Finite Group Databasebeta

← #27156638

Subgroups — #27156638

Order 12416.

200 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 44 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#20261707C211yesmaximal, characteristic
2#20261677C211yesmaximal, characteristic
3#20260697C211yesmaximal, characteristic
4C2×#13095520C412yescomplemented
5C2×C4⋊#87604—21no—
6C2×C22⋊#87604C412yes—
7#15690867—21no—
8#15690867—21no—
9(C2×#87604)⋊C4—21no—
10C22×C4×D97C2211yescharacteristic
11C2×#13095494—21no—
12C22⋊#11715895C411yescharacteristic, complemented
13#15690722—21no—
14C22⋊C8×C97C411yescharacteristic, complemented
15C22×#87604—22no—
16#13095520—42no—
17C2×#11715897—22no—
18C4⋊#87604—42no—
19C22⋊#87604—42no—
20C23×D97C2×C411yescharacteristic
21C22×Dic97C2×C411yescharacteristic
22C2×C4×D97—42no—
23C2×C4×D97D411yescharacteristic
24C2×C4×D97D411yescharacteristic
25C2×C4×D97D411yescharacteristic
26C2×C4×D97Q811yescharacteristic
27#13095494—42no—
28C2×#11715895—21no—
29C22×C4×C97C2×C411yescharacteristic
30C2×C8×C97—21no—
31C23×C97C4211yescharacteristic
32C22×D97—21no—
33C22×D97—21no—
34C22×D97—41no—
35C22×D97—41no—
36C22×D97C22⋊C411yescharacteristic
37C22×D97C4⋊C411yescharacteristic
38C2×#87604—42no—
39C2×#87604—42no—
40C2×#87604—42no—
41C2×C4×C97—41no—
42C2×C4×C97C22⋊C411yescharacteristic
43C2×C4×C97C4⋊C411yescharacteristic
44C2×Dic97—22no—
45C2×Dic97C22⋊C411yescharacteristic
46C2×Dic97C4⋊C411yescharacteristic
47C4×D97—22no—
48C4×D97—22no—
49C4×D97—41no—
50C4×D97—41no—
51#11715897—42no—
52C8×C97—41no—
53#11715895—41no—
54C22×C97—21no—
55C22×C97—21no—
56C22×C97—21no—
57C22×C97#9511yescharacteristic
58C2×D97—21no—
59C2×D97—21no—
60C2×D97—21no—
61C2×D97—21no—
62C2×D97—21no—
63C2×D97—42no—
64C2×D97—42no—
65C2×D97#10011yescharacteristic
66C2×D97#10111yescharacteristic
67#87604—82no—
68C4×C97—21no—
69C4×C97—21no—
70Dic97—21no—
71Dic97#9912yes—
72C2×C97—41no—
73C2×C97#34211yescharacteristic
74C2×C97#34311yescharacteristic
75C2×C97#34411yescharacteristic
76D97—21no—
77D97—21no—
78D97—41no—
79#1305—971nomaximal, Sylow
80C97#130511yescharacteristic, complemented, Sylow
81#407—971no—
82#407—971no—
83#380—971no—
84C23×C4—971no—
85C2×M16—972no—
86C2×M16—1941no—
87C2×C4⋊C4—1941no—
88C2×C22⋊C4—972no—
89C22⋊C8—971no—
90C22⋊C8—1941no—
91C22⋊C8—1941no—
92C22⋊C8—1941no—
93C22⋊C8#8760411yescharacteristic, complemented
94#95—1941no—
95C24—971no—
96C22×C4—971no—
97C22×C4—971no—
98C22×C4—971no—
99C22×C4—971no—
100C22×C4—971no—
101C22×C4—3882no—
102C22×C4—1942no—
103C22×C4C2×#8760411yescharacteristic
104M16—3882no—
105M16—3882no—
106C2×C8—21no—
107C2×C8—1942no—
108C2×C8—1941no—
109C4⋊C4—3882no—
110C22⋊C4—3882no—
111C23—971no—
112C23—971no—
113C23—3881no—
114C23—3881no—
115C23—1941no—
116C23—1941no—
117C23C4×#8760411yescharacteristic
118C2×C4—41no—
119C2×C4—971no—
120C2×C4—971no—
121C2×C4—3881no—
122C2×C4—3881no—
123C2×C4—3882no—
124C2×C4—3882no—
125C2×C4—3882no—
126C2×C4—1942no—
127C2×C4—1942no—
128C2×C4—1942no—
129C2×C4C22⋊#8760411yescharacteristic
130C2×C4C4⋊#8760411yescharacteristic
131C8—41no—
132C8—3881no—
133C8—3882no—
134C22—21no—
135C22—21no—
136C22—21no—
137C22—971no—
138C22—971no—
139C22—3882no—
140C22—3882no—
141C22—1941no—
142C22—1941no—
143C22—1941no—
144C22—1941no—
145C22—1941no—
146C22(C2×#87604)⋊C411yescharacteristic
147C4—21no—
148C4—21no—
149C4—7762no—
150C4—972no—
151C4—1941no—
152C2—41no—
153C2—3881no—
154C2—1941no—
155C2—1941no—
156C2#2026161211yescharacteristic
157C2#2026161411yescharacteristic
158C2#2026161711yescharacteristic
subgroupG/H
centerC22(C2×#87604)⋊C4
commutatorC23×C97C42
FrattiniC22×C4C2×#87604
FittingC22⋊C8×C97C4
socleC22×C97#95
prime radicalC23×D97C2×C4
cosocleC22×C4×D97C22
hypercenterC22⋊C8#87604
hypocenterC97#1305
LCSG ▷ C23×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×#13095520C4semidirect
12. C22⋊#11715895C4semidirect
14. C22⋊C8×C97C4semidirect
80. C97#1305semidirect
93. C22⋊C8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617072
#202616772
#202606972
#130597