Finite Group Databasebeta

← #27156987

Subgroups — #27156987

Order 12416.

252 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261755C212yesmaximal, complemented
2#20261580C212yesmaximal, complemented
3#20261578C212yesmaximal
4C2×#15690769C211yesmaximal, characteristic
5#15690877—22no—
6C22⋊#11715897C2214yes—
7#15690865—22no—
8#15690769—22no—
9#15690769—22no—
10C2×C4⋊Dic97C411yescharacteristic
11C2×C4⋊Dic97C2211yescharacteristic
12C2×C22⋊Dic97C411yescharacteristic
13C2×C22⋊Dic97C2211yescharacteristic
14C2×C4×Dic97C411yescharacteristic
15C2×C4×Dic97C2211yescharacteristic
16C23×C4×C97C411yescharacteristic
17C2×#11715897—24no—
18C22×Dic97C2×C411yescharacteristic
19C22×Dic97C2×C411yescharacteristic
20C22×Dic97C2×C411yescharacteristic
21C22×Dic97C2311yescharacteristic
22C4⋊Dic97—22no—
23C4⋊Dic97—22no—
24C22⋊Dic97—22no—
25C22⋊Dic97—22no—
26C4×Dic97—21no—
27C4×Dic97—21no—
28C4×Dic97D412yes—
29C4×Dic97D412yes—
30C24×C97C2×C411yescharacteristic
31C22×C4×C97—22no—
32C22×C4×C97—21no—
33C22×C4×C97—21no—
34C22×C4×C97—22no—
35C22×C4×C97C2×C411yescharacteristic
36C22×C4×C97C2×C411yescharacteristic
37C23×C97—21no—
38C23×C97—21no—
39C23×C97—41no—
40C23×C97—41no—
41C23×C97M1612yes—
42C23×C97C22×C411yescharacteristic
43C2×C4×C97—24no—
44C2×C4×C97—24no—
45C2×C4×C97—21no—
46C2×C4×C97—21no—
47C2×C4×C97—21no—
48C2×C4×C97—21no—
49C2×C4×C97C22⋊C411yescharacteristic
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97M1612yes—
52C2×Dic97—21no—
53C2×Dic97—21no—
54C2×Dic97—41no—
55C2×Dic97—41no—
56C2×Dic97—41no—
57C2×Dic97—41no—
58C2×Dic97C22⋊C412yes—
59C2×Dic97C2×D412yes—
60#11715897—44no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97—22no—
64C22×C97—42no—
65C22×C97—42no—
66C22×C97—41no—
67C22×C97—41no—
68C22×C97C2×C22⋊C411yescharacteristic
69C22×C97C2×M1611yescharacteristic
70C22×C97C2×M1611yescharacteristic
71C4×C97—21no—
72C4×C97—21no—
73C4×C97—22no—
74Dic97—22no—
75Dic97—22no—
76Dic97—41no—
77Dic97—41no—
78C2×C97—21no—
79C2×C97—21no—
80C2×C97—42no—
81C2×C97#40711yescharacteristic
82C2×C97#41011yescharacteristic
83C2×C97#41311yescharacteristic
84#1464—971nomaximal, Sylow
85C97#146411yescharacteristic, complemented, Sylow
86C2×#117—971no—
87#432—972no—
88#324—972no—
89#323—972no—
90C23×C4#8760411yescharacteristic
91#117—1942no—
92#117—1942no—
93C2×C4⋊C4—971no—
94C2×C4⋊C4—971no—
95C2×C22⋊C4—971no—
96C2×C22⋊C4—971no—
97C2×C42—971no—
98C2×C42—971no—
99C4⋊C8—1942no—
100C22⋊C8—974no—
101#97—1942no—
102C24C2×#8760411yescharacteristic
103C22×C4—22no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—22no—
107C22×C4—971no—
108C22×C4—971no—
109C22×C4—971no—
110C22×C4—971no—
111C22×C4C2×#8760411yescharacteristic
112C22×C4C2×#8760411yescharacteristic
113C2×C8—1944no—
114C4⋊C4—1942no—
115C4⋊C4—1942no—
116C22⋊C4—1942no—
117C22⋊C4—1942no—
118C42—972no—
119C42—972no—
120C42—1941no—
121C42—1941no—
122C23—21no—
123C23—21no—
124C23—41no—
125C23—41no—
126C23#1309552012yes—
127C23C22×#8760411yescharacteristic
128C2×C4—24no—
129C2×C4—24no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—972no—
135C2×C4—972no—
136C2×C4—3881no—
137C2×C4—3881no—
138C2×C4—3881no—
139C2×C4—3881no—
140C2×C4—1941no—
141C2×C4—1941no—
142C2×C4C22⋊#8760411yescharacteristic
143C2×C4C22⋊#8760411yescharacteristic
144C2×C4#1309552012yes—
145C8—3884no—
146C22—21no—
147C22—21no—
148C22—22no—
149C22—42no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22C2×C22⋊#8760411yescharacteristic
154C22C2×#1309552011yescharacteristic
155C22C2×#1309552011yescharacteristic
156C4—21no—
157C4—21no—
158C4—22no—
159C4—3881no—
160C4—3881no—
161C4—1942no—
162C4—1942no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2026170511yescharacteristic
167C2#2026171411yescharacteristic
168C2#2026172111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC23×C97C22×C4
FrattiniC23C22×#87604
FittingC23×C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×Dic97C23
hypercenterC23×C4#87604
hypocenterC97#1464
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. #20261755C2semidirect
2. #20261580C2semidirect
85. C97#1464semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202615802
#202615782
C2×#156907692
#146497