Finite Group Databasebeta

← #27157867

Subgroups — #27157867

Order 12416.

308 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 78 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
1C2×#15690873C214yesmaximal
2#20261694C211yesmaximal, characteristic
3#20261679C211yesmaximal, characteristic
4C4⋊Q8×D97C211yesmaximal, characteristic
5C2×#13095519C2212yes—
6C2×C4⋊#87604C2212yes—
7#15690873—28no—
8#15690865—22no—
9(C2×#87604)⋊C4—22no—
10C2×Q8×D97C2212yes—
11#15690810C411yescharacteristic, complemented
12#15690809—22no—
13#15690808C412yescomplemented
14C4⋊C4×D97—22no—
15C42×D97C2211yescharacteristic
16C4⋊Q8×C97C411yescharacteristic, complemented
17C22×#87604—22no—
18#13095519—22no—
19#13095519D414yes—
20C2×#11715897—22no—
21C4⋊#87604—24no—
22C2×Dic194—22no—
23C2×Dic194C2×C412yes—
24Q8×D97—24no—
25Q8×D97—24no—
26C2×C4×D97—22no—
27C2×C4×D97D412yes—
28C2×C4×D97C2311yescharacteristic
29C4⋊Dic97—22no—
30#13095490—24no—
31C4×Dic97C2×C411yescharacteristic
32C4×Dic97D412yes—
33C2×Q8×C97C2×C412yes—
34C4⋊C4×C97—22no—
35C42×C97C2×C411yescharacteristic
36C22×D97C2×D411yescharacteristic
37C2×#87604—42no—
38C2×#87604—44no—
39Q8×C97—24no—
40C2×C4×C97—22no—
41C2×C4×C97C22⋊C412yes—
42C2×C4×C97C22×C411yescharacteristic
43C2×Dic97—22no—
44C2×Dic97C22⋊C412yes—
45C2×Dic97C2×D411yescharacteristic
46C4×D97—24no—
47C4×D97—42no—
48C4×D97—42no—
49C4×D97C2×D412yes—
50C4×D97#4112yes—
51Dic194—24no—
52Dic194—44no—
53#11715897—24no—
54C22×C97C2×C22⋊C411yescharacteristic
55C2×D97#12411yescharacteristic
56C2×D97#12711yescharacteristic
57C2×D97#13714yes—
58#87604—82no—
59C4×C97—22no—
60C4×C97—42no—
61C4×C97C4×D412yes—
62Dic97—22no—
63Dic97—42no—
64Dic97#12112yes—
65C2×C97#39011yescharacteristic
66C2×C97#41912yes—
67D97#49112yes—
68D97#49712yes—
69#1960—971nomaximal, Sylow
70C97#196011yescharacteristic, complemented, Sylow
71C2×C4⋊Q8—971no—
72C2×#103—974no—
73C2×#97—971no—
74#382—971no—
75C22×Q8—972no—
76C22×C8—972no—
77C4⋊Q8—972no—
78C4⋊Q8—971no—
79C4⋊Q8—1942no—
80C4⋊Q8#8760411yescharacteristic, complemented
81C2×C4⋊C4—972no—
82C2×C4⋊C4—1942no—
83C2×C42—971no—
84#103—1948no—
85#97—1942no—
86#95—1942no—
87C2×Q8—972no—
88C2×Q8—1942no—
89C2×Q8—1944no—
90C2×Q8—1944no—
91C2×Q8C2×#8760412yes—
92C22×C4—971no—
93C22×C4—972no—
94C22×C4—1942no—
95C22×C4—1942no—
96C2×C8—974no—
97C2×C8—1942no—
98C2×C8—1942no—
99C4⋊C4—22no—
100C4⋊C4—1944no—
101C4⋊C4—1944no—
102C4⋊C4—1942no—
103C42—972no—
104C42—971no—
105C42C2×#8760411yescharacteristic
106C23—971no—
107Q8—24no—
108Q8—3884no—
109Q8—1944no—
110C2×C4—22no—
111C2×C4—971no—
112C2×C4—972no—
113C2×C4—972no—
114C2×C4—972no—
115C2×C4—3882no—
116C2×C4—3882no—
117C2×C4—3882no—
118C2×C4—3884no—
119C2×C4—1944no—
120C2×C4—1942no—
121C2×C4C22⋊#8760412yes—
122C2×C4C22×#8760411yescharacteristic
123C8—1944no—
124C22—971no—
125C22—971no—
126C22—974no—
127C22C2×C22⋊#8760411yescharacteristic
128C4—22no—
129C4—42no—
130C4—7762no—
131C4—972no—
132C4—3882no—
133C4—1942no—
134C4D4×#8760412yes—
135C2—972no—
136C2—972no—
137C2#2026168711yescharacteristic
138C2#2026172812yes—
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#1960
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #15690810C4semidirect
13. #15690808C4semidirect
16. C4⋊Q8×C97C4semidirect
70. C97#1960semidirect
80. C4⋊Q8#87604semidirect

Maximal subgroups

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

subgroupindex
C2×#156908732
#202616942
#202616792
C4⋊Q8×D972
#196097