Finite Group Databasebeta

← #27140796

Subgroups — #27140796

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#20261471C211yesmaximal, characteristic, complemented
2#20261111C211yesmaximal, characteristic
3#20260936C212yesmaximal, complemented
4#20260465C212yesmaximal, complemented
5#331×C97C211yesmaximal, characteristic, complemented
6C2×#13095500—22no—
7C2×#13095496—22no—
8#15690807C2211yescharacteristic
9#15690797—22no—
10#15690791—22no—
11C4×Dic194C2212yes—
12D4×Dic97—22no—
13#15690732—22no—
14C4×#11715895C2211yescharacteristic
15#127×C97C2211yescharacteristic
16C4⋊C8×C97C2212yes—
17C2×Dic194D412yes—
18C2×D388—21no—
19C2×D388—22no—
20C22×Dic97—22no—
21#13095500—44no—
22#13095496—44no—
23C2×#11715895—21no—
24C4⋊Dic97D412yes—
25#13095490—22no—
26C22⋊Dic97—24no—
27C22⋊Dic97—22no—
28C4×Dic97—22no—
29C2×D4×C97—22no—
30C2×D4×C97D412yes—
31C2×C8×C97—22no—
32C42×C97C2311yescharacteristic
33C23×C97—22no—
34C22×D97—41no—
35D4×C97—42no—
36D4×C97—42no—
37D4×C97—42no—
38C2×C4×C97C2×D412yes—
39C2×C4×C97C2×D411yescharacteristic
40C2×Dic97—22no—
41C2×Dic97—22no—
42C2×Dic97—82no—
43D388—42no—
44D388—42no—
45D388—42no—
46Dic194—22no—
47Dic194—22no—
48C8×C97—42no—
49#11715895—22no—
50C22×C97—44no—
51C22×C97—42no—
52C22×C97#12011yescharacteristic
53C2×D97—82no—
54C2×D97—81no—
55C4×C97—21no—
56C4×C97#13512yes—
57C4×C97AGL(1,ℤ/8)12yes—
58Dic97—42no—
59Dic97—42no—
60C2×C97—82no—
61C2×C97#44912yes—
62C2×C97#45311yescharacteristic
63D97—161no—
64#1620—971nomaximal, Sylow
65C97#162011yescharacteristic, complemented, Sylow
66#559—971no—
67#493—971no—
68#460—972no—
69#331D9711yescharacteristic, complemented
70Q8⋊C8—972no—
71C2×SD16—1942no—
72C2×D8—1942no—
73#127—971no—
74#127C2×D9711yescharacteristic
75#124—1942no—
76#123—1942no—
77C4×Q8—972no—
78C4×D4—1942no—
79C4⋊C8C2×D9712yes—
80#102—1942no—
81C4×C8—971no—
82C2×Q8—972no—
83C2×D4—22no—
84C2×D4—1942no—
85C2×D4—1941no—
86C2×D4D38812yes—
87C22×C4—1942no—
88SD16—3884no—
89D8—3884no—
90C2×C8—22no—
91C2×C8—1941no—
92C4⋊C4—972no—
93C4⋊C4—1942no—
94C22⋊C4—1944no—
95C22⋊C4—1942no—
96C42—1942no—
97C42C22×D9711yescharacteristic
98C23—22no—
99C23—3881no—
100Q8—1942no—
101Q8—1942no—
102D4—42no—
103D4—42no—
104D4—42no—
105D4—3882no—
106D4—3882no—
107D4—3882no—
108C2×C4—7762no—
109C2×C4—1942no—
110C2×C4—1942no—
111C2×C4D4×D9712yes—
112C2×C4C2×D38811yescharacteristic
113C8—42no—
114C8—1942no—
115C22—44no—
116C22—42no—
117C22—7762no—
118C22—7761no—
119C22#1569078011yescharacteristic
120C4—21no—
121C4—3882no—
122C4—3882no—
123C4#1569083112yes—
124C4#1569083412yes—
125C2—82no—
126C2—15521no—
127C2#2026088112yes—
128C2#2026090511yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#331×C97C2
socleC22×C97#120
prime radical#20261111C2
hypercenter#331D97
hypocenterC97#1620
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
1. #20261471C2semidirect
3. #20260936C2semidirect
4. #20260465C2semidirect
5. #331×C97C2semidirect
65. C97#1620semidirect
69. #331D97semidirect

Maximal subgroups

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

subgroupindex
#202614712
#202611112
#202609362
#202604652
#331×C972
#162097