Finite Group Databasebeta

← #30638018

Subgroups — #30638018

Order 16256.

262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550618C211yesmaximal, characteristic, complemented
2#22550263C212yesmaximal, complemented
3#22549616C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#496×C127C211yesmaximal, characteristic, complemented
6C2×#14523236—22no—
7C2×#14523233—22no—
8#16861198—21no—
9#16861197C2212yes—
10#16861195—22no—
11#16861179—22no—
12#16861177—21no—
13#16861120C2212yescomplemented
14C2×SD16×C127—21no—
15C2×D8×C127—21no—
16#127×C127C2211yescharacteristic, complemented
17#124×C127C2211yescharacteristic
18#97×C127C2211yescharacteristic, complemented
19C23×D127—21no—
20C2×Dic254D412yes—
21C2×D508—21no—
22C2×D508D412yes—
23C2×C22⋊D127—22no—
24#14523236—44no—
25#14523233—44no—
26C2×C127⋊C8—22no—
27C22⋊Dic127—21no—
28(C2×D127)⋊C4—22no—
29(C2×D127)⋊C4—22no—
30(C2×D127)⋊C4—22no—
31C4×Dic127—21no—
32C2×Q8×C127D411yescharacteristic
33C2×D4×C127—22no—
34C2×D4×C127—21no—
35C2×D4×C127D411yescharacteristic
36SD16×C127—42no—
37D8×C127—42no—
38C2×C8×C127—21no—
39C22⋊C4×C127—22no—
40C42×C127C2311yescharacteristic
41C22×D127—22no—
42C22×D127—82no—
43C22×D127—81no—
44Dic254—42no—
45D508—42no—
46D508—81no—
47C22⋊D127—82no—
48C2×Dic127—22no—
49C127⋊C8—24no—
50C23×C127—21no—
51C23×C127—41no—
52Q8×C127—41no—
53D4×C127—41no—
54D4×C127—41no—
55D4×C127—41no—
56D4×C127—42no—
57D4×C127—42no—
58C2×C4×C127—21no—
59C2×C4×C127C2×D412yes—
60C2×C4×C127C2×D411yescharacteristic
61C8×C127—22no—
62C22×C127—41no—
63C22×C127—42no—
64C22×C127—82no—
65C22×C127—81no—
66C22×C127#12011yescharacteristic
67C2×D127—42no—
68C2×D127—42no—
69C2×D127—42no—
70C2×D127—84no—
71C4×C127—21no—
72C4×C127—22no—
73C4×C127—41no—
74Dic127—42no—
75C2×C127—81no—
76C2×C127—161no—
77C2×C127#45312yes—
78C2×C127#45311yescharacteristic
79D127—82no—
80#1655—1271nomaximal, Sylow
81C127#165511yescharacteristic, complemented, Sylow
82H(3,𝔽4)—1271no—
83#496—1272no—
84#496D12711yescharacteristic, complemented
85#329—1271no—
86#329—1272no—
87C2×SD16—21no—
88C2×SD16—2542no—
89C2×D8—21no—
90C2×D8—2542no—
91#127C2×D12711yescharacteristic, complemented
92#124—1272no—
93#124—2542no—
94#124—2541no—
95#124C2×D12711yescharacteristic
96#120—2542no—
97#120—2541no—
98#97—1272no—
99#97C2×D12711yescharacteristic, complemented
100C24—2541no—
101C2×Q8—1272no—
102C2×Q8C22⋊D12711yescharacteristic
103C2×D4—22no—
104C2×D4—21no—
105C2×D4—1272no—
106C2×D4—2541no—
107C2×D4—2542no—
108C2×D4C22⋊D12711yescharacteristic
109SD16—42no—
110SD16—5084no—
111D8—42no—
112D8—5084no—
113C2×C8—21no—
114C2×C8—2542no—
115C22⋊C4—22no—
116C22⋊C4—2542no—
117C22⋊C4—2541no—
118C22⋊C4—2542no—
119C22⋊C4—2542no—
120C42—2541no—
121C42C22×D12711yescharacteristic
122C23—21no—
123C23—41no—
124C23—10162no—
125C23—10161no—
126C23—2542no—
127Q8—41no—
128Q8—5082no—
129D4—41no—
130D4—41no—
131D4—42no—
132D4—41no—
133D4—42no—
134D4—10161no—
135D4—10162no—
136D4—5082no—
137C2×C4—21no—
138C2×C4—2542no—
139C2×C4D4×D12712yes—
140C2×C4C2×C22⋊D12711yescharacteristic
141C8—22no—
142C8—2544no—
143C22—42no—
144C22—41no—
145C22—82no—
146C22—81no—
147C22—10164no—
148C22—5082no—
149C22—5082no—
150C22—5082no—
151C22#1686117711yescharacteristic
152C4—22no—
153C4—21no—
154C4—41no—
155C4—5082no—
156C2—81no—
157C2—161no—
158C2—10162no—
159C2#2255004212yes—
160C2#2255004311yescharacteristic
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#496×C127C2
socleC22×C127#120
hypercenter#496D127
hypocenterC127#1655
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #22550618C2semidirect
2. #22550263C2semidirect
3. #22549616C2semidirect
4. #22549615C2semidirect
5. #496×C127C2semidirect
13. #16861120C22semidirect
16. #127×C127C22semidirect
18. #97×C127C22semidirect
81. C127#1655semidirect
84. #496D127semidirect
91. #127C2×D127semidirect
99. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506182
#225502632
#225496162
#225496152
#496×C1272
#1655127