Finite Group Databasebeta

← #30638025

Subgroups — #30638025

Order 16256.

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

normal subgroups only

This group has 230 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, complemented
2#22550265C212yesmaximal, complemented
3#22550242C211yesmaximal, characteristic, complemented
4#22549615C212yesmaximal, complemented
5#330×C127C211yesmaximal, characteristic, complemented
6C2×#14523237—22no—
7C2×D1016—22no—
8#16861206C2211yescharacteristic
9#16861197C2212yes—
10#16861194—22no—
11#16861187—22no—
12#16861184—22no—
13#16861129—22no—
14#16861120C2211yescharacteristic
15#125×C127C2211yescharacteristic
16#97×C127C2212yes—
17C2×Dic254D412yes—
18C2×D508—22no—
19C2×D508—22no—
20C2×D508—21no—
21C2×D508D412yes—
22C2×C4×D127—22no—
23#14523237—44no—
24D1016—44no—
25C2×C127⋊C8—21no—
26Dic127⋊C4—22no—
27(C2×D127)⋊C4—22no—
28(C2×D127)⋊C4—22no—
29(C2×D127)⋊C4—22no—
30C2×C8×C127—22no—
31C4⋊C4×C127—22no—
32C4⋊C4×C127D412yes—
33C42×C127C2311yescharacteristic
34C22×D127—22no—
35C22×D127—41no—
36Dic254—42no—
37D508—42no—
38D508—42no—
39D508—42no—
40D508—42no—
41D508—82no—
42C2×Dic127—22no—
43C4×D127—82no—
44C127⋊C8—41no—
45C2×C4×C127—22no—
46C2×C4×C127C2×D412yes—
47C2×C4×C127C2×D411yescharacteristic
48C8×C127—24no—
49C22×C127#12011yescharacteristic
50C2×D127—42no—
51C2×D127—42no—
52C2×D127—42no—
53C2×D127—82no—
54C2×D127—81no—
55C4×C127—22no—
56C4×C127—21no—
57C4×C127—42no—
58Dic127—42no—
59C2×C127#45312yes—
60C2×C127#45411yescharacteristic
61D127—82no—
62D127—161no—
63#1656—1271nomaximal, Sylow
64C127#165611yescharacteristic, complemented, Sylow
65#562—1271no—
66#496—1272no—
67#490—1271no—
68#330D12711yescharacteristic, complemented
69#329—1272no—
70C2×SD16—2542no—
71C2×D8—2542no—
72#127—1271no—
73#125C2×D12711yescharacteristic
74#124—1272no—
75#123—2542no—
76C22⋊Q8—2542no—
77#121—2542no—
78#102—2542no—
79#97—1271no—
80#97C2×D12712yes—
81C2×Q8—1272no—
82C2×D4—1272no—
83C2×D4—2542no—
84C2×D4—2542no—
85C2×D4—2541no—
86C22×C4—2542no—
87SD16—5084no—
88D8—5084no—
89C2×C8—22no—
90C2×C8—2541no—
91C4⋊C4—22no—
92C4⋊C4—2542no—
93C4⋊C4D50812yes—
94C22⋊C4—2542no—
95C22⋊C4—2542no—
96C22⋊C4—2542no—
97C42C22×D12711yescharacteristic
98C23—5081no—
99C23—2542no—
100Q8—5082no—
101D4—10162no—
102D4—5082no—
103D4—5082no—
104D4—5082no—
105D4—5082no—
106C2×C4—22no—
107C2×C4—10162no—
108C2×C4—2542no—
109C2×C4D4×D12712yes—
110C2×C4C2×D50811yescharacteristic
111C8—24no—
112C8—5081no—
113C22—10162no—
114C22—10161no—
115C22—5082no—
116C22—5082no—
117C22—5082no—
118C22#1686117911yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—5082no—
123C2—20321no—
124C2—10162no—
125C2#2255004512yes—
126C2#2255005111yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#330×C127C2
socleC22×C127#120
hypercenter#330D127
hypocenterC127#1656
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #22550624C2semidirect
2. #22550265C2semidirect
3. #22550242C2semidirect
4. #22549615C2semidirect
5. #330×C127C2semidirect
64. C127#1656semidirect
68. #330D127semidirect

Maximal subgroups

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

subgroupindex
#225506242
#225502652
#225502422
#225496152
#330×C1272
#1656127