Finite Group Databasebeta

← #27140817

Subgroups — #27140817

Order 12416.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261451C211yesmaximal, characteristic, complemented
2#20260927C212yesmaximal, complemented
3#20260484C211yesmaximal, characteristic, complemented
4#20260464C212yesmaximal, complemented
5#492×C97C211yesmaximal, characteristic, complemented
6C2×#13095496—22no—
7#15690799—22no—
8#15690785—22no—
9C4×D388C2212yes—
10#15690768—22no—
11C4⋊#11715895C2212yes—
12#15690731—22no—
13C2×SD16×C97—22no—
14C4⋊Q8×C97C2211yescharacteristic
15#127×C97C2211yescharacteristic
16C4×C8×C97C2211yescharacteristic
17C2×D388—22no—
18C2×D388D412yes—
19C2×C4×D97—22no—
20#13095496—44no—
21C2×#11715895—22no—
22C4⋊Dic97D412yes—
23#13095488—24no—
24#13095488—22no—
25C4×Dic97—22no—
26C2×Q8×C97D412yes—
27C2×D4×C97—22no—
28C2×D4×C97—21no—
29SD16×C97—44no—
30C2×C8×C97—21no—
31C4⋊C4×C97—22no—
32C42×C97C2311yescharacteristic
33C23×C97—41no—
34C22×D97—22no—
35Q8×C97—42no—
36D4×C97—42no—
37D4×C97—42no—
38D4×C97—42no—
39C2×C4×C97—22no—
40C2×C4×C97C2×D412yes—
41C2×C4×C97C2×D411yescharacteristic
42C2×Dic97—22no—
43D388—22no—
44D388—22no—
45D388—82no—
46C4×D97—42no—
47C4×D97—42no—
48C8×C97—22no—
49#11715895—42no—
50C22×C97—82no—
51C22×C97—81no—
52C22×C97#12011yescharacteristic
53C2×D97—42no—
54C2×D97—42no—
55C2×D97—42no—
56C4×C97—21no—
57C4×C97—42no—
58C4×C97#13512yes—
59C4×C97AGL(1,ℤ/8)12yes—
60Dic97—42no—
61C2×C97—161no—
62C2×C97#44912yes—
63C2×C97#45311yescharacteristic
64D97—82no—
65#1624—971nomaximal, Sylow
66C97#162411yescharacteristic, complemented, Sylow
67#555—971no—
68#492D9711yescharacteristic, complemented
69#459—972no—
70#333—971no—
71D4⋊C8—972no—
72C2×SD16—22no—
73C2×D8—1942no—
74C4⋊Q8C2×D9711yescharacteristic
75#127C2×D9711yescharacteristic
76#124—1942no—
77#121—1942no—
78C4×D4—972no—
79#117—1942no—
80C4⋊C8—972no—
81#102—1942no—
82C4×C8C2×D9711yescharacteristic
83C2×Q8C22⋊D9712yes—
84C2×D4—22no—
85C2×D4—21no—
86C2×D4—972no—
87C2×D4—1942no—
88C22×C4—1942no—
89SD16—44no—
90D8—3884no—
91C2×C8—21no—
92C2×C8—1942no—
93C4⋊C4—22no—
94C4⋊C4—972no—
95C22⋊C4—1944no—
96C22⋊C4—1942no—
97C42—1942no—
98C42C22×D9711yescharacteristic
99C23—41no—
100C23—1942no—
101Q8—42no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—7762no—
106D4—1942no—
107D4—1942no—
108C2×C4—22no—
109C2×C4—3882no—
110C2×C4—3882no—
111C2×C4—1942no—
112C2×C4D4×D9712yes—
113C2×C4C2×C22⋊D9711yescharacteristic
114C8—22no—
115C8—3882no—
116C22—82no—
117C22—81no—
118C22—3882no—
119C22—3882no—
120C22—3882no—
121C22#1569077811yescharacteristic
122C4—21no—
123C4—42no—
124C4—3882no—
125C4#1569083112yes—
126C4#1569083712yes—
127C2—7762no—
128C2—161no—
129C2#2026087912yes—
130C2#2026090211yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#492×C97C2
socleC22×C97#120
hypercenter#492D97
hypocenterC97#1624
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. #20261451C2semidirect
2. #20260927C2semidirect
3. #20260484C2semidirect
4. #20260464C2semidirect
5. #492×C97C2semidirect
66. C97#1624semidirect
68. #492D97semidirect

Maximal subgroups

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

subgroupindex
#202614512
#202609272
#202604842
#202604642
#492×C972
#162497