Finite Group Databasebeta

← #27156970

Subgroups — #27156970

Order 12416.

264 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 76 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#20261752C212yesmaximal, complemented
2#20261580C212yesmaximal, complemented
3#20261579C212yesmaximal
4C2×#15690768C211yesmaximal, characteristic
5#15690877—22no—
6C22⋊#11715897C2214yes—
7C4×#11715897—22no—
8C22×C4×D97C411yescharacteristic
9#15690768—22no—
10#15690768—22no—
11C2×C4⋊Dic97C2211yescharacteristic
12C2×#13095488C411yescharacteristic
13C2×#13095488C2211yescharacteristic
14C2×C4×Dic97C411yescharacteristic
15C2×C4×Dic97C2211yescharacteristic
16C2×C4⋊C4×C97C411yescharacteristic
17C2×#11715897—24no—
18C23×D97C2×C411yescharacteristic
19C22×Dic97C2×C411yescharacteristic
20C22×Dic97C2×C411yescharacteristic
21C22×Dic97C2311yescharacteristic
22C2×C4×D97—21no—
23C2×C4×D97—21no—
24C2×C4×D97—22no—
25C2×C4×D97C814yescomplemented
26C4⋊Dic97—22no—
27#13095488—22no—
28#13095488—22no—
29C4×Dic97—21no—
30C4×Dic97—21no—
31C4×Dic97D412yes—
32C4×Dic97D412yes—
33C22×C4×C97C2×C411yescharacteristic
34C22×C4×C97C2×C411yescharacteristic
35C22×C4×C97C2×C411yescharacteristic
36C4⋊C4×C97—22no—
37C23×C97C22×C411yescharacteristic
38C22×D97—21no—
39C22×D97—21no—
40C22×D97—41no—
41C22×D97—41no—
42C22×D97C2×C812yes—
43C2×C4×C97—21no—
44C2×C4×C97—21no—
45C2×C4×C97—21no—
46C2×C4×C97—41no—
47C2×C4×C97—41no—
48C2×C4×C97C22⋊C412yes—
49C2×C4×C97C2×C812yes—
50C2×Dic97—21no—
51C2×Dic97—41no—
52C2×Dic97—41no—
53C2×Dic97C22⋊C411yescharacteristic
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C2×C812yes—
56C2×Dic97M1612yes—
57C2×Dic97C2×D412yes—
58C4×D97—24no—
59C4×D97—24no—
60#11715897—44no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97C2×C22⋊C411yescharacteristic
64C22×C97C22×C811yescharacteristic
65C22×C97C2×M1611yescharacteristic
66C2×D97—22no—
67C2×D97—42no—
68C2×D97—41no—
69C2×D97—41no—
70C2×D97—42no—
71C4×C97—22no—
72C4×C97—22no—
73C4×C97—41no—
74Dic97—22no—
75Dic97—41no—
76Dic97C22⋊C812yes—
77Dic97C22⋊C812yes—
78C2×C97—21no—
79C2×C97—21no—
80C2×C97C2×C22⋊C811yescharacteristic
81C2×C97#41011yescharacteristic
82C2×C97#41311yescharacteristic
83D97—42no—
84#1458—971nomaximal, Sylow
85C97#145811yescharacteristic, complemented, Sylow
86C2×#117—971no—
87#431—972no—
88#324—972no—
89#323—972no—
90C23×C4—971no—
91#117—1942no—
92#117—1942no—
93C2×C4⋊C4—971no—
94C2×C4⋊C4#8760411yescharacteristic
95C2×C22⋊C4—971no—
96C2×C22⋊C4—971no—
97C2×C42—971no—
98C2×C42—971no—
99C4⋊C8—1942no—
100C22⋊C8—974no—
101C4×C8—1942no—
102C24—971no—
103C22×C4—971no—
104C22×C4—971no—
105C22×C4—974no—
106C22×C4—971no—
107C22×C4—1941no—
108C22×C4—1941no—
109C22×C4—1942no—
110C22×C4C2×#8760411yescharacteristic
111C22×C4C2×#8760411yescharacteristic
112C22×C4C2×#8760411yescharacteristic
113C2×C8—1944no—
114C4⋊C4—22no—
115C4⋊C4—1942no—
116C22⋊C4—1942no—
117C22⋊C4—1942no—
118C42—972no—
119C42—972no—
120C42—1941no—
121C42—1941no—
122C23—972no—
123C23—3881no—
124C23—3881no—
125C23—1941no—
126C23—1941no—
127C23C22×#8760411yescharacteristic
128C2×C4—21no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—41no—
133C2×C4—971no—
134C2×C4—971no—
135C2×C4—972no—
136C2×C4—972no—
137C2×C4—972no—
138C2×C4—3881no—
139C2×C4—3881no—
140C2×C4—1944no—
141C2×C4—1944no—
142C2×C4—1941no—
143C2×C4C22⋊#8760412yes—
144C2×C4#1309551912yes—
145C8—3884no—
146C22—21no—
147C22—21no—
148C22—3882no—
149C22—3882no—
150C22—3881no—
151C22—3881no—
152C22—1942no—
153C22C2×C22⋊#8760411yescharacteristic
154C22C2×#1309551911yescharacteristic
155C22D97⋊M1611yescharacteristic
156C4—22no—
157C4—22no—
158C4—41no—
159C4—972no—
160C4—972no—
161C4—3881no—
162C4—1942no—
163C2—21no—
164C2—21no—
165C2—3882no—
166C2#2026170311yescharacteristic
167C2#2026171611yescharacteristic
168C2#2026172211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC22×C97C22×C8
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×D97C2×C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1458
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
1. #20261752C2semidirect
2. #20261580C2semidirect
25. C2×C4×D97C8semidirect
85. C97#1458semidirect

Maximal subgroups

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

subgroupindex
#202617522
#202615802
#202615792
C2×#156907682
#145897