Finite Group Databasebeta

← #27157198

Subgroups — #27157198

Order 12416.

212 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 62 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#20261756C211yesmaximal, characteristic
2C4×#13095520C211yesmaximal, characteristic
3#20261592C214yesmaximal, complemented
4C2×#15690801C211yesmaximal, characteristic
5C2×#13095520—21no—
6C4⋊#11715897—21no—
7C22⋊#11715897—21no—
8#15690865C2212yes—
9#15690865C2212yes—
10C4×#11715897—21no—
11#15690801—21no—
12#15690801—21no—
13#15690801C412yes—
14#15690801C2212yes—
15C2×C4⋊Dic97—21no—
16C2×C4⋊Dic97C411yescharacteristic
17C2×#13095490—21no—
18C2×C4×Dic97C2211yescharacteristic
19C2×C4⋊C4×C97C411yescharacteristic
20#13095520—24no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C22×Dic97—21no—
24C22×Dic97D411yescharacteristic
25C22×Dic97D411yescharacteristic
26C4⋊Dic97—21no—
27C4⋊Dic97—22no—
28C4⋊Dic97—22no—
29C4⋊Dic97C2×C412yes—
30#13095490—22no—
31#13095490—22no—
32C4×Dic97C2×C411yescharacteristic
33C4×Dic97D411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97C2311yescharacteristic
36C22×C4×C97—21no—
37C22×C4×C97C2×C411yescharacteristic
38C4⋊C4×C97—21no—
39C4⋊C4×C97C2×C412yes—
40C23×C97C22⋊C411yescharacteristic
41C2×C4×C97—22no—
42C2×C4×C97—22no—
43C2×C4×C97—42no—
44C2×C4×C97C22⋊C411yescharacteristic
45C2×C4×C97C22×C411yescharacteristic
46C2×Dic97—22no—
47C2×Dic97—22no—
48C2×Dic97—41no—
49C2×Dic97—42no—
50C2×Dic97C22⋊C411yescharacteristic
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C2×D411yescharacteristic
53C2×Dic97C2×D411yescharacteristic
54#11715897—24no—
55#11715897—42no—
56C22×C97—21no—
57C22×C97—21no—
58C22×C97#10412yes—
59C22×C97C2×C22⋊C411yescharacteristic
60C4×C97—21no—
61C4×C97—42no—
62C4×C97#10112yes—
63Dic97—21no—
64Dic97—21no—
65Dic97—21no—
66Dic97—21no—
67Dic97—42no—
68C2×C97—22no—
69C2×C97C2×#10111yescharacteristic
70C2×C97C2×#10411yescharacteristic
71C2×C97#42111yescharacteristic
72#1528—971nomaximal, Sylow
73C97#152811yescharacteristic, complemented, Sylow
74C2×#125—971no—
75#432—971no—
76C4×M16—971no—
77#330—974no—
78C2×M16—1941no—
79#125—972no—
80#125—972no—
81#125—1941no—
82#125—1941no—
83C2×C4⋊C4—971no—
84C2×C4⋊C4—1941no—
85C2×C4⋊C4—1941no—
86C2×C4⋊C4#8760411yescharacteristic
87C2×C42—971no—
88C4⋊C8—1941no—
89C22⋊C8—1941no—
90#97—972no—
91#97—972no—
92C4×C8—1941no—
93C22×C4—21no—
94C22×C4—971no—
95C22×C4—971no—
96C22×C4—1941no—
97C22×C4C2×#8760411yescharacteristic
98M16—1944no—
99C2×C8—1942no—
100C2×C8—1942no—
101C4⋊C4—21no—
102C4⋊C4—972no—
103C4⋊C4—1942no—
104C4⋊C4—1942no—
105C4⋊C4—1942no—
106C4⋊C4—1942no—
107C4⋊C4—1941no—
108C4⋊C4C2×#8760412yes—
109C42—971no—
110C42—971no—
111C42—971no—
112C42—971no—
113C23C22⋊#8760411yescharacteristic
114C2×C4—22no—
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—971no—
118C2×C4—971no—
119C2×C4—971no—
120C2×C4—971no—
121C2×C4—3882no—
122C2×C4—3881no—
123C2×C4—1942no—
124C2×C4—1942no—
125C2×C4C22⋊#8760411yescharacteristic
126C2×C4C22×#8760411yescharacteristic
127C8—3882no—
128C8—1944no—
129C22—21no—
130C22—21no—
131C22#1569087412yes—
132C22C2×C22⋊#8760411yescharacteristic
133C4—21no—
134C4—42no—
135C4—3882no—
136C4—1941no—
137C4—1941no—
138C4—1941no—
139C4—1941no—
140C4#1569087112yes—
141C2—22no—
142C2C2×#1569087111yescharacteristic
143C2C2×#1569087411yescharacteristic
144C2#2026173411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C22⋊C4
cosocleC4×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1528
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20261592C2semidirect
73. C97#1528semidirect

Maximal subgroups

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

subgroupindex
#202617562
C4×#130955202
#202615922
C2×#156908012
#152897