Finite Group Databasebeta

← #27157210

Subgroups — #27157210

Order 12416.

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

normal subgroups only

This group has 212 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#20261694C211yesmaximal, characteristic
3#20261592C214yesmaximal, complemented
4#20261290C211yesmaximal, characteristic
5C2×#13095519—21no—
6C4⋊#11715897—21no—
7#15690867—21no—
8#15690865—21no—
9#15690865C2212yes—
10#15690865C2212yes—
11#15690810C411yescharacteristic
12#15690801C2212yes—
13#15690790—21no—
14#15690786—21no—
15Q8×Dic97—21no—
16#15690775—21no—
17#15690768C412yes—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic
20#13095519—24no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C2×Dic194—21no—
24C2×C4×D97C2×C411yescharacteristic
25C2×C4×D97D411yescharacteristic
26C2×C4×D97D411yescharacteristic
27C4⋊Dic97—21no—
28C4⋊Dic97—22no—
29C4⋊Dic97C2×C412yes—
30#13095490—22no—
31#13095488—22no—
32C4×Dic97—22no—
33C4×Dic97D411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97C2311yescharacteristic
36C2×Q8×C97—21no—
37C4⋊C4×C97—21no—
38C4⋊C4×C97C2×C412yes—
39C42×C97C2×C411yescharacteristic
40C22×D97C22⋊C411yescharacteristic
41Q8×C97—42no—
42C2×C4×C97—22no—
43C2×C4×C97C22⋊C411yescharacteristic
44C2×C4×C97C22⋊C411yescharacteristic
45C2×C4×C97C22×C411yescharacteristic
46C2×Dic97—22no—
47C2×Dic97C22⋊C411yescharacteristic
48C2×Dic97C2×D411yescharacteristic
49C2×Dic97C2×D411yescharacteristic
50C4×D97—22no—
51C4×D97—22no—
52C4×D97—41no—
53Dic194—42no—
54#11715897—24no—
55#11715897—42no—
56C22×C97C2×C22⋊C411yescharacteristic
57C2×D97—21no—
58C2×D97—21no—
59C2×D97#10112yes—
60C4×C97—21no—
61C4×C97—21no—
62C4×C97—42no—
63C4×C97#10412yes—
64Dic97—21no—
65Dic97—21no—
66Dic97—21no—
67Dic97—42no—
68C2×C97C2×#10111yescharacteristic
69C2×C97C2×#10411yescharacteristic
70C2×C97#42111yescharacteristic
71D97—22no—
72#1531—971nomaximal, Sylow
73C97#153111yescharacteristic, complemented, Sylow
74#533—971no—
75#432—971no—
76C2×#97—971no—
77#330—974no—
78C22×C8—1941no—
79C4⋊Q8—971no—
80C4⋊Q8#8760411yescharacteristic
81#125—972no—
82C22⋊Q8—1941no—
83C22⋊Q8—1941no—
84C4×Q8—1941no—
85C4×Q8—1941no—
86#117—972no—
87C2×C42—971no—
88C4⋊C8—1941no—
89C22⋊C8—1941no—
90#97—972no—
91#97—972no—
92#97—1941no—
93C2×Q8—21no—
94C2×Q8—1941no—
95C22×C4—971no—
96C22×C4—971no—
97C22×C4—971no—
98C2×C8—1942no—
99C2×C8—1942no—
100C2×C8—1944no—
101C4⋊C4—21no—
102C4⋊C4—972no—
103C4⋊C4—1942no—
104C4⋊C4—1942no—
105C4⋊C4—1941no—
106C4⋊C4C2×#8760412yes—
107C22⋊C4—1942no—
108C42—971no—
109C42—971no—
110C42—971no—
111C42—1942no—
112C42C2×#8760411yescharacteristic
113C23—971no—
114Q8—42no—
115Q8—3882no—
116C2×C4—22no—
117C2×C4—971no—
118C2×C4—971no—
119C2×C4—971no—
120C2×C4—3881no—
121C2×C4—1942no—
122C2×C4—1942no—
123C2×C4—1942no—
124C2×C4C22⋊#8760411yescharacteristic
125C2×C4C22⋊#8760411yescharacteristic
126C2×C4C22×#8760411yescharacteristic
127C8—3882no—
128C8—1944no—
129C22—972no—
130C22—1941no—
131C22—1941no—
132C22C2×C22⋊#8760411yescharacteristic
133C4—21no—
134C4—21no—
135C4—42no—
136C4—3882no—
137C4—1941no—
138C4—1941no—
139C4—1941no—
140C4#1569087512yes—
141C2—1942no—
142C2D97⋊#10111yescharacteristic
143C2C2×#1569087511yescharacteristic
144C2#2026173311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4⋊Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenterC4⋊Q8#87604
hypocenterC97#1531
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
3. #20261592C2semidirect
73. C97#1531semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202616942
#202615922
#202612902
#153197