Finite Group Databasebeta

← #27157436

Subgroups — #27157436

Order 12416.

420 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C2×#15690889C212yesmaximal
2#20261613C214yesmaximal
3C2×#15690768C211yesmaximal, characteristic
4#15690889—24no—
5#15690889C418yescomplemented
6C2×C4⋊#87604—22no—
7C2×C22⋊#87604C2214yes—
8C2×C4×#87604—22no—
9C22×C4×D97C2211yescharacteristic
10#15690768—22no—
11#15690768—22no—
12C2×C4⋊Dic97C411yescharacteristic, complemented
13C2×#13095488C2212yes—
14C2×C4×Dic97C412yescomplemented
15C2×C4⋊C4×C97C411yescharacteristic, complemented
16C22×#87604—24no—
17C4⋊#87604—28no—
18C22⋊#87604—24no—
19C22⋊#87604C2×C418yes—
20C4×#87604—28no—
21C23×D97C2311yescharacteristic
22C22×Dic97C2×C412yes—
23C22×Dic97C2×C411yescharacteristic
24C2×C4×D97C2×C414yes—
25C2×C4×D97D412yes—
26C2×C4×D97D412yes—
27C2×C4×D97D412yes—
28C2×C4×D97Q812yes—
29C4⋊Dic97—22no—
30#13095488—24no—
31C4×Dic97—22no—
32C4×Dic97—22no—
33C22×C4×C97C2×C412yes—
34C22×C4×C97C2×C411yescharacteristic
35C4⋊C4×C97—22no—
36C23×C97C22×C411yescharacteristic
37C22×D97—41no—
38C22×D97—41no—
39C22×D97C22×C412yes—
40C22×D97C2×D412yes—
41C22×D97C2×D411yescharacteristic
42C22×D97C2×Q811yescharacteristic
43C2×#87604—28no—
44C2×#87604—44no—
45C2×#87604—44no—
46C2×C4×C97—22no—
47C2×C4×C97—42no—
48C2×C4×C97C4212yes—
49C2×C4×C97C22⋊C412yes—
50C2×C4×C97C4⋊C412yes—
51C2×Dic97—22no—
52C2×Dic97—42no—
53C2×Dic97C4212yes—
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C4⋊C412yes—
57C4×D97C22⋊C414yes—
58C4×D97C22⋊C414yes—
59C4×D97C4⋊C414yes—
60C4×D97C4⋊C414yes—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97C2×C4211yescharacteristic
64C22×C97C2×C22⋊C411yescharacteristic
65C22×C97C2×C4⋊C411yescharacteristic
66C2×D97—22no—
67C2×D97—22no—
68C2×D97—42no—
69C2×D97—41no—
70C2×D97—41no—
71C2×D97C2×C22⋊C412yes—
72C2×D97C2×C4⋊C412yes—
73#87604—48no—
74C4×C97—42no—
75C4×C97#9514yes—
76Dic97—42no—
77Dic97#9512yes—
78Dic97#9512yes—
79C2×C97—21no—
80C2×C97—21no—
81C2×C97C2×#9511yescharacteristic
82C2×C97#41011yescharacteristic
83C2×C97#41011yescharacteristic
84D97—22no—
85D97—22no—
86#1740—971nomaximal, Sylow
87C97#174011yescharacteristic, complemented, Sylow
88C2×#117—971no—
89C2×#117—972no—
90#342—974no—
91C23×C4—971no—
92#117—978no—
93#117—1942no—
94#117—1942no—
95#117—1944no—
96C2×C4⋊C4—971no—
97C2×C4⋊C4—1942no—
98C2×C4⋊C4#8760411yescharacteristic, complemented
99C2×C22⋊C4—974no—
100C2×C22⋊C4—972no—
101C2×C42—972no—
102C2×C42—1942no—
103C24—971no—
104C22×C4—971no—
105C22×C4—974no—
106C22×C4—972no—
107C22×C4—972no—
108C22×C4—972no—
109C22×C4—972no—
110C22×C4—972no—
111C22×C4—1944no—
112C22×C4C2×#8760412yes—
113C22×C4C2×#8760411yescharacteristic
114C4⋊C4—22no—
115C4⋊C4—1948no—
116C4⋊C4—1942no—
117C22⋊C4—978no—
118C22⋊C4—1944no—
119C22⋊C4—1944no—
120C42—1942no—
121C42—1942no—
122C42—1948no—
123C23—972no—
124C23—971no—
125C23—971no—
126C23—972no—
127C23—3881no—
128C23—3881no—
129C23C22×#8760411yescharacteristic
130C2×C4—22no—
131C2×C4—42no—
132C2×C4—972no—
133C2×C4—971no—
134C2×C4—971no—
135C2×C4—972no—
136C2×C4—974no—
137C2×C4—974no—
138C2×C4—974no—
139C2×C4—974no—
140C2×C4—3884no—
141C2×C4—3884no—
142C2×C4—3882no—
143C2×C4—1948no—
144C2×C4—1942no—
145C2×C4C4×#8760412yes—
146C2×C4C22⋊#8760412yes—
147C2×C4C4⋊#8760412yes—
148C22—21no—
149C22—21no—
150C22—972no—
151C22—972no—
152C22—3882no—
153C22—3881no—
154C22—3881no—
155C22—1942no—
156C22—1942no—
157C22C2×C4×#8760411yescharacteristic
158C22C2×C22⋊#8760411yescharacteristic
159C22C2×C4⋊#8760411yescharacteristic
160C4—42no—
161C4—972no—
162C4—972no—
163C4—3888no—
164C4—3882no—
165C4(C2×#87604)⋊C414yes—
166C2—21no—
167C2—21no—
168C2—1942no—
169C2—1942no—
170C2C2×(C2×#87604)⋊C411yescharacteristic
171C2#2026171511yescharacteristic
172C2#2026171611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC22×C97C2×C42
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radical, cosocleC23×D97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1740
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #15690889C4semidirect
12. C2×C4⋊Dic97C4semidirect
14. C2×C4×Dic97C4semidirect
15. C2×C4⋊C4×C97C4semidirect
87. C97#1740semidirect
98. C2×C4⋊C4#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908892
#202616132
C2×#156907682
#174097