Finite Group Databasebeta

← #30650496

Subgroups — #30650496

Order 16256.

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

normal subgroups only

This group has 464 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#22550682C212yesmaximal, complemented
2#22550677C212yesmaximal, complemented
3#22550143C212yesmaximal, complemented
4#22550136C212yesmaximal, complemented
5#22550053C211yesmaximal, characteristic, complemented
6#22550048C211yesmaximal, characteristic, complemented
7#22550043C212yesmaximal, complemented
8#22549999C212yesmaximal, complemented
9#496×C127C211yesmaximal, characteristic, complemented
10#16861257C2211yescharacteristic, complemented
11#16861254C2211yescharacteristic
12#16861254C2212yescomplemented
13#16861240C2212yescomplemented
14#16861238—22no—
15#16861236—22no—
16#16861235C2212yes—
17#16861235C2212yescomplemented
18#16861235C2212yescomplemented
19#16861233—22no—
20#16861233—22no—
21#16861231C2212yes—
22#16861230—22no—
23#16861228—22no—
24SD16×D127—22no—
25D8×D127—22no—
26#16861215C2212yescomplemented
27#16861215C2212yescomplemented
28#16861145C2212yescomplemented
29#16861137C2212yescomplemented
30#16861137C2212yescomplemented
31#16861128C2211yescharacteristic, complemented
32#16861126C2212yescomplemented
33#16861126C2211yescharacteristic, complemented
34C2×SD16×C127C2212yescomplemented
35C2×D8×C127C2212yescomplemented
36#127×C127C2211yescharacteristic, complemented
37#124×C127C2211yescharacteristic, complemented
38#97×C127C2211yescharacteristic, complemented
39#14523249—22no—
40#14523248—41no—
41#14523248C2312yes—
42#14523248C2312yes—
43#14523247—22no—
44#14523247—22no—
45#14523247—22no—
46Q8×D127—22no—
47C2×C22⋊D127—41no—
48C2×C22⋊D127—42no—
49D4×D127—22no—
50D4×D127—22no—
51D4×D127—22no—
52Dic508D412yescomplemented
53#14523238—22no—
54#14523237D412yescomplemented
55#14523237D412yescomplemented
56#14523236—22no—
57#14523236—22no—
58#14523236—22no—
59#14523235—22no—
60D1016D412yescomplemented
61#14523233—22no—
62#14523233—22no—
63#14523233—22no—
64#14523232C2312yes—
65#14523232C2312yes—
66#14523231—22no—
67#14523231—22no—
68C8×D127—22no—
69C8×D127—22no—
70C2×Q8×C127C2311yescharacteristic
71C2×D4×C127—41no—
72C2×D4×C127C2312yes—
73C2×D4×C127C2311yescharacteristic
74SD16×C127—22no—
75SD16×C127—22no—
76D8×C127—22no—
77D8×C127—22no—
78C2×C8×C127C2312yes—
79C22⋊C4×C127—41no—
80C42×C127C2311yescharacteristic
81C22×D127—42no—
82C22×D127—41no—
83Dic254—41no—
84Dic254C2×D412yes—
85Dic254C2×D412yes—
86D508—41no—
87D508C2×D412yes—
88D508C2×D412yes—
89C22⋊D127—22no—
90C22⋊D127—22no—
91C22⋊D127—42no—
92C22⋊D127—42no—
93C22⋊D127—42no—
94C2×Dic127—42no—
95C2×Dic127—41no—
96C4×D127—22no—
97C4×D127—22no—
98C4×D127—42no—
99C127⋊C8—22no—
100C127⋊C8—22no—
101C23×C127—21no—
102C23×C127—22no—
103Q8×C127—22no—
104D4×C127—22no—
105D4×C127—22no—
106D4×C127—22no—
107D4×C127—82no—
108C2×C4×C127—21no—
109C2×C4×C127—21no—
110C2×C4×C127C2411yescharacteristic
111C8×C127C2×D412yes—
112C8×C127C2×D412yes—
113C22×C127—41no—
114C22×C127—42no—
115C22×C127—82no—
116C22×C127—81no—
117C22×C127H(5,𝔽2)11yescharacteristic
118C2×D127—22no—
119C2×D127—22no—
120C2×D127—162no—
121C2×D127—161no—
122C4×C127—41no—
123C4×C127—41no—
124C4×C127C22×D412yes—
125Dic127—22no—
126Dic127—22no—
127C2×C127—21no—
128C2×C127—82no—
129C2×C127—81no—
130C2×C127D4211yescharacteristic
131D127—42no—
132D127—42no—
133#3288—1271nomaximal, Sylow
134C127#328811yescharacteristic, complemented, Sylow
135#577—1272no—
136#576—1272no—
137#496D12711yescharacteristic, complemented
138#472—1272no—
139#471—1272no—
140#455—1271no—
141#454—1271no—
142#453—1272no—
143#444—1272no—
1442−1+4—1271no—
145H(5,𝔽2)—1271no—
146H(5,𝔽2)—1272no—
147#137—1272no—
148#137—2542no—
149AGL(1,ℤ/8)—1272no—
150AGL(1,ℤ/8)—1272no—
151AGL(1,ℤ/8)—1272no—
152AGL(1,ℤ/8)—2542no—
153AGL(1,ℤ/8)—2542no—
154AGL(1,ℤ/8)—2542no—
155#135—1272no—
156#135—2542no—
157#135—2542no—
158C2×SD16—2542no—
159C2×SD16C2×D12712yescomplemented
160C2×D8—2542no—
161C2×D8C2×D12712yescomplemented
162#131—1272no—
163#131—1272no—
164#127C2×D12711yescharacteristic, complemented
165#124C2×D12711yescharacteristic, complemented
166#108—1272no—
167#104—1272no—
168#104—1272no—
169#101—1271no—
170#100—1272no—
171#100—1271no—
172#97C2×D12711yescharacteristic, complemented
173#41—1272no—
174#41—1272no—
175#41—5081no—
176#41—2542no—
177#41—2542no—
178#41—2542no—
179#41—2542no—
180C2×Q8—2542no—
181C2×Q8C22×D12711yescharacteristic
182C2×D4—41no—
183C2×D4—5081no—
184C2×D4—5082no—
185C2×D4—2542no—
186C2×D4—2542no—
187C2×D4—2542no—
188C2×D4C22×D12711yescharacteristic
189C2×D4C22×D12712yes—
190Q16—1272no—
191Q16—2542no—
192SD16—22no—
193SD16—22no—
194SD16—1272no—
195SD16—1272no—
196SD16—2542no—
197SD16—2542no—
198SD16—2542no—
199SD16—2542no—
200D8—22no—
201D8—22no—
202D8—1272no—
203D8—2542no—
204D8—2542no—
205D8—2542no—
206M16—1272no—
207M16—1272no—
208M16—2542no—
209M16—2542no—
210C2×C8—2542no—
211C2×C8—2542no—
212C2×C8C22×D12712yes—
213C22⋊C4—41no—
214C42C22×D12711yescharacteristic
215C23—21no—
216C23—22no—
217C23—5082no—
218C23—5081no—
219Q8—22no—
220Q8—1272no—
221Q8—1272no—
222Q8—5081no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—1272no—
228D4—1272no—
229D4—5081no—
230D4—5082no—
231D4—5082no—
232D4—5082no—
233D4—2542no—
234D4—2542no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—5082no—
238C2×C4—5081no—
239C2×C4—5082no—
240C2×C4—2542no—
241C2×C4—2542no—
242C2×C4C23×D12711yescharacteristic
243C8—2542no—
244C8—2542no—
245C8D4×D12712yes—
246C8D4×D12712yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—20321no—
252C22—20322no—
253C22—2542no—
254C22—2542no—
255C22#1686125411yescharacteristic
256C4—41no—
257C4—41no—
258C4—2542no—
259C4—2542no—
260C4C2×D4×D12712yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—5082no—
265C2—5082no—
266C2#2255051111yescharacteristic
subgroupG/H
centerC2#22550511
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#496×C127C2
socleC2×C127D42
hypercenter#496D127
hypocenterC127#3288
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #22550682C2semidirect
2. #22550677C2semidirect
3. #22550143C2semidirect
4. #22550136C2semidirect
5. #22550053C2semidirect
6. #22550048C2semidirect
7. #22550043C2semidirect
8. #22549999C2semidirect
9. #496×C127C2semidirect
10. #16861257C22semidirect
12. #16861254C22semidirect
13. #16861240C22semidirect
17. #16861235C22semidirect
18. #16861235C22semidirect
26. #16861215C22semidirect
27. #16861215C22semidirect
28. #16861145C22semidirect
29. #16861137C22semidirect
30. #16861137C22semidirect
31. #16861128C22semidirect
32. #16861126C22semidirect
33. #16861126C22semidirect
34. C2×SD16×C127C22semidirect
35. C2×D8×C127C22semidirect
36. #127×C127C22semidirect
37. #124×C127C22semidirect
38. #97×C127C22semidirect
52. Dic508D4semidirect
54. #14523237D4semidirect
55. #14523237D4semidirect
60. D1016D4semidirect
134. C127#3288semidirect
137. #496D127semidirect
159. C2×SD16C2×D127semidirect
161. C2×D8C2×D127semidirect
164. #127C2×D127semidirect
165. #124C2×D127semidirect
172. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506822
#225506772
#225501432
#225501362
#225500532
#225500482
#225500432
#225499992
#496×C1272
#3288127