Finite Group Databasebeta

← #27157008

Subgroups — #27157008

Order 12416.

204 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 60 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#20261755C211yesmaximal, characteristic
2#20261751C211yesmaximal, characteristic
3#20261581C212yesmaximal
4#20261580C212yesmaximal
5C2×#15690800C211yesmaximal, characteristic
6#15690877—21no—
7C4⋊#11715897—21no—
8C22⋊#11715897C2212yes—
9C22⋊#11715897C2212yes—
10#15690865—21no—
11C4×#11715897—21no—
12#15690800—21no—
13#15690800—21no—
14#15690800—21no—
15#15690800—21no—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×#13095490C411yescharacteristic
18C2×#13095490C411yescharacteristic
19C2×#13095490C411yescharacteristic
20C2×#13095490C2211yescharacteristic
21C2×C4×Dic97C2211yescharacteristic
22C2×C4⋊C4×C97C411yescharacteristic
23C2×#11715897—22no—
24C2×#11715897—22no—
25C22×Dic97C2×C411yescharacteristic
26C22×Dic97C2×C411yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2311yescharacteristic
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31#13095490—21no—
32#13095490—21no—
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39C4×Dic97D411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C4×Dic97D411yescharacteristic
43C22×C4×C97C2×C411yescharacteristic
44C22×C4×C97C2×C411yescharacteristic
45C22×C4×C97C2×C411yescharacteristic
46C4⋊C4×C97—21no—
47C4⋊C4×C97—21no—
48C23×C97C22×C411yescharacteristic
49C2×C4×C97—22no—
50C2×C4×C97—21no—
51C2×C4×C97—21no—
52C2×C4×C97—22no—
53C2×C4×C97—41no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×Dic97—22no—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—22no—
60C2×Dic97—41no—
61C2×Dic97—41no—
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C22⋊C411yescharacteristic
64C2×Dic97C2×D411yescharacteristic
65C2×Dic97C2×D411yescharacteristic
66#11715897—42no—
67#11715897—42no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97C2×C22⋊C411yescharacteristic
71C22×C97#13112yes—
72C4×C97—21no—
73C4×C97—21no—
74C4×C97—41no—
75C4×C97—41no—
76Dic97—21no—
77Dic97—21no—
78Dic97—21no—
79Dic97—41no—
80Dic97—41no—
81Dic97#10112yes—
82C2×C97—22no—
83C2×C97#40811yescharacteristic
84C2×C97#41011yescharacteristic
85C2×C97C2×#10111yescharacteristic
86#1470—971nomaximal, Sylow
87C97#147011yescharacteristic, complemented, Sylow
88C2×#125—971no—
89#432—971no—
90#431—971no—
91#324—972no—
92#324—972no—
93#125—1941no—
94#125—1941no—
95#125—1941no—
96#125—1941no—
97C2×C4⋊C4—971no—
98C2×C4⋊C4—971no—
99C2×C4⋊C4—971no—
100C2×C4⋊C4—971no—
101C2×C4⋊C4—971no—
102C2×C4⋊C4#8760411yescharacteristic
103C2×C42—971no—
104C4⋊C8—1941no—
105C4⋊C8—1941no—
106C22⋊C8—972no—
107C22⋊C8—972no—
108#97—1941no—
109C4×C8—1941no—
110C22×C4—971no—
111C22×C4—971no—
112C22×C4—971no—
113C22×C4—971no—
114C22×C4C2×#8760411yescharacteristic
115C22×C4C2×#8760411yescharacteristic
116C22×C4C2×#8760411yescharacteristic
117C2×C8—1942no—
118C2×C8—1942no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—1941no—
122C4⋊C4—1941no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4—1941no—
126C4⋊C4—1941no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C42—971no—
132C42—971no—
133C42—971no—
134C42—971no—
135C23C22×#8760411yescharacteristic
136C2×C4—22no—
137C2×C4—21no—
138C2×C4—21no—
139C2×C4—22no—
140C2×C4—41no—
141C2×C4—971no—
142C2×C4—971no—
143C2×C4—971no—
144C2×C4—971no—
145C2×C4—3881no—
146C2×C4—3881no—
147C2×C4—1942no—
148C2×C4—1941no—
149C2×C4—1941no—
150C2×C4—1942no—
151C2×C4C22⋊#8760411yescharacteristic
152C2×C4C22⋊#8760411yescharacteristic
153C8—3882no—
154C8—3882no—
155C22—21no—
156C22—21no—
157C22C2×C22⋊#8760411yescharacteristic
158C22#1569089712yes—
159C4—21no—
160C4—21no—
161C4—41no—
162C4—41no—
163C4—972no—
164C4—3881no—
165C4—3881no—
166C4—1941no—
167C4—1941no—
168C4—1941no—
169C2—22no—
170C2#2026170811yescharacteristic
171C2#2026171411yescharacteristic
172C2D97⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutator, prime radicalC23×C97C22×C4
FrattiniC23C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
cosocleC22×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1470
LCSG ▷ C23×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
87. C97#1470semidirect

Maximal subgroups

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

subgroupindex
#202617552
#202617512
#202615812
#202615802
C2×#156908002
#147097