16861240C22semidirect
20. #16861235C22semidirect
31. #16861215C22semidirect
32. #16861215C22semidirect
33. #16861145C22semidirect
34. #16861145C22semidirect
36. #16861137C22semidirect
37. #16861137C22semidirect
38. #16861128C22semidirect
39. #16861126C22semidirect
40. C2×Q16×C127C22semidirect
41. C2×SD16×C127C22semidirect
43. #124×C127C22semidirect
44. C4×C8×C127C22semidirect
150. C127#3290semidirect
154. #495D127semidirect
178. C2×Q16C2×D127semidirect
180. C2×SD16C2×D127semidirect
186. #124C2×D127semidirect
194. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506892
#225506822
#225506772
#225501482
#225501432
#225501362
#225500532
#225500482
#225499942
#495×C1272
#3290127