Finite Group Databasebeta

← #27157275

Subgroups — #27157275

Order 12416.

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

normal subgroups only

This group has 210 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#20261738C212yesmaximal
2#20261594C214yesmaximal, complemented
3#20261288C211yesmaximal, characteristic
4C2×#13095520—22no—
5#15690877C2214yes—
6C4⋊#11715897—22no—
7#15690809C2212yes—
8#15690801C412yes—
9#15690787—22no—
10C4×Dic194—22no—
11#15690769C411yescharacteristic
12C2×C4×Dic97C2211yescharacteristic
13#117×C97C411yescharacteristic
14#13095520—44no—
15C2×#11715897—24no—
16C2×Dic194—22no—
17C22×Dic97D412yes—
18C4⋊Dic97—22no—
19C4⋊Dic97C2×C412yes—
20#13095490—22no—
21#13095490—22no—
22C22⋊Dic97—21no—
23C4×Dic97—21no—
24C4×Dic97C2×C411yescharacteristic
25C4×Dic97D412yes—
26C4×Dic97C2311yescharacteristic
27C22×C4×C97C2×C411yescharacteristic
28C4⋊C4×C97C2×C412yes—
29C22⋊C4×C97—21no—
30C42×C97—21no—
31C23×C97C22⋊C411yescharacteristic
32C2×C4×C97—22no—
33C2×C4×C97—22no—
34C2×C4×C97C22⋊C411yescharacteristic
35C2×C4×C97C22×C411yescharacteristic
36C2×Dic97—22no—
37C2×Dic97—22no—
38C2×Dic97—22no—
39C2×Dic97C22⋊C412yes—
40C2×Dic97C2×D412yes—
41Dic194—42no—
42Dic194—42no—
43#11715897—44no—
44C22×C97—21no—
45C22×C97—22no—
46C22×C97C2×C22⋊C411yescharacteristic
47C4×C97—21no—
48C4×C97—42no—
49C4×C97#10112yes—
50Dic97—22no—
51Dic97—42no—
52Dic97AGL(1,ℤ/8)12yes—
53Dic97#13712yes—
54C2×C97—41no—
55C2×C97C2×#10111yescharacteristic
56C2×C97#41712yes—
57#1561—971nomaximal, Sylow
58C97#156111yescharacteristic, complemented, Sylow
59#533—971no—
60#423—972no—
61#332—974no—
62C2×M16—1942no—
63C4⋊Q8—972no—
64#125—972no—
65C22⋊Q8—1942no—
66C4×Q8—1942no—
67#117—971no—
68#117#8760411yescharacteristic
69C2×C42—971no—
70C4⋊C8—974no—
71C4⋊C8—1942no—
72C2×Q8—1942no—
73C22×C4—972no—
74C22×C4C2×#8760411yescharacteristic
75M16—3884no—
76C2×C8—1944no—
77C4⋊C4—972no—
78C4⋊C4—1942no—
79C4⋊C4—1942no—
80C4⋊C4—1942no—
81C4⋊C4C2×#8760412yes—
82C22⋊C4—21no—
83C22⋊C4—1941no—
84C42—21no—
85C42—972no—
86C42—971no—
87C42—971no—
88C42—1941no—
89C23C22⋊#8760411yescharacteristic
90Q8—3882no—
91Q8—3882no—
92C2×C4—22no—
93C2×C4—22no—
94C2×C4—972no—
95C2×C4—972no—
96C2×C4—1942no—
97C2×C4—1942no—
98C2×C4—1942no—
99C2×C4C22⋊#8760411yescharacteristic
100C2×C4C22×#8760411yescharacteristic
101C8—3884no—
102C22—21no—
103C22—22no—
104C22C2×C22⋊#8760411yescharacteristic
105C4—21no—
106C4—42no—
107C4—972no—
108C4—972no—
109C4—3882no—
110C4—1942no—
111C4#1569087112yes—
112C2—41no—
113C2C2×#1569087111yescharacteristic
114C2#2026172612yes—
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#117×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C22⋊C4
cosocleC4×Dic97C23
hypercenter#117#87604
hypocenterC97#1561
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #20261594C2semidirect
58. C97#1561semidirect

Maximal subgroups

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

subgroupindex
#202617382
#202615942
#202612882
#156197