Finite Group Databasebeta

← #27157100

Subgroups — #27157100

Order 12416.

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

normal subgroups only

This group has 202 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#20261753C211yesmaximal, characteristic
3#20261585C214yesmaximal, complemented
4#20261290C211yesmaximal, characteristic
5#15690877—21no—
6C4⋊#11715897C2212yes—
7#15690867—21no—
8#15690867—21no—
9#15690865—21no—
10C4×#11715897C2212yes—
11#15690810C411yescharacteristic
12#15690801—21no—
13#15690790C412yes—
14#15690786—21no—
15Q8×Dic97—21no—
16#15690775C2212yes—
17#15690768—21no—
18C42×D97C2211yescharacteristic
19C4⋊Q8×C97C411yescharacteristic
20C2×#11715897—22no—
21C2×#11715897—22no—
22C2×Dic194C2×C412yes—
23C2×C4×D97C2×C411yescharacteristic
24C2×C4×D97D411yescharacteristic
25C2×C4×D97D411yescharacteristic
26C4⋊Dic97—21no—
27C4⋊Dic97—21no—
28C4⋊Dic97—22no—
29#13095490—22no—
30#13095488—22no—
31C4×Dic97—22no—
32C4×Dic97D411yescharacteristic
33C4×Dic97D411yescharacteristic
34C4×Dic97C2311yescharacteristic
35C2×Q8×C97—21no—
36C4⋊C4×C97—21no—
37C4⋊C4×C97C2×C412yes—
38C42×C97C2×C411yescharacteristic
39C22×D97C22⋊C411yescharacteristic
40Q8×C97—42no—
41C2×C4×C97—22no—
42C2×C4×C97C22⋊C411yescharacteristic
43C2×C4×C97C22⋊C411yescharacteristic
44C2×C4×C97C22×C411yescharacteristic
45C2×Dic97—22no—
46C2×Dic97C22⋊C411yescharacteristic
47C2×Dic97C2×D411yescharacteristic
48C2×Dic97C2×D411yescharacteristic
49C4×D97—22no—
50C4×D97—22no—
51C4×D97—41no—
52Dic194—24no—
53#11715897—42no—
54#11715897—42no—
55C22×C97C2×C22⋊C411yescharacteristic
56C2×D97—21no—
57C2×D97—21no—
58C2×D97—21no—
59C4×C97—21no—
60C4×C97—21no—
61C4×C97—42no—
62C4×C97#13112yes—
63Dic97—21no—
64Dic97—21no—
65Dic97—42no—
66Dic97#13712yes—
67C2×C97#40811yescharacteristic
68C2×C97#41911yescharacteristic
69C2×C97#42111yescharacteristic
70D97—41no—
71#1497—971nomaximal, Sylow
72C97#149711yescharacteristic, complemented, Sylow
73#533—971no—
74#432—971no—
75#431—971no—
76Q8⋊C8—974no—
77C4⋊Q8—971no—
78C4⋊Q8#8760411yescharacteristic
79#125—1941no—
80C22⋊Q8—972no—
81C22⋊Q8—1941no—
82C4×Q8—972no—
83C4×Q8—1941no—
84#117—1941no—
85C2×C42—971no—
86C4⋊C8—972no—
87C4⋊C8—1941no—
88C22⋊C8—1941no—
89C22⋊C8—1941no—
90#97—1941no—
91C4×C8—972no—
92C2×Q8—21no—
93C2×Q8—972no—
94C22×C4—971no—
95C22×C4—971no—
96C22×C4—971no—
97C2×C8—1942no—
98C2×C8—1942no—
99C4⋊C4—21no—
100C4⋊C4—1942no—
101C4⋊C4—1942no—
102C4⋊C4—1941no—
103C4⋊C4—1941no—
104C4⋊C4C2×#8760412yes—
105C22⋊C4—1942no—
106C42—971no—
107C42—971no—
108C42—971no—
109C42—1942no—
110C42C2×#8760411yescharacteristic
111C23—971no—
112Q8—42no—
113Q8—1944no—
114C2×C4—22no—
115C2×C4—971no—
116C2×C4—971no—
117C2×C4—971no—
118C2×C4—3881no—
119C2×C4—1942no—
120C2×C4—1942no—
121C2×C4—1942no—
122C2×C4C22⋊#8760411yescharacteristic
123C2×C4C22⋊#8760411yescharacteristic
124C2×C4C22×#8760411yescharacteristic
125C8—3882no—
126C8—3882no—
127C22—1941no—
128C22—1941no—
129C22—1941no—
130C22C2×C22⋊#8760411yescharacteristic
131C4—21no—
132C4—21no—
133C4—42no—
134C4—972no—
135C4—3882no—
136C4—1941no—
137C4—1941no—
138C4#1569089812yes—
139C2—3881no—
140C2#2026171011yescharacteristic
141C2#2026172811yescharacteristic
142C2#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#1497
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
3. #20261585C2semidirect
72. C97#1497semidirect

Maximal subgroups

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

subgroupindex
#202617572
#202617532
#202615852
#202612902
#149797