Finite Group Databasebeta

← #30650523

Subgroups — #30650523

Order 16256.

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

normal subgroups only

This group has 460 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#22550682C214yesmaximal, complemented
2#22550136C214yesmaximal, complemented
3#22550058C212yesmaximal, complemented
4#22550043C212yesmaximal, complemented
5#22549994C212yesmaximal, complemented
6#492×C127C211yesmaximal, characteristic, complemented
7#16861257C2212yes—
8#16861254C2212yescomplemented
9#16861240C2214yescomplemented
10#16861238—24no—
11#16861236—24no—
12#16861235C2214yes—
13#16861231C2212yes—
14#16861230—24no—
15SD16×D127—24no—
16#16861215C2214yescomplemented
17#16861145C2212yescomplemented
18#16861137C2214yescomplemented
19#16861128C2212yescomplemented
20#16861126C2212yescomplemented
21C2×SD16×C127C2214yescomplemented
22C4⋊Q8×C127C2211yescharacteristic, complemented
23#127×C127C2211yescharacteristic
24C4×C8×C127C2211yescharacteristic, complemented
25#14523249—24no—
26#14523248—42no—
27#14523248C2314yes—
28#14523247—24no—
29Q8×D127—24no—
30C2×C22⋊D127—42no—
31D4×D127—24no—
32Dic508D412yescomplemented
33#14523238—24no—
34#14523237D414yescomplemented
35#14523236—24no—
36#14523235—24no—
37D1016D412yescomplemented
38#14523233—24no—
39#14523232C2314yes—
40#14523231—24no—
41C8×D127—24no—
42C2×Q8×C127C2312yes—
43C2×D4×C127—41no—
44C2×D4×C127C2312yes—
45SD16×C127—24no—
46SD16×C127—24no—
47C2×C8×C127C2312yes—
48C4⋊C4×C127—41no—
49C42×C127C2311yescharacteristic
50C22×D127—42no—
51Dic254—42no—
52Dic254C2×D414yes—
53D508—42no—
54D508C2×D414yes—
55C22⋊D127—24no—
56C22⋊D127—44no—
57C2×Dic127—42no—
58C4×D127—24no—
59C4×D127—44no—
60C127⋊C8—24no—
61C23×C127—22no—
62Q8×C127—24no—
63D4×C127—24no—
64D4×C127—82no—
65C2×C4×C127—21no—
66C2×C4×C127—22no—
67C2×C4×C127C2411yescharacteristic
68C8×C127C2×D412yes—
69C8×C127C2×D412yes—
70C22×C127—42no—
71C22×C127—82no—
72C22×C127H(5,𝔽2)11yescharacteristic
73C2×D127—24no—
74C2×D127—162no—
75C4×C127—22no—
76C4×C127—42no—
77C4×C127C22×D412yes—
78Dic127—24no—
79C2×C127—21no—
80C2×C127—82no—
81C2×C127D4211yescharacteristic
82D127—44no—
83#3291—1271nomaximal, Sylow
84C127#329111yescharacteristic, complemented, Sylow
85#577—1274no—
86#492D12711yescharacteristic, complemented
87#471—1274no—
88#456—1272no—
89#453—1272no—
90#443—1272no—
912−1+4—1272no—
92H(5,𝔽2)—1272no—
93#137—1274no—
94#137—2544no—
95AGL(1,ℤ/8)—1274no—
96AGL(1,ℤ/8)—2544no—
97#135—1272no—
98#135—2544no—
99C2×SD16—2544no—
100C2×SD16C2×D12714yescomplemented
101#131—1274no—
102C4⋊Q8C2×D12711yescharacteristic, complemented
103#127C2×D12711yescharacteristic
104#108—1272no—
105#104—1274no—
106#101—1272no—
107#100—1272no—
108C4×C8C2×D12711yescharacteristic, complemented
109#41—1274no—
110#41—5082no—
111#41—2544no—
112#41—2544no—
113C2×Q8—2544no—
114C2×Q8C22×D12712yes—
115C2×D4—41no—
116C2×D4—5082no—
117C2×D4—2544no—
118C2×D4C22×D12712yes—
119Q16—1272no—
120Q16—2544no—
121SD16—24no—
122SD16—24no—
123SD16—1274no—
124SD16—2544no—
125SD16—2544no—
126D8—1272no—
127D8—2544no—
128M16—1274no—
129M16—2544no—
130C2×C8—2544no—
131C2×C8C22×D12712yes—
132C4⋊C4—41no—
133C42C22×D12711yescharacteristic
134C23—22no—
135C23—5082no—
136Q8—24no—
137Q8—1274no—
138Q8—5082no—
139D4—24no—
140D4—82no—
141D4—1274no—
142D4—5082no—
143D4—5084no—
144D4—2544no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—5084no—
148C2×C4—5082no—
149C2×C4—2544no—
150C2×C4C23×D12711yescharacteristic
151C8—2544no—
152C8D4×D12712yes—
153C8D4×D12712yes—
154C22—42no—
155C22—82no—
156C22—20322no—
157C22—2544no—
158C22#1686125411yescharacteristic
159C4—22no—
160C4—42no—
161C4—2544no—
162C4C2×D4×D12712yes—
163C2—21no—
164C2—82no—
165C2—5084no—
166C2#2255051111yescharacteristic
subgroupG/H
centerC2#22550511
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#492×C127C2
socleC2×C127D42
hypercenter#492D127
hypocenterC127#3291
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C2×C4 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #22550682C2semidirect
2. #22550136C2semidirect
3. #22550058C2semidirect
4. #22550043C2semidirect
5. #22549994C2semidirect
6. #492×C127C2semidirect
8. #16861254C22semidirect
9. #16861240C22semidirect
16. #16861215C22semidirect
17. #16861145C22semidirect
18. #16861137C22semidirect
19. #16861128C22semidirect
20. #16861126C22semidirect
21. C2×SD16×C127C22semidirect
22. C4⋊Q8×C127C22semidirect
24. C4×C8×C127C22semidirect
32. Dic508D4semidirect
34. #14523237D4semidirect
37. D1016D4semidirect
84. C127#3291semidirect
86. #492D127semidirect
100. C2×SD16C2×D127semidirect
102. C4⋊Q8C2×D127semidirect
108. C4×C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506822
#225501362
#225500582
#225500432
#225499942
#492×C1272
#3291127