Finite Group Databasebeta

← #27156836

Subgroups — #27156836

Order 12416.

162 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 40 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#20261721C211yesmaximal, characteristic
2#20261714C211yesmaximal, characteristic
3#20260701C211yesmaximal, characteristic
4D97⋊M16—21no—
5#15690889—21no—
6#15690871C412yescomplemented
7#15690870—21no—
8#15690869C412yes—
9#15690868—21no—
10C2×#13095511C2211yescharacteristic
11C2×#13095495C411yescharacteristic, complemented
12#15690722—21no—
13#15690722—21no—
14C22×C8×C97C411yescharacteristic, complemented
15#13095521—41no—
16#13095520—22no—
17#13095519—41no—
18C4⋊#87604—41no—
19C22⋊#87604—22no—
20C4×#87604—41no—
21#13095511—41no—
22#13095511—41no—
23C2×Dic194C2×C411yescharacteristic
24C2×D388C2×C411yescharacteristic
25C2×C22⋊D97D411yescharacteristic
26C2×C22⋊D97Q811yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29#13095495—41no—
30C2×#11715895—21no—
31C22×C4×C97C2×C411yescharacteristic
32C2×C8×C97—21no—
33C2×C8×C97—41no—
34C23×C97C4⋊C411yescharacteristic
35C22×D97C22⋊C411yescharacteristic
36C22×D97C4⋊C411yescharacteristic
37C2×#87604—42no—
38C2×C4×C97—21no—
39C2×C4×C97—21no—
40C2×C4×C97C4211yescharacteristic
41C2×C4×C97C22⋊C411yescharacteristic
42C22⋊D97—41no—
43C22⋊D97—41no—
44C2×Dic97C22⋊C411yescharacteristic
45C2×Dic97C4⋊C411yescharacteristic
46D388—41no—
47C4×D97—21no—
48C4×D97—21no—
49C4×D97—21no—
50C4×D97—21no—
51Dic194—41no—
52#11715897—42no—
53C8×C97—41no—
54#11715895—41no—
55C22×C97—21no—
56C22×C97—41no—
57C22×C97#9511yescharacteristic
58C2×D97—21no—
59C2×D97—21no—
60C2×D97—41no—
61C2×D97—41no—
62#87604—82no—
63C4×C97—21no—
64C4×C97#9911yescharacteristic
65C4×C97#9911yescharacteristic
66Dic97—21no—
67Dic97—21no—
68C2×C97—21no—
69C2×C97—41no—
70C2×C97#34211yescharacteristic
71D97—41no—
72D97—41no—
73#1395—971nomaximal, Sylow
74C97#139511yescharacteristic, complemented, Sylow
75#413—971no—
76#410—971no—
77#408—971no—
78C2×#41—971no—
79C2×M16—971no—
80C2×M16—1941no—
81C22×C8#8760411yescharacteristic, complemented
82#117—1941no—
83#101—972no—
84#100—1941no—
85#99—972no—
86#99—1941no—
87C22⋊C8—1941no—
88C22⋊C8—1941no—
89#41—3881no—
90#41—3881no—
91C2×Q8—971no—
92C2×D4—971no—
93C2×D4—971no—
94C2×D4—971no—
95C22×C4—971no—
96C22×C4—971no—
97C22×C4C2×#8760411yescharacteristic
98M16—3881no—
99M16—3881no—
100M16—1942no—
101C2×C8—21no—
102C2×C8—41no—
103C2×C8—3881no—
104C2×C8—1941no—
105C4⋊C4—3881no—
106C22⋊C4—1942no—
107C42—3881no—
108C23—971no—
109C23—971no—
110C23C4⋊#8760411yescharacteristic
111Q8—3881no—
112D4—3881no—
113D4—3881no—
114D4—3881no—
115C2×C4—21no—
116C2×C4—21no—
117C2×C4—971no—
118C2×C4—971no—
119C2×C4—3882no—
120C2×C4—1941no—
121C2×C4—1941no—
122C2×C4—1941no—
123C2×C4—1941no—
124C2×C4C4×#8760411yescharacteristic
125C2×C4C22⋊#8760411yescharacteristic
126C8—41no—
127C8—3882no—
128C8—3881no—
129C22—21no—
130C22—41no—
131C22—3881no—
132C22—3881no—
133C22—1941no—
134C22—1941no—
135C22(C2×#87604)⋊C411yescharacteristic
136C4—21no—
137C4—7762no—
138C4—1941no—
139C4—1941no—
140C4#1569086811yescharacteristic
141C4#1569086911yescharacteristic
142C2—21no—
143C2—41no—
144C2—3881no—
145C2—3881no—
146C2#2026161311yescharacteristic
subgroupG/H
centerC4#15690869
commutatorC2×C4×C97C42
FrattiniC22×C4C2×#87604
FittingC22×C8×C97C4
socleC2×C97#342
prime radical, cosocleC2×#13095511C22
hypercenterC22×C8#87604
hypocenterC97#1395
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
6. #15690871C4semidirect
11. C2×#13095495C4semidirect
14. C22×C8×C97C4semidirect
74. C97#1395semidirect
81. C22×C8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617212
#202617142
#202607012
#139597