Finite Group Databasebeta

← #30637846

Subgroups — #30637846

Order 16256.

314 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 65 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#22550248C211yesmaximal, characteristic, complemented
2#22550166C212yesmaximal, complemented
3#22549620C211yesmaximal, characteristic, complemented
4D4⋊(C127⋊C8)C212yesmaximal, complemented
5D42×C127C211yesmaximal, characteristic
6C2×#14523236—22no—
7#16861209C2211yescharacteristic, complemented
8#16861140—22no—
9C4⋊(C127⋊C8)C2212yescomplemented
10#16861130—22no—
11C4×C127⋊C8C2211yescharacteristic, complemented
12C22×D4×C127—22no—
13#127×C127C2211yescharacteristic
14#121×C127—22no—
15#120×C127—22no—
16C4×D4×C127C2212yes—
17C2×Dic254—21no—
18#14523236—44no—
19C2×C127⋊C8—21no—
20C2×C127⋊C8—22no—
21C4⋊Dic127—22no—
22C24×C127—22no—
23C2×D4×C127—24no—
24C2×D4×C127—22no—
25C2×D4×C127—44no—
26C2×D4×C127—44no—
27C2×D4×C127D412yes—
28C2×D4×C127D412yes—
29C22×C4×C127—22no—
30C4⋊C4×C127D412yes—
31C22⋊C4×C127—22no—
32C22⋊C4×C127—22no—
33C42×C127C2311yescharacteristic
34Dic254—42no—
35C2×Dic127—41no—
36C127⋊C8—22no—
37C127⋊C8—42no—
38C23×C127—22no—
39C23×C127—22no—
40C23×C127—48no—
41C23×C127—82no—
42D4×C127—22no—
43D4×C127—42no—
44D4×C127—48no—
45D4×C127—42no—
46D4×C127—42no—
47D4×C127SD1614yescomplemented
48C2×C4×C127—22no—
49C2×C4×C127—42no—
50C2×C4×C127—42no—
51C2×C4×C127C2×D411yescharacteristic
52C2×C4×C127C2×D412yes—
53C22×C127—24no—
54C22×C127—42no—
55C22×C127—42no—
56C22×C127—44no—
57C22×C127—42no—
58C22×C127—88no—
59C22×C127#12011yescharacteristic
60C4×C127—21no—
61C4×C127—42no—
62C4×C127C2×SD1612yes—
63C4×C127AGL(1,ℤ/8)12yes—
64Dic127—81no—
65C2×C127—44no—
66C2×C127—82no—
67C2×C127#45012yes—
68C2×C127#45311yescharacteristic
69#1629—1271nomaximal, Sylow
70C127#162911yescharacteristic, complemented, Sylow
71D42D12711yescharacteristic
72#492—1271no—
73#476—1272no—
74#331—1271no—
75D4⋊C8—1272no—
76C22×D4—22no—
77C2×SD16—2542no—
78C4⋊Q8—1271no—
79#127C2×D12711yescharacteristic
80#121—22no—
81#120—22no—
82C4×D4C2×D12712yes—
83#106—2542no—
84C4⋊C8—1272no—
85#102—2542no—
86C4×C8—1271no—
87C24—22no—
88C2×Q8—2541no—
89C2×D4—24no—
90C2×D4—22no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4C22⋊D12712yes—
94C2×D4C22⋊D12712yes—
95C22×C4—22no—
96SD16—5084no—
97C2×C8—2542no—
98C2×C8—2541no—
99C4⋊C4—2542no—
100C4⋊C4C22⋊D12712yes—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42C22×D12711yescharacteristic
104C23—22no—
105C23—22no—
106C23—48no—
107C23—82no—
108Q8—5082no—
109D4—22no—
110D4—42no—
111D4—48no—
112D4—42no—
113D4—42no—
114D4#1452323614yescomplemented
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—5081no—
119C2×C4C2×C22⋊D12711yescharacteristic
120C2×C4C2×C22⋊D12712yes—
121C8—5082no—
122C8—2542no—
123C22—24no—
124C22—42no—
125C22—42no—
126C22—44no—
127C22—42no—
128C22—88no—
129C22#1686117811yescharacteristic
130C4—21no—
131C4—42no—
132C4—10161no—
133C4C2×#1452323612yes—
134C4#1686123512yes—
135C2—44no—
136C2—82no—
137C2#2255002712yes—
138C2#2255004411yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC42×C127C23
FrattiniC42C22×D127
Fitting, prime radicalD42×C127C2
socleC22×C127#120
hypercenterD42D127
hypocenterC127#1629
LCSG ▷ C42×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

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

NH ≅ G/Ntype
1. #22550248C2semidirect
2. #22550166C2semidirect
3. #22549620C2semidirect
4. D4⋊(C127⋊C8)C2semidirect
7. #16861209C22semidirect
9. C4⋊(C127⋊C8)C22semidirect
11. C4×C127⋊C8C22semidirect
47. D4×C127SD16semidirect
70. C127#1629semidirect
114. D4#14523236semidirect

Maximal subgroups

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

subgroupindex
#225502482
#225501662
#225496202
D4⋊(C127⋊C8)2
D42×C1272
#1629127