Finite Group Databasebeta

← #27156637

Subgroups — #27156637

Order 12416.

200 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 46 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#20261705C211yesmaximal, characteristic
2#20261705C211yesmaximal, characteristic
3#20260593C211yesmaximal, characteristic
4C2×#13095520—21no—
5C2×#13095520C412yescomplemented
6C22⋊#11715897—21no—
7C22⋊#11715897—21no—
8C22⋊#11715897—21no—
9C22⋊#11715897C412yescomplemented
10C23×Dic97C2211yescharacteristic
11C2×#13095490—21no—
12C2×C22⋊Dic97C411yescharacteristic
13#15690717—21no—
14C2×C22⋊C4×C97C411yescharacteristic
15#13095520—42no—
16#13095520—42no—
17C2×#11715897—22no—
18C2×#11715897—22no—
19C22×Dic97—21no—
20C22×Dic97—42no—
21C22×Dic97C2×C411yescharacteristic
22C22×Dic97C2×C411yescharacteristic
23C22×Dic97D411yescharacteristic
24C22×Dic97D411yescharacteristic
25C22×Dic97D411yescharacteristic
26C22×Dic97Q811yescharacteristic
27#13095490—42no—
28C22⋊Dic97—41no—
29C24×C97C2×C411yescharacteristic
30C22×C4×C97—21no—
31C22⋊C4×C97—41no—
32C23×C97—21no—
33C23×C97—21no—
34C23×C97—41no—
35C23×C97—41no—
36C23×C97C4211yescharacteristic
37C23×C97C22⋊C411yescharacteristic
38C23×C97C4⋊C411yescharacteristic
39C2×C4×C97—41no—
40C2×C4×C97—41no—
41C2×C4×C97—41no—
42C2×Dic97—22no—
43C2×Dic97—22no—
44C2×Dic97—22no—
45C2×Dic97—41no—
46C2×Dic97—41no—
47C2×Dic97—41no—
48C2×Dic97—41no—
49C2×Dic97—41no—
50C2×Dic97—41no—
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C22⋊C411yescharacteristic
53C2×Dic97C4⋊C411yescharacteristic
54C2×Dic97C4⋊C411yescharacteristic
55#11715897—42no—
56#11715897—42no—
57C22×C97—21no—
58C22×C97—21no—
59C22×C97—21no—
60C22×C97—21no—
61C22×C97—21no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97—21no—
65C22×C97—42no—
66C22×C97—42no—
67C22×C97#9511yescharacteristic
68C22×C97#10011yescharacteristic
69C22×C97#10111yescharacteristic
70C4×C97—81no—
71Dic97—21no—
72Dic97—21no—
73Dic97—21no—
74Dic97—81no—
75Dic97#9912yes—
76C2×C97—21no—
77C2×C97—21no—
78C2×C97—41no—
79C2×C97—41no—
80C2×C97#34211yescharacteristic
81C2×C97#34311yescharacteristic
82C2×C97#34411yescharacteristic
83#1305—971nomaximal, Sylow
84C97#130511yescharacteristic, complemented, Sylow
85#407—971no—
86#407—971no—
87#380—971no—
88C23×C4—971no—
89C2×M16—972no—
90C2×M16—1941no—
91C2×C4⋊C4—1941no—
92C2×C22⋊C4—971no—
93C2×C22⋊C4#8760411yescharacteristic
94C22⋊C8—972no—
95C22⋊C8—1941no—
96C22⋊C8—1941no—
97C22⋊C8—1941no—
98#95—1941no—
99C24C2×#8760411yescharacteristic
100C22×C4—21no—
101C22×C4—971no—
102C22×C4—971no—
103C22×C4—971no—
104C22×C4—971no—
105C22×C4—971no—
106C22×C4—971no—
107C22×C4—3882no—
108C22×C4—1941no—
109M16—3882no—
110M16—3882no—
111C2×C8—1942no—
112C2×C8—1942no—
113C4⋊C4—3882no—
114C22⋊C4—41no—
115C22⋊C4—3881no—
116C23—21no—
117C23—21no—
118C23—41no—
119C23—41no—
120C23C4×#8760411yescharacteristic
121C23C22⋊#8760411yescharacteristic
122C23C4⋊#8760411yescharacteristic
123C2×C4—41no—
124C2×C4—41no—
125C2×C4—41no—
126C2×C4—971no—
127C2×C4—971no—
128C2×C4—971no—
129C2×C4—971no—
130C2×C4—3881no—
131C2×C4—3881no—
132C2×C4—3881no—
133C2×C4—3881no—
134C2×C4—3881no—
135C2×C4—3881no—
136C2×C4—1942no—
137C2×C4—1942no—
138C2×C4—1942no—
139C8—3882no—
140C8—3882no—
141C22—21no—
142C22—21no—
143C22—21no—
144C22—21no—
145C22—21no—
146C22—21no—
147C22—21no—
148C22—21no—
149C22—42no—
150C22—42no—
151C22(C2×#87604)⋊C411yescharacteristic
152C22#1569087011yescharacteristic
153C22#1569087111yescharacteristic
154C4—81no—
155C4—7761no—
156C4—972no—
157C4—1941no—
158C4—1941no—
159C4—1941no—
160C2—21no—
161C2—21no—
162C2—41no—
163C2—41no—
164C2#2026161211yescharacteristic
165C2#2026161511yescharacteristic
166C2#2026161611yescharacteristic
subgroupG/H
centerC22(C2×#87604)⋊C4
commutatorC23×C97C42
FrattiniC24C2×#87604
FittingC2×C22⋊C4×C97C4
socleC22×C97#95
prime radicalC24×C97C2×C4
cosocleC23×Dic97C22
hypercenterC2×C22⋊C4#87604
hypocenterC97#1305
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
5. C2×#13095520C4semidirect
9. C22⋊#11715897C4semidirect
84. C97#1305semidirect

Maximal subgroups

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

subgroupindex
#202617052
#202605932
#130597