Finite Group Databasebeta

← #27140820

Subgroups — #27140820

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#20261109C211yesmaximal, characteristic, complemented
3#20260931C212yesmaximal, complemented
4#20260462C212yesmaximal, complemented
5#333×C97C211yesmaximal, characteristic, complemented
6C2×#13095498—22no—
7C2×D776—22no—
8#15690807C2211yescharacteristic
9#15690799—22no—
10#15690785—22no—
11C4×D388C2212yes—
12#15690768—22no—
13#15690732—22no—
14C4×#11715895C2211yescharacteristic
15C4⋊Q8×C97C2211yescharacteristic
16C4⋊C8×C97C2212yes—
17C2×D388—22no—
18C2×D388—21no—
19C2×D388—22no—
20C2×D388D412yes—
21C2×C4×D97—22no—
22#13095498—44no—
23D776—44no—
24C2×#11715895—21no—
25C4⋊Dic97D412yes—
26#13095488—24no—
27#13095488—22no—
28C4×Dic97—22no—
29C2×Q8×C97D412yes—
30C2×C8×C97—22no—
31C4⋊C4×C97—22no—
32C42×C97C2311yescharacteristic
33C22×D97—22no—
34C22×D97—41no—
35Q8×C97—42no—
36C2×C4×C97—22no—
37C2×C4×C97C2×D412yes—
38C2×C4×C97C2×D411yescharacteristic
39C2×Dic97—22no—
40D388—22no—
41D388—22no—
42D388—42no—
43D388—42no—
44D388—42no—
45D388—82no—
46C4×D97—42no—
47C4×D97—42no—
48C8×C97—42no—
49#11715895—22no—
50C22×C97#12011yescharacteristic
51C2×D97—42no—
52C2×D97—42no—
53C2×D97—42no—
54C2×D97—82no—
55C2×D97—81no—
56C4×C97—21no—
57C4×C97—42no—
58C4×C97#13512yes—
59C4×C97AGL(1,ℤ/8)12yes—
60Dic97—42no—
61C2×C97#44912yes—
62C2×C97#45311yescharacteristic
63D97—82no—
64D97—161no—
65#1624—971nomaximal, Sylow
66C97#162411yescharacteristic, complemented, Sylow
67#555—971no—
68#492—971no—
69#459—972no—
70#333D9711yescharacteristic, complemented
71D4⋊C8—972no—
72C2×SD16—1942no—
73C2×D8—1942no—
74C4⋊Q8C2×D9711yescharacteristic
75#127—971no—
76#124—1942no—
77#121—1942no—
78C4×D4—972no—
79#117—1942no—
80C4⋊C8C2×D9712yes—
81#102—1942no—
82C4×C8—971no—
83C2×Q8D38812yes—
84C2×D4—972no—
85C2×D4—1942no—
86C2×D4—1942no—
87C2×D4—1941no—
88C22×C4—1942no—
89SD16—3884no—
90D8—3884no—
91C2×C8—22no—
92C2×C8—1941no—
93C4⋊C4—22no—
94C4⋊C4—972no—
95C22⋊C4—1944no—
96C22⋊C4—1942no—
97C42—1942no—
98C42C22×D9711yescharacteristic
99C23—3881no—
100C23—1942no—
101Q8—42no—
102D4—7762no—
103D4—3882no—
104D4—3882no—
105D4—3882no—
106D4—1942no—
107D4—1942no—
108C2×C4—22no—
109C2×C4—3882no—
110C2×C4—3882no—
111C2×C4—1942no—
112C2×C4D4×D9712yes—
113C2×C4C2×D38811yescharacteristic
114C8—42no—
115C8—1942no—
116C22—7762no—
117C22—7761no—
118C22—3882no—
119C22—3882no—
120C22—3882no—
121C22#1569078011yescharacteristic
122C4—21no—
123C4—42no—
124C4—3882no—
125C4#1569083312yes—
126C4#1569083712yes—
127C2—7762no—
128C2—15521no—
129C2#2026088312yes—
130C2#2026090511yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#333×C97C2
socleC22×C97#120
hypercenter#333D97
hypocenterC97#1624
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
1. #20261451C2semidirect
2. #20261109C2semidirect
3. #20260931C2semidirect
4. #20260462C2semidirect
5. #333×C97C2semidirect
66. C97#1624semidirect
70. #333D97semidirect

Maximal subgroups

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

subgroupindex
#202614512
#202611092
#202609312
#202604622
#333×C972
#162497