Finite Group Databasebeta

← #27157128

Subgroups — #27157128

Order 12416.

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

normal subgroups only

This group has 228 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#20261757C211yesmaximal, characteristic
2#20261671C211yesmaximal, characteristic
3#20261588C214yesmaximal
4#20261290C211yesmaximal, characteristic
5C2×C4×#87604—21no—
6C4⋊#11715897—21no—
7#15690867C2212yes—
8#15690865—21no—
9(C2×#87604)⋊C4—21no—
10(C2×#87604)⋊C4C2212yes—
11#15690810C411yescharacteristic, complemented
12#15690801—21no—
13#15690790C2212yes—
14#15690786—21no—
15Q8×Dic97—21no—
16#15690775C412yescomplemented
17#15690768—21no—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic, complemented
20C22×#87604—22no—
21C2×#11715897—22no—
22C4×#87604—24no—
23C2×Dic194C2×C412yes—
24C2×C4×D97D411yescharacteristic
25C2×C4×D97D411yescharacteristic
26C2×C4×D97C2311yescharacteristic
27C4⋊Dic97—21no—
28C4⋊Dic97—21no—
29C4⋊Dic97—22no—
30#13095490—22no—
31#13095488—22no—
32C4×Dic97—22no—
33C4×Dic97C2×C411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97D411yescharacteristic
36C2×Q8×C97—21no—
37C4⋊C4×C97—21no—
38C4⋊C4×C97C2×C412yes—
39C42×C97C2×C411yescharacteristic
40C22×D97C2×D411yescharacteristic
41C2×#87604—24no—
42C2×#87604—42no—
43C2×#87604—42no—
44Q8×C97—42no—
45C2×C4×C97—22no—
46C2×C4×C97C22⋊C411yescharacteristic
47C2×C4×C97C22⋊C411yescharacteristic
48C2×C4×C97C22×C411yescharacteristic
49C2×Dic97—22no—
50C2×Dic97C22⋊C411yescharacteristic
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C2×D411yescharacteristic
53C4×D97—22no—
54C4×D97—22no—
55C4×D97—41no—
56Dic194—42no—
57#11715897—42no—
58C22×C97C2×C22⋊C411yescharacteristic
59C2×D97—21no—
60C2×D97—21no—
61C2×D97#13712yes—
62#87604—44no—
63C4×C97—21no—
64C4×C97—21no—
65C4×C97—42no—
66C4×C97#10412yes—
67Dic97—21no—
68Dic97—21no—
69Dic97—21no—
70Dic97—42no—
71C2×C97#41011yescharacteristic
72C2×C97#41911yescharacteristic
73C2×C97C2×#10411yescharacteristic
74D97—22no—
75#1506—971nomaximal, Sylow
76C97#150611yescharacteristic, complemented, Sylow
77#533—971no—
78#432—971no—
79#376—971no—
80#328—974no—
81C4⋊Q8—971no—
82C4⋊Q8#8760411yescharacteristic, complemented
83#125—1941no—
84C22⋊Q8—972no—
85C22⋊Q8—1941no—
86C4×Q8—972no—
87C4×Q8—1941no—
88#117—1941no—
89C2×C42—971no—
90C2×C42—1941no—
91C4⋊C8—1941no—
92C22⋊C8—972no—
93#97—1941no—
94#95—972no—
95#95—1941no—
96C2×Q8—21no—
97C2×Q8—972no—
98C22×C4—971no—
99C22×C4—971no—
100C22×C4—971no—
101C22×C4—1942no—
102C2×C8—1942no—
103C4⋊C4—21no—
104C4⋊C4—1942no—
105C4⋊C4—1942no—
106C4⋊C4—1941no—
107C4⋊C4—1941no—
108C4⋊C4C2×#8760412yes—
109C22⋊C4—1942no—
110C42—971no—
111C42—971no—
112C42—971no—
113C42—1942no—
114C42—1944no—
115C42C2×#8760411yescharacteristic
116C23—971no—
117Q8—42no—
118Q8—3882no—
119C2×C4—22no—
120C2×C4—971no—
121C2×C4—971no—
122C2×C4—971no—
123C2×C4—3881no—
124C2×C4—3882no—
125C2×C4—3882no—
126C2×C4—1942no—
127C2×C4—1944no—
128C2×C4—1942no—
129C2×C4—1942no—
130C2×C4C22⋊#8760411yescharacteristic
131C2×C4C22⋊#8760411yescharacteristic
132C2×C4C22×#8760411yescharacteristic
133C8—3882no—
134C22—972no—
135C22—1941no—
136C22—1941no—
137C22C2×C22⋊#8760411yescharacteristic
138C4—21no—
139C4—21no—
140C4—42no—
141C4—3884no—
142C4—3882no—
143C4—1941no—
144C4—1941no—
145C4—1941no—
146C4#1569087412yes—
147C2—1942no—
148C2#2026171611yescharacteristic
149C2#2026172811yescharacteristic
150C2C2×#1569087411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1506
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #15690810C4semidirect
16. #15690775C4semidirect
19. C4⋊Q8×C97C4semidirect
76. C97#1506semidirect
82. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616712
#202615882
#202612902
#150697