Finite Group Databasebeta

← #30637847

Subgroups — #30637847

Order 16256.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550511C211yesmaximal, characteristic
2#22550248C211yesmaximal, characteristic, complemented
3#22550167C212yesmaximal, complemented
4#22549603C212yesmaximal, complemented
5#331×C127C211yesmaximal, characteristic, complemented
6C2×D4×D127—22no—
7C2×#14523236—22no—
8#16861209C2211yescharacteristic, complemented
9#16861180—22no—
10#16861177—22no—
11C4×D508C2212yes—
12#16861141—22no—
13#16861131—22no—
14C4×C127⋊C8C2211yescharacteristic, complemented
15#127×C127C2211yescharacteristic
16C4⋊C8×C127C2212yescomplemented
17C23×D127—22no—
18C2×Dic254—21no—
19C2×D508D412yes—
20C2×C22⋊D127—24no—
21D4×D127—44no—
22D4×D127—44no—
23C2×C4×D127—22no—
24#14523236—44no—
25C2×C127⋊C8—21no—
26C4⋊Dic127—22no—
27C4⋊Dic127D412yes—
28C22⋊Dic127—22no—
29(C2×D127)⋊C4—22no—
30C2×D4×C127—22no—
31C2×D4×C127D412yes—
32C2×C8×C127—22no—
33C42×C127C2311yescharacteristic
34C22×D127—22no—
35C22×D127—48no—
36C22×D127—82no—
37Dic254—42no—
38D508—22no—
39D508SD1614yescomplemented
40C22⋊D127—48no—
41C2×Dic127—22no—
42C2×Dic127—41no—
43C4×D127—42no—
44C4×D127—42no—
45C127⋊C8—22no—
46C23×C127—22no—
47D4×C127—42no—
48D4×C127—42no—
49D4×C127—42no—
50C2×C4×C127C2×D412yes—
51C2×C4×C127C2×D411yescharacteristic
52C8×C127—42no—
53C22×C127—44no—
54C22×C127—42no—
55C22×C127#12011yescharacteristic
56C2×D127—24no—
57C2×D127—42no—
58C2×D127—42no—
59C2×D127—88no—
60C4×C127—21no—
61C4×C127C2×SD1612yes—
62C4×C127AGL(1,ℤ/8)12yes—
63Dic127—42no—
64Dic127—81no—
65C2×C127—82no—
66C2×C127#45012yes—
67C2×C127#45311yescharacteristic
68D127—44no—
69#1629—1271nomaximal, Sylow
70C127#162911yescharacteristic, complemented, Sylow
71D42—1271no—
72#492—1271no—
73#476—1272no—
74#331D12711yescharacteristic, complemented
75D4⋊C8—1272no—
76C22×D4—2542no—
77C2×SD16—2542no—
78C4⋊Q8—1271no—
79#127C2×D12711yescharacteristic
80#121—2542no—
81#120—2542no—
82C4×D4—1272no—
83#106—2542no—
84C4⋊C8C2×D12712yescomplemented
85#102—2542no—
86C4×C8—1271no—
87C24—2542no—
88C2×Q8—2541no—
89C2×D4—22no—
90C2×D4—1272no—
91C2×D4—5084no—
92C2×D4—5084no—
93C2×D4—2544no—
94C2×D4D50812yes—
95C22×C4—2542no—
96SD16—5084no—
97C2×C8—22no—
98C2×C8—2541no—
99C4⋊C4—1272no—
100C4⋊C4—2542no—
101C22⋊C4—2542no—
102C22⋊C4—2542no—
103C42C22×D12711yescharacteristic
104C23—22no—
105C23—10162no—
106C23—5088no—
107C23—2542no—
108Q8—5082no—
109D4—42no—
110D4—42no—
111D4—42no—
112D4—1274no—
113D4—5088no—
114D4—2542no—
115C2×C4—5081no—
116C2×C4—5082no—
117C2×C4—5082no—
118C2×C4—2542no—
119C2×C4D4×D12712yes—
120C2×C4C2×D50811yescharacteristic
121C8—42no—
122C8—2542no—
123C22—44no—
124C22—42no—
125C22—10168no—
126C22—5082no—
127C22—5082no—
128C22—2544no—
129C22#1686117911yescharacteristic
130C4—21no—
131C4—10161no—
132C4—5082no—
133C4SD16×D12712yes—
134C4#1686123312yes—
135C2—82no—
136C2—5084no—
137C2#2255002812yes—
138C2#2255004611yescharacteristic
subgroupG/H
centerC22#16861179
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting#331×C127C2
socleC22×C127#120
prime radical#22550511C2
hypercenter#331D127
hypocenterC127#1629
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #22550248C2semidirect
3. #22550167C2semidirect
4. #22549603C2semidirect
5. #331×C127C2semidirect
8. #16861209C22semidirect
14. C4×C127⋊C8C22semidirect
16. C4⋊C8×C127C22semidirect
39. D508SD16semidirect
70. C127#1629semidirect
74. #331D127semidirect
84. C4⋊C8C2×D127semidirect

Maximal subgroups

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

subgroupindex
#225505112
#225502482
#225501672
#225496032
#331×C1272
#1629127