Finite Group Databasebeta

← #27156716

Subgroups — #27156716

Order 12416.

146 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 30 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#20261613C211yesmaximal, characteristic
2#20261579C211yesmaximal, characteristic
3#20260661C211yesmaximal, characteristic
4C2×C22⋊#87604—21no—
5C22⋊#11715897—21no—
6C2×#13095488—21no—
7C2×#13095488C2211yescharacteristic
8#15690717—21no—
9#15690717C411yescharacteristic, complemented
10C2×C4⋊C4×C97C411yescharacteristic, complemented
11C22×#87604—41no—
12C2×#11715897—41no—
13C22⋊#87604—22no—
14C22⋊#87604—41no—
15C23×D97D411yescharacteristic
16C22×Dic97—21no—
17C22×Dic97D411yescharacteristic
18#13095488—41no—
19#13095488—41no—
20#13095488—41no—
21C22×C4×C97—21no—
22C22×C4×C97C2×C411yescharacteristic
23C4⋊C4×C97—41no—
24C23×C97C22⋊C411yescharacteristic
25C22×D97—21no—
26C22×D97—21no—
27C22×D97—41no—
28C22×D97—41no—
29C22×D97D811yescharacteristic
30C22×D97SD1611yescharacteristic
31C2×#87604—42no—
32C2×#87604—81no—
33C2×#87604—81no—
34C2×C4×C97—21no—
35C2×C4×C97—41no—
36C2×C4×C97—41no—
37C2×C4×C97—41no—
38C2×C4×C97—41no—
39C2×Dic97—21no—
40C2×Dic97—41no—
41C2×Dic97—41no—
42C2×Dic97—41no—
43C2×Dic97—41no—
44#11715897—81no—
45C22×C97—21no—
46C22×C97—21no—
47C22×C97#9911yescharacteristic
48C22×C97#10211yescharacteristic
49C22×C97#10411yescharacteristic
50C2×D97—21no—
51C2×D97—21no—
52C2×D97—41no—
53C2×D97—41no—
54C2×D97—41no—
55C2×D97—41no—
56C2×D97—81no—
57#87604—82no—
58C4×C97—41no—
59C4×C97—81no—
60Dic97—41no—
61Dic97—81no—
62C2×C97—21no—
63C2×C97—21no—
64C2×C97#32711yescharacteristic
65C2×C97#35211yescharacteristic
66C2×C97#35411yescharacteristic
67D97—41no—
68D97—41no—
69#1344—971nomaximal, Sylow
70C97#134411yescharacteristic, complemented, Sylow
71#397—971no—
72#342—971no—
73#323—971no—
74C2×C4⋊C4#8760411yescharacteristic, complemented
75C2×C22⋊C4—971no—
76C2×C22⋊C4—1941no—
77C2×C22⋊C4—1941no—
78C22⋊C8—1941no—
79#95—971no—
80#95—1941no—
81C24—971no—
82C22×C4—21no—
83C22×C4—971no—
84C22×C4—3881no—
85C22×C4—1941no—
86C22×C4C2×#8760411yescharacteristic
87C2×C8—3881no—
88C4⋊C4—41no—
89C22⋊C4—3881no—
90C22⋊C4—3881no—
91C22⋊C4—3881no—
92C22⋊C4—3881no—
93C22⋊C4—1942no—
94C23—971no—
95C23—971no—
96C23—3881no—
97C23—3881no—
98C23—1941no—
99C23—1941no—
100C23C22⋊#8760411yescharacteristic
101C2×C4—21no—
102C2×C4—41no—
103C2×C4—41no—
104C2×C4—41no—
105C2×C4—41no—
106C2×C4—7761no—
107C2×C4—7761no—
108C2×C4—3881no—
109C2×C4—3881no—
110C2×C4—3882no—
111C2×C4—3881no—
112C2×C4—3881no—
113C2×C4—1941no—
114C8—7761no—
115C22—21no—
116C22—21no—
117C22—7761no—
118C22—3881no—
119C22—3881no—
120C22—3881no—
121C22—3881no—
122C22—1941no—
123C22—1941no—
124C22#1569086811yescharacteristic
125C22#1569087211yescharacteristic
126C22#1569087411yescharacteristic
127C4—41no—
128C4—81no—
129C4—7761no—
130C4—7762no—
131C4—3881no—
132C2—21no—
133C2—21no—
134C2—3881no—
135C2—3881no—
136C2#2026158611yescharacteristic
137C2#2026163711yescharacteristic
138C2#2026164111yescharacteristic
subgroupG/H
centerC22#15690868
commutatorC22×C4×C97C2×C4
FrattiniC22×C4C2×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97#99
prime radicalC23×D97D4
cosocleC2×#13095488C22
hypercenterC2×C4⋊C4#87604
hypocenterC97#1344
LCSG ▷ C22×C4×C97 ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
9. #15690717C4semidirect
10. C2×C4⋊C4×C97C4semidirect
70. C97#1344semidirect
74. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
#202616132
#202615792
#202606612
#134497