Finite Group Databasebeta

← #27140066

Subgroups — #27140066

Order 12416.

312 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 93 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#20260699C212yesmaximal
2#20260461C214yesmaximal, complemented
3C22×C4⋊C4×C97C211yesmaximal, characteristic
4C2×#13095495—22no—
5C22⋊#11715895—22no—
6C22⋊#11715895C2214yes—
7C23×C4×C97C411yescharacteristic
8C23×C4×C97C411yescharacteristic
9C23×C4×C97C2211yescharacteristic
10C2×C4⋊C4×C97—22no—
11C2×C4⋊C4×C97—22no—
12C2×C4⋊C4×C97C412yes—
13C2×C4⋊C4×C97C2212yes—
14#13095495—44no—
15C2×#11715895—24no—
16C24×C97C2×C411yescharacteristic
17C22×C4×C97—21no—
18C22×C4×C97—21no—
19C22×C4×C97—24no—
20C22×C4×C97—21no—
21C22×C4×C97—21no—
22C22×C4×C97—42no—
23C22×C4×C97—42no—
24C22×C4×C97C2×C412yes—
25C22×C4×C97C2×C412yes—
26C22×C4×C97C2×C411yescharacteristic
27C22×C4×C97D412yes—
28C22×C4×C97D412yes—
29C22×C4×C97C2311yescharacteristic
30C4⋊C4×C97—24no—
31C4⋊C4×C97—24no—
32C23×C97—22no—
33C23×C97—22no—
34C23×C97—22no—
35C23×C97C22⋊C411yescharacteristic
36C23×C97C22⋊C411yescharacteristic
37C23×C97C22×C411yescharacteristic
38C2×C4×C97—24no—
39C2×C4×C97—24no—
40C2×C4×C97—24no—
41C2×C4×C97—22no—
42C2×C4×C97—22no—
43C2×C4×C97—22no—
44C2×C4×C97—41no—
45C2×C4×C97—42no—
46C2×C4×C97—41no—
47C2×C4×C97—42no—
48C2×C4×C97—42no—
49C2×C4×C97—42no—
50C2×C4×C97C22⋊C412yes—
51C2×C4×C97M1614yescomplemented
52C2×C4×C97C2×D412yes—
53#11715895—44no—
54C22×C97—28no—
55C22×C97—21no—
56C22×C97—21no—
57C22×C97—21no—
58C22×C97—21no—
59C22×C97—21no—
60C22×C97—21no—
61C22×C97#9912yes—
62C22×C97#10112yes—
63C22×C97C2×C22⋊C411yescharacteristic
64C22×C97C2×M1612yes—
65C4×C97—24no—
66C4×C97—22no—
67C4×C97—22no—
68C4×C97—42no—
69C2×C97—22no—
70C2×C97—22no—
71C2×C97—22no—
72C2×C97#40711yescharacteristic
73C2×C97C2×#9911yescharacteristic
74C2×C97C2×#10111yescharacteristic
75#1463—971nomaximal, Sylow
76C97#146311yescharacteristic, complemented, Sylow
77C22×C4⋊C4D9711yescharacteristic
78#407—972no—
79#324—974no—
80C23×C4Dic9711yescharacteristic
81C23×C4Dic9711yescharacteristic
82C23×C4C2×D9711yescharacteristic
83C2×M16—1942no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4Dic9712yes—
87C2×C4⋊C4C2×D9712yes—
88C22⋊C8—974no—
89C22⋊C8—1942no—
90C24C2×Dic9711yescharacteristic
91C22×C4—21no—
92C22×C4—21no—
93C22×C4—24no—
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—42no—
97C22×C4—42no—
98C22×C4C2×Dic9712yes—
99C22×C4C2×Dic9712yes—
100C22×C4C2×Dic9711yescharacteristic
101C22×C4C22⋊D9712yes—
102C22×C4C22⋊D9712yes—
103C22×C4C22×D9711yescharacteristic
104M16—3884no—
105C2×C8—1944no—
106C4⋊C4—24no—
107C4⋊C4—24no—
108C23—22no—
109C23—22no—
110C23—22no—
111C23C22⋊Dic9711yescharacteristic
112C23C22⋊Dic9711yescharacteristic
113C23C22×Dic9711yescharacteristic
114C2×C4—24no—
115C2×C4—24no—
116C2×C4—22no—
117C2×C4—22no—
118C2×C4—24no—
119C2×C4—22no—
120C2×C4—41no—
121C2×C4—42no—
122C2×C4—41no—
123C2×C4—42no—
124C2×C4—42no—
125C2×C4—42no—
126C2×C4C22⋊Dic9712yes—
127C2×C4#1309549514yescomplemented
128C2×C4C2×C22⋊D9712yes—
129C8—3884no—
130C22—28no—
131C22—21no—
132C22—21no—
133C22—21no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22#1569072512yes—
138C22#1569072912yes—
139C22C2×C22⋊Dic9711yescharacteristic
140C22C2×#1309549512yes—
141C4—24no—
142C4—22no—
143C4—22no—
144C4—42no—
145C2—22no—
146C2—22no—
147C2—22no—
148C2#2026069911yescharacteristic
149C2C2×#1569072511yescharacteristic
150C2C2×#1569072911yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC22×C4⋊C4×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×C4×C97C23
hypercenterC22×C4⋊C4D97
hypocenterC97#1463
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #20260461C2semidirect
51. C2×C4×C97M16semidirect
76. C97#1463semidirect
127. C2×C4#13095495semidirect

Maximal subgroups

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

subgroupindex
#202606992
#202604612
C22×C4⋊C4×C972
#146397