Finite Group Databasebeta

← #46781806

Subgroups — #46781806

Order 33664.

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

normal subgroups only

This group has 240 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#30682314C211yesmaximal, characteristic
2#30682138C212yesmaximal, complemented
3#30681682C211yesmaximal, characteristic, complemented
4Q8⋊(C263⋊C8)C212yesmaximal, complemented
5#559×C263C211yesmaximal, characteristic, complemented
6C2×#16862363—22no—
7C2×#16862361—22no—
8#22554467C2211yescharacteristic
9C4⋊(C263⋊C8)C2212yes—
10#22554390—22no—
11C4×C263⋊C8C2211yescharacteristic
12#127×C263C2211yescharacteristic
13#124×C263—22no—
14#123×C263—22no—
15C4×Q8×C263C2212yes—
16C4×D4×C263—22no—
17C2×D1052—22no—
18C2×D1052—21no—
19#16862363—44no—
20#16862361—44no—
21C2×C263⋊C8—21no—
22C2×C263⋊C8—22no—
23C2×Q8×C263D412yes—
24C2×D4×C263—22no—
25C2×D4×C263D412yes—
26C22×C4×C263—22no—
27C4⋊C4×C263—22no—
28C4⋊C4×C263D412yes—
29C22⋊C4×C263—24no—
30C22⋊C4×C263—22no—
31C42×C263—22no—
32C42×C263C2311yescharacteristic
33C22×D263—41no—
34D1052—42no—
35D1052—42no—
36D1052—42no—
37C263⋊C8—22no—
38C263⋊C8—42no—
39C23×C263—22no—
40Q8×C263—22no—
41Q8×C263—22no—
42D4×C263—42no—
43D4×C263—42no—
44D4×C263—42no—
45C2×C4×C263—22no—
46C2×C4×C263—22no—
47C2×C4×C263—82no—
48C2×C4×C263C2×D411yescharacteristic
49C2×C4×C263C2×D412yes—
50C22×C263—44no—
51C22×C263—42no—
52C22×C263#12011yescharacteristic
53C2×D263—82no—
54C2×D263—81no—
55C4×C263—21no—
56C4×C263—42no—
57C4×C263—42no—
58C4×C263#13512yes—
59C4×C263AGL(1,ℤ/8)12yes—
60C2×C263—82no—
61C2×C263#44912yes—
62C2×C263#45311yescharacteristic
63D263—161no—
64C263#162011yescharacteristic, complemented, Sylow
65#1620—2631nomaximal, Sylow
66#559D26311yescharacteristic, complemented
67#493—2631no—
68#460—2632no—
69#331—2631no—
70Q8⋊C8—2632no—
71C2×SD16—5262no—
72C2×D8—5262no—
73#127—2631no—
74#127C2×D26311yescharacteristic
75#124—22no—
76#123—22no—
77C4×Q8C2×D26312yes—
78C4×D4—22no—
79C4⋊C8—2632no—
80#102—5262no—
81C4×C8—2631no—
82C2×Q8C22⋊D26312yes—
83C2×D4—22no—
84C2×D4—5262no—
85C2×D4—5261no—
86C2×D4C22⋊D26312yes—
87C22×C4—22no—
88SD16—10524no—
89D8—10524no—
90C2×C8—5261no—
91C2×C8—5262no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D26312yes—
94C22⋊C4—24no—
95C22⋊C4—22no—
96C42—22no—
97C42C22×D26311yescharacteristic
98C23—22no—
99C23—10521no—
100Q8—22no—
101Q8—22no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—10522no—
106D4—10522no—
107D4—10522no—
108C2×C4—22no—
109C2×C4—22no—
110C2×C4—82no—
111C2×C4C2×C22⋊D26311yescharacteristic
112C2×C4C2×C22⋊D26312yes—
113C8—5262no—
114C8—10522no—
115C22—44no—
116C22—42no—
117C22—21042no—
118C22—21041no—
119C22#2255443911yescharacteristic
120C4—21no—
121C4—42no—
122C4—42no—
123C4#2255449012yes—
124C4#2255449512yes—
125C2—82no—
126C2—42081no—
127C2#3068208312yes—
128C2#3068210611yescharacteristic
subgroupG/H
centerC22#22554439
commutator, cosocleC42×C263C23
FrattiniC42C22×D263
Fitting#559×C263C2
socleC22×C263#120
prime radical#30682314C2
hypercenter#559D263
hypocenterC263#1620
LCSG ▷ C42×C263 ▷ C22×C263 ▷ C263
UCSC1 ◁ C22 ◁ C42 ◁ #559

Complements

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

NH ≅ G/Ntype
2. #30682138C2semidirect
3. #30681682C2semidirect
4. Q8⋊(C263⋊C8)C2semidirect
5. #559×C263C2semidirect
64. C263#1620semidirect
66. #559D263semidirect

Maximal subgroups

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

subgroupindex
#306823142
#306821382
#306816822
Q8⋊(C263⋊C8)2
#559×C2632
#1620263