Finite Group Databasebeta

← #30638051

Subgroups — #30638051

Order 16256.

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

normal subgroups only

This group has 196 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#22550624C211yesmaximal, characteristic
2#22550238C212yesmaximal, complemented
3#22549618C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#491×C127C211yesmaximal, characteristic, complemented
6#16861197C2212yes—
7#16861194—22no—
8#16861187—22no—
9#16861184—22no—
10#16861132—22no—
11#16861129—22no—
12#16861120C2212yes—
13C4⋊Q8×C127C2211yescharacteristic
14#125×C127C2211yescharacteristic
15#103×C127—22no—
16#97×C127C2211yescharacteristic
17C2×Dic254D412yes—
18C2×D508—22no—
19C2×D508D412yes—
20C2×C4×D127—22no—
21C2×C127⋊C8—22no—
22Dic127⋊C4—22no—
23(C2×D127)⋊C4—22no—
24(C2×D127)⋊C4—22no—
25(C2×D127)⋊C4—22no—
26C2×Q8×C127—21no—
27C2×C8×C127—21no—
28C4⋊C4×C127—22no—
29C4⋊C4×C127—22no—
30C4⋊C4×C127D412yes—
31C42×C127C2311yescharacteristic
32C22×D127—22no—
33Dic254—42no—
34D508—42no—
35D508—82no—
36C2×Dic127—22no—
37C4×D127—82no—
38C127⋊C8—42no—
39Q8×C127—42no—
40C2×C4×C127—22no—
41C2×C4×C127—41no—
42C2×C4×C127C2×D412yes—
43C2×C4×C127C2×D411yescharacteristic
44C8×C127—41no—
45C22×C127#12011yescharacteristic
46C2×D127—42no—
47C2×D127—42no—
48C2×D127—42no—
49C4×C127—21no—
50C4×C127—22no—
51C4×C127—42no—
52C4×C127—81no—
53Dic127—42no—
54C2×C127#45411yescharacteristic
55C2×C127#45512yes—
56D127—82no—
57#1662—1271nomaximal, Sylow
58C127#166211yescharacteristic, complemented, Sylow
59#562—1271no—
60#491D12711yescharacteristic, complemented
61#489—1272no—
62#330—1271no—
63#329—1272no—
64C4⋊Q8C2×D12711yescharacteristic
65#125C2×D12711yescharacteristic
66#124—1272no—
67#123—2542no—
68C22⋊Q8—2542no—
69#121—2542no—
70#103—22no—
71#103—2542no—
72#102—2542no—
73#97—1272no—
74#97C2×D12711yescharacteristic
75C2×Q8—21no—
76C2×Q8—1272no—
77C2×D4—1272no—
78C2×D4—2542no—
79C22×C4—2542no—
80C2×C8—21no—
81C2×C8—2542no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—2542no—
85C4⋊C4C22⋊D12712yes—
86C22⋊C4—2542no—
87C22⋊C4—2542no—
88C22⋊C4—2542no—
89C42C22×D12711yescharacteristic
90C23—2542no—
91Q8—42no—
92Q8—5082no—
93D4—10162no—
94D4—5082no—
95C2×C4—22no—
96C2×C4—41no—
97C2×C4—10162no—
98C2×C4—2542no—
99C2×C4D4×D12712yes—
100C2×C4C2×C22⋊D12711yescharacteristic
101C8—41no—
102C8—5082no—
103C22—5082no—
104C22—5082no—
105C22—5082no—
106C22#1686117711yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—81no—
111C4—5082no—
112C2—10162no—
113C2#2255004811yescharacteristic
114C2#2255005212yes—
subgroupG/H
centerC22#16861177
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#491×C127C2
socleC22×C127#120
prime radical#22550624C2
hypercenter#491D127
hypocenterC127#1662
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
2. #22550238C2semidirect
3. #22549618C2semidirect
4. #22549615C2semidirect
5. #491×C127C2semidirect
58. C127#1662semidirect
60. #491D127semidirect

Maximal subgroups

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

subgroupindex
#225506242
#225502382
#225496182
#225496152
#491×C1272
#1662127