Finite Group Databasebeta

← #30638023

Subgroups — #30638023

Order 16256.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#22550266C212yesmaximal, complemented
2#22550242C211yesmaximal, characteristic, complemented
3#22549618C211yesmaximal, characteristic, complemented
4#22549616C212yesmaximal, complemented
5#562×C127C211yesmaximal, characteristic, complemented
6C2×#14523235—22no—
7C2×#14523233—22no—
8#16861206C2211yescharacteristic
9#16861129—22no—
10#16861120C2211yescharacteristic
11#16861120C2212yes—
12#125×C127C2211yescharacteristic
13#124×C127C2212yes—
14#123×C127—22no—
15C22⋊Q8×C127—22no—
16#121×C127—22no—
17C2×D508—22no—
18C2×D508—21no—
19#14523235—44no—
20#14523233—44no—
21C2×C127⋊C8—22no—
22C2×C127⋊C8—21no—
23C2×Q8×C127D412yes—
24C2×D4×C127—22no—
25C2×D4×C127D412yes—
26C22×C4×C127—22no—
27C4⋊C4×C127—22no—
28C4⋊C4×C127—22no—
29C4⋊C4×C127D412yes—
30C22⋊C4×C127—22no—
31C22⋊C4×C127—22no—
32C22⋊C4×C127—22no—
33C42×C127C2311yescharacteristic
34C22×D127—41no—
35D508—42no—
36D508—42no—
37D508—42no—
38C127⋊C8—24no—
39C127⋊C8—41no—
40C23×C127—22no—
41Q8×C127—42no—
42D4×C127—42no—
43D4×C127—82no—
44C2×C4×C127—22no—
45C2×C4×C127—22no—
46C2×C4×C127—82no—
47C2×C4×C127C2×D411yescharacteristic
48C2×C4×C127C2×D412yes—
49C22×C127—42no—
50C22×C127—42no—
51C22×C127—42no—
52C22×C127#12011yescharacteristic
53C2×D127—82no—
54C2×D127—81no—
55C4×C127—22no—
56C4×C127—21no—
57C4×C127—42no—
58C4×C127—42no—
59C2×C127—82no—
60C2×C127#45312yes—
61C2×C127#45411yescharacteristic
62D127—161no—
63#1656—1271nomaximal, Sylow
64C127#165611yescharacteristic, complemented, Sylow
65#562D12711yescharacteristic, complemented
66#496—1272no—
67#490—1271no—
68#330—1271no—
69#329—1272no—
70C2×SD16—2542no—
71C2×D8—2542no—
72#127—1271no—
73#125C2×D12711yescharacteristic
74#124C2×D12712yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—2542no—
79#97—1271no—
80#97—1272no—
81C2×Q8C22⋊D12712yes—
82C2×D4—22no—
83C2×D4—2541no—
84C2×D4—2542no—
85C2×D4C22⋊D12712yes—
86C22×C4—22no—
87SD16—5084no—
88D8—5084no—
89C2×C8—2541no—
90C2×C8—2542no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D12712yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D12711yescharacteristic
98C23—22no—
99C23—5081no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—5082no—
104D4—5082no—
105D4—5082no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D12711yescharacteristic
110C2×C4C2×C22⋊D12712yes—
111C8—5081no—
112C8—2544no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—10162no—
117C22—10161no—
118C22#1686117811yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—20321no—
125C2#2255004412yes—
126C2#2255004911yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#562×C127C2
socleC22×C127#120
hypercenter#562D127
hypocenterC127#1656
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #22550266C2semidirect
2. #22550242C2semidirect
3. #22549618C2semidirect
4. #22549616C2semidirect
5. #562×C127C2semidirect
64. C127#1656semidirect
65. #562D127semidirect

Maximal subgroups

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

subgroupindex
#225502662
#225502422
#225496182
#225496162
#562×C1272
#1656127