Finite Group Databasebeta

← #30636733

Subgroups — #30636733

Order 16256.

168 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 37 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#22549645C211yesmaximal, characteristic, complemented
2#22549602C211yesmaximal, characteristic, complemented
3#394×C127C211yesmaximal, characteristic
4C2×#14523232—21no—
5C22⋊(C127⋊C8)—21no—
6C22×D4×C127C411yescharacteristic
7C2×C4⋊C4×C127C2211yescharacteristic
8C2×C22⋊C4×C127—21no—
9C2×C22⋊C4×C127—21no—
10#95×C127C411yescharacteristic
11#14523232—22no—
12#14523232—41no—
13C2×C127⋊C8—41no—
14C2×C127⋊C8—41no—
15C24×C127—21no—
16C2×D4×C127—41no—
17C2×D4×C127—42no—
18C22×C4×C127—21no—
19C22×C4×C127C2×C411yescharacteristic
20C22×C4×C127D411yescharacteristic
21C22×C4×C127D411yescharacteristic
22C4⋊C4×C127—41no—
23C22⋊C4×C127—42no—
24C22⋊C4×C127—41no—
25C22⋊C4×C127—41no—
26C127⋊C8—42no—
27C127⋊C8—81no—
28C23×C127—22no—
29C23×C127—41no—
30C23×C127—41no—
31C23×C127—41no—
32C23×C127—41no—
33C23×C127—41no—
34C23×C127—41no—
35C23×C127C22⋊C411yescharacteristic
36D4×C127—42no—
37D4×C127—42no—
38C2×C4×C127—21no—
39C2×C4×C127—21no—
40C2×C4×C127—21no—
41C2×C4×C127—21no—
42C2×C4×C127—41no—
43C2×C4×C127—41no—
44C2×C4×C127—41no—
45C2×C4×C127—41no—
46C2×C4×C127—41no—
47C2×C4×C127D811yescharacteristic
48C2×C4×C127SD1611yescharacteristic
49C22×C127—21no—
50C22×C127—21no—
51C22×C127—42no—
52C22×C127—42no—
53C22×C127—42no—
54C22×C127—81no—
55C22×C127—81no—
56C22×C127—81no—
57C22×C127—81no—
58C22×C127#9911yescharacteristic
59C22×C127#10211yescharacteristic
60C22×C127#10411yescharacteristic
61C4×C127—21no—
62C4×C127—21no—
63C4×C127—41no—
64C4×C127—41no—
65C4×C127—81no—
66C2×C127—21no—
67C2×C127—21no—
68C2×C127—82no—
69C2×C127#32711yescharacteristic
70C2×C127AΣL(1,𝔽16)11yescharacteristic
71C2×C127#35511yescharacteristic
72#1345—1271nomaximal, Sylow
73C127#134511yescharacteristic, complemented, Sylow
74#394D12711yescharacteristic
75#343—1271no—
76#324—1271no—
77C22×D4Dic12711yescharacteristic
78C2×M16—2541no—
79C2×C4⋊C4C2×D12711yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C22⋊C8—2541no—
83#95Dic12711yescharacteristic
84C24—21no—
85C2×D4—41no—
86C2×D4—42no—
87C22×C4—21no—
88C22×C4C2×Dic12711yescharacteristic
89C22×C4C22⋊D12711yescharacteristic
90C22×C4C22⋊D12711yescharacteristic
91M16—5081no—
92M16—2542no—
93C2×C8—5081no—
94C2×C8—5081no—
95C4⋊C4—41no—
96C22⋊C4—42no—
97C22⋊C4—41no—
98C22⋊C4—41no—
99C23—22no—
100C23—41no—
101C23—41no—
102C23—41no—
103C23—41no—
104C23—41no—
105C23—41no—
106C23C22⋊Dic12711yescharacteristic
107D4—42no—
108D4—42no—
109C2×C4—21no—
110C2×C4—21no—
111C2×C4—21no—
112C2×C4—21no—
113C2×C4—41no—
114C2×C4—41no—
115C2×C4—41no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4#1452323311yescharacteristic
119C2×C4#1452323611yescharacteristic
120C8—10161no—
121C8—5082no—
122C22—21no—
123C22—21no—
124C22—42no—
125C22—42no—
126C22—42no—
127C22—81no—
128C22—81no—
129C22—81no—
130C22—81no—
131C22#1686112411yescharacteristic
132C22#1686113011yescharacteristic
133C22#1686113611yescharacteristic
134C4—21no—
135C4—21no—
136C4—41no—
137C4—41no—
138C4—81no—
139C2—21no—
140C2—21no—
141C2—82no—
142C2#2254961011yescharacteristic
143C2#2254966311yescharacteristic
144C2#2254967311yescharacteristic
subgroupG/H
centerC22#16861124
commutatorC22×C4×C127C2×C4
FrattiniC2×C4⋊C4C2×D127
Fitting#394×C127C2
socleC22×C127#99
prime radicalC22×D4×C127C4
cosocleC2×C4⋊C4×C127C22
hypercenter#394D127
hypocenterC127#1345
LCSG ▷ C22×C4×C127 ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #22549645C2semidirect
2. #22549602C2semidirect
73. C127#1345semidirect

Maximal subgroups

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

subgroupindex
#225496452
#225496022
#394×C1272
#1345127