Finite Group Databasebeta

← #27156838

Subgroups — #27156838

Order 12416.

162 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 38 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#20261716C211yesmaximal, characteristic
2#20261710C211yesmaximal, characteristic
3#20260718C211yesmaximal, characteristic
4C2×#13095521C411yescharacteristic, complemented
5C2×#13095519C411yescharacteristic, complemented
6#15690889—21no—
7#15690869—21no—
8#15690869C412yes—
9#15690867—21no—
10#15690867—21no—
11C2×#13095512C2211yescharacteristic
12M16×D97—21no—
13#15690729C411yescharacteristic, complemented
14#15690726—21no—
15#101×C97C411yescharacteristic, complemented
16#13095521—41no—
17#13095519—41no—
18C2×#11715897—21no—
19C2×#11715897—21no—
20C4⋊#87604—41no—
21C22⋊#87604—22no—
22C4×#87604—41no—
23#13095512—41no—
24#13095512—41no—
25C2×D388C2×C411yescharacteristic
26C2×D388D411yescharacteristic
27C2×D388Q811yescharacteristic
28C2×C4×D97C2×C411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97D411yescharacteristic
31#13095495—21no—
32#13095494—41no—
33C8×D97—41no—
34C2×Q8×C97C2×C411yescharacteristic
35M16×C97—21no—
36C22×D97C22⋊C411yescharacteristic
37C22×D97C4⋊C411yescharacteristic
38C22×D97C4⋊C411yescharacteristic
39C2×#87604—42no—
40Q8×C97—41no—
41C2×C4×C97C4211yescharacteristic
42C2×C4×C97C22⋊C411yescharacteristic
43C2×C4×C97C4⋊C411yescharacteristic
44C2×Dic97C22⋊C411yescharacteristic
45D388—41no—
46D388—41no—
47D388—41no—
48C4×D97—21no—
49C4×D97—21no—
50C4×D97—21no—
51C4×D97—21no—
52C4×D97—21no—
53C4×D97—21no—
54#11715897—41no—
55#11715897—41no—
56C8×C97—41no—
57#11715895—41no—
58C22×C97#9511yescharacteristic
59C2×D97—21no—
60C2×D97—21no—
61C2×D97—21no—
62C2×D97—41no—
63C2×D97—41no—
64C2×D97—41no—
65#87604—82no—
66C4×C97—21no—
67C4×C97—21no—
68C4×C97—21no—
69Dic97#9911yescharacteristic
70Dic97#9911yescharacteristic
71C2×C97—21no—
72C2×C97#34211yescharacteristic
73D97—41no—
74D97—41no—
75D97—41no—
76#1395—971nomaximal, Sylow
77C97#139511yescharacteristic, complemented, Sylow
78#413—971no—
79#410—971no—
80#408—971no—
81C2×#41—971no—
82C2×M16—971no—
83C2×M16—1941no—
84C22×C8—971no—
85#117—1941no—
86#101—971no—
87#101#8760411yescharacteristic, complemented
88#100—1941no—
89#99—972no—
90#99—1941no—
91C22⋊C8—1941no—
92C22⋊C8—1941no—
93#41—3881no—
94#41—3881no—
95C2×Q8C2×#8760411yescharacteristic
96C2×D4—971no—
97C2×D4—971no—
98C2×D4—971no—
99C22×C4—971no—
100C22×C4—971no—
101C22×C4—971no—
102M16—21no—
103M16—3881no—
104M16—3881no—
105M16—1941no—
106C2×C8—3881no—
107C2×C8—3881no—
108C2×C8—1941no—
109C2×C8—1941no—
110C4⋊C4—3881no—
111C22⋊C4—1942no—
112C42—3881no—
113C23—971no—
114C23—971no—
115C23—971no—
116Q8—41no—
117D4—3881no—
118D4—3881no—
119D4—3881no—
120C2×C4—971no—
121C2×C4—3882no—
122C2×C4—1941no—
123C2×C4—1941no—
124C2×C4—1941no—
125C2×C4—1941no—
126C2×C4—1941no—
127C2×C4—1941no—
128C2×C4C4×#8760411yescharacteristic
129C2×C4C22⋊#8760411yescharacteristic
130C2×C4C4⋊#8760411yescharacteristic
131C8—41no—
132C8—3881no—
133C8—3881no—
134C8—3881no—
135C22—3881no—
136C22—3881no—
137C22—3881no—
138C22—1941no—
139C22—1941no—
140C22—1941no—
141C22(C2×#87604)⋊C411yescharacteristic
142C4—21no—
143C4—21no—
144C4—21no—
145C4—7762no—
146C4—971no—
147C4—971no—
148C2—21no—
149C2—3881no—
150C2—3881no—
151C2—3881no—
152C2#2026161211yescharacteristic
subgroupG/H
centerC2#20261612
commutatorC2×C4×C97C42
FrattiniC2×Q8C2×#87604
Fitting#101×C97C4
socleC2×C97#342
prime radical, cosocleC2×#13095512C22
hypercenter#101#87604
hypocenterC97#1395
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #101

Complements

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

NH ≅ G/Ntype
4. C2×#13095521C4semidirect
5. C2×#13095519C4semidirect
13. #15690729C4semidirect
15. #101×C97C4semidirect
77. C97#1395semidirect
87. #101#87604semidirect

Maximal subgroups

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

subgroupindex
#202617162
#202617102
#202607182
#139597