Finite Group Databasebeta

← #30636662

Subgroups — #30636662

Order 16256.

170 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 41 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#22549602C212yesmaximal, complemented
2#390×C127C211yesmaximal, characteristic
3C22⋊(C127⋊C8)—22no—
4C22×D4×C127C411yescharacteristic
5C2×C4⋊C4×C127C2211yescharacteristic
6C2×C22⋊C4×C127—22no—
7C2×C42×C127C411yescharacteristic
8C2×C127⋊C8—42no—
9C24×C127—21no—
10C2×D4×C127—21no—
11C2×D4×C127—41no—
12C2×D4×C127—41no—
13C2×D4×C127C812yescomplemented
14C22×C4×C127—21no—
15C22×C4×C127C2×C411yescharacteristic
16C22×C4×C127D412yes—
17C4⋊C4×C127—41no—
18C22⋊C4×C127—44no—
19C42×C127—41no—
20C127⋊C8—82no—
21C23×C127—22no—
22C23×C127—42no—
23C23×C127—42no—
24C23×C127—41no—
25C23×C127—41no—
26C23×C127C22⋊C411yescharacteristic
27D4×C127—42no—
28D4×C127—42no—
29C2×C4×C127—21no—
30C2×C4×C127—21no—
31C2×C4×C127—22no—
32C2×C4×C127—22no—
33C2×C4×C127—41no—
34C2×C4×C127—41no—
35C2×C4×C127—42no—
36C2×C4×C127C2×C811yescharacteristic
37C2×C4×C127M1611yescharacteristic
38C22×C127—21no—
39C22×C127—21no—
40C22×C127—44no—
41C22×C127—42no—
42C22×C127—82no—
43C22×C127—81no—
44C22×C127—81no—
45C22×C127C22⋊C811yescharacteristic
46C22×C127#9911yescharacteristic
47C22×C127#10011yescharacteristic
48C4×C127—21no—
49C4×C127—21no—
50C4×C127—42no—
51C4×C127—42no—
52C2×C127—21no—
53C2×C127—21no—
54C2×C127—82no—
55C2×C127#32311yescharacteristic
56C2×C127#35311yescharacteristic
57C2×C127#35511yescharacteristic
58#1318—1271nomaximal, Sylow
59C127#131811yescharacteristic, complemented, Sylow
60#390D12711yescharacteristic
61#324—1272no—
62C22×D4Dic12711yescharacteristic
63C2×C4⋊C4C2×D12711yescharacteristic
64C2×C22⋊C4—22no—
65C2×C42Dic12711yescharacteristic
66C22⋊C8—2542no—
67C24—21no—
68C2×D4—21no—
69C2×D4—41no—
70C2×D4—41no—
71C2×D4C127⋊C812yescomplemented
72C22×C4—21no—
73C22×C4C2×Dic12711yescharacteristic
74C22×C4C22⋊D12712yes—
75C2×C8—5082no—
76C4⋊C4—41no—
77C22⋊C4—44no—
78C42—41no—
79C23—22no—
80C23—42no—
81C23—42no—
82C23—41no—
83C23—41no—
84C23C22⋊Dic12711yescharacteristic
85D4—42no—
86D4—42no—
87C2×C4—21no—
88C2×C4—21no—
89C2×C4—22no—
90C2×C4—22no—
91C2×C4—41no—
92C2×C4—41no—
93C2×C4—42no—
94C2×C4C2×C127⋊C811yescharacteristic
95C2×C4#1452323211yescharacteristic
96C8—10162no—
97C22—21no—
98C22—21no—
99C22—44no—
100C22—42no—
101C22—82no—
102C22—81no—
103C22—81no—
104C22C22⋊(C127⋊C8)11yescharacteristic
105C22#1686112411yescharacteristic
106C22#1686112611yescharacteristic
107C4—21no—
108C4—21no—
109C4—42no—
110C4—42no—
111C2—21no—
112C2—21no—
113C2—82no—
114C2#2254960011yescharacteristic
115C2#2254966911yescharacteristic
116C2#2254967311yescharacteristic
subgroupG/H
centerC22#16861124
commutatorC2×C4×C127C2×C8
FrattiniC2×C4⋊C4C2×D127
Fitting#390×C127C2
socleC22×C127#99
prime radicalC22×D4×C127C4
cosocleC2×C4⋊C4×C127C22
hypercenter#390D127
hypocenterC127#1318
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #22549602C2semidirect
13. C2×D4×C127C8semidirect
59. C127#1318semidirect
71. C2×D4C127⋊C8semidirect

Maximal subgroups

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

subgroupindex
#225496022
#390×C1272
#1318127