Finite Group Databasebeta

← #30638034

Subgroups — #30638034

Order 16256.

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

normal subgroups only

This group has 228 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
2#22550238C212yesmaximal, complemented
3#22549616C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#497×C127C211yesmaximal, characteristic, complemented
6#16861198—21no—
7#16861197C2212yes—
8#16861195—22no—
9#16861179—22no—
10#16861177—21no—
11#16861132—22no—
12#16861129—22no—
13#16861120C2212yescomplemented
14C2×Q16×C127—21no—
15C2×SD16×C127—21no—
16C4⋊Q8×C127C2211yescharacteristic, complemented
17#124×C127C2211yescharacteristic
18#97×C127C2211yescharacteristic, complemented
19C23×D127—21no—
20C2×Dic254D412yes—
21C2×D508—21no—
22C2×D508D412yes—
23C2×C22⋊D127—22no—
24C2×C127⋊C8—22no—
25C22⋊Dic127—21no—
26(C2×D127)⋊C4—22no—
27(C2×D127)⋊C4—22no—
28(C2×D127)⋊C4—22no—
29C4×Dic127—21no—
30C2×Q8×C127—21no—
31C2×Q8×C127D411yescharacteristic
32C2×D4×C127D411yescharacteristic
33Q16×C127—42no—
34SD16×C127—42no—
35C2×C8×C127—21no—
36C4⋊C4×C127—22no—
37C22⋊C4×C127—22no—
38C42×C127C2311yescharacteristic
39C22×D127—22no—
40C22×D127—81no—
41C22×D127—82no—
42Dic254—42no—
43D508—42no—
44D508—81no—
45C22⋊D127—82no—
46C2×Dic127—22no—
47C127⋊C8—42no—
48C23×C127—21no—
49Q8×C127—41no—
50Q8×C127—41no—
51Q8×C127—41no—
52D4×C127—41no—
53C2×C4×C127—21no—
54C2×C4×C127—41no—
55C2×C4×C127C2×D412yes—
56C2×C4×C127C2×D411yescharacteristic
57C8×C127—22no—
58C22×C127—41no—
59C22×C127—42no—
60C22×C127#12011yescharacteristic
61C2×D127—42no—
62C2×D127—42no—
63C2×D127—42no—
64C2×D127—84no—
65C4×C127—21no—
66C4×C127—22no—
67C4×C127—41no—
68C4×C127—81no—
69Dic127—42no—
70C2×C127—81no—
71C2×C127#45311yescharacteristic
72C2×C127#45512yes—
73D127—82no—
74#1659—1271nomaximal, Sylow
75C127#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)—1271no—
77#497D12711yescharacteristic, complemented
78#489—1272no—
79#329—1271no—
80#329—1272no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D12711yescharacteristic, complemented
84#124—1272no—
85#124—2542no—
86#124—2541no—
87#124C2×D12711yescharacteristic
88#120—2542no—
89#120—2541no—
90#103—2542no—
91#102—2542no—
92#97—1272no—
93#97C2×D12711yescharacteristic, complemented
94C24—2541no—
95C2×Q8—21no—
96C2×Q8—1272no—
97C2×Q8C22⋊D12711yescharacteristic
98C2×D4—1272no—
99C2×D4—2541no—
100C2×D4—2542no—
101C2×D4C22⋊D12711yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—2542no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—2542no—
109C22⋊C4—2541no—
110C22⋊C4—2542no—
111C22⋊C4—2542no—
112C42—2541no—
113C42C22×D12711yescharacteristic
114C23—21no—
115C23—10161no—
116C23—10162no—
117C23—2542no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—5082no—
122D4—41no—
123D4—10161no—
124D4—10162no—
125D4—5082no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—2542no—
129C2×C4D4×D12712yes—
130C2×C4C2×C22⋊D12711yescharacteristic
131C8—22no—
132C8—5082no—
133C22—41no—
134C22—42no—
135C22—10164no—
136C22—5082no—
137C22—5082no—
138C22—5082no—
139C22#1686117711yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—5082no—
145C2—81no—
146C2—10162no—
147C2#2255004311yescharacteristic
148C2#2255005212yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#497×C127C2
socleC22×C127#120
prime radical#22550618C2
hypercenter#497D127
hypocenterC127#1659
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #22550238C2semidirect
3. #22549616C2semidirect
4. #22549615C2semidirect
5. #497×C127C2semidirect
13. #16861120C22semidirect
16. C4⋊Q8×C127C22semidirect
18. #97×C127C22semidirect
75. C127#1659semidirect
77. #497D127semidirect
83. C4⋊Q8C2×D127semidirect
93. #97C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225506182
#225502382
#225496162
#225496152
#497×C1272
#1659127