Finite Group Databasebeta

← #27157080

Subgroups — #27157080

Order 12416.

252 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261756C211yesmaximal, characteristic
2C4×#13095520C211yesmaximal, characteristic
3#20261585C214yesmaximal, complemented
4C2×#15690775C211yesmaximal, characteristic
5C2×#13095520—21no—
6C4⋊#11715897C2212yes—
7C22⋊#11715897—21no—
8#15690865—21no—
9#15690865—21no—
10C4×#11715897C2212yes—
11C22×Dic194C411yescharacteristic
12#15690775—21no—
13#15690775—21no—
14#15690775C412yes—
15#15690775C2212yes—
16C2×#13095490—21no—
17C2×C4×Dic97—21no—
18C2×C4×Dic97C2211yescharacteristic
19C2×C4⋊C4×C97C411yescharacteristic
20#13095520—24no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C2×Dic194—21no—
24C2×Dic194—24no—
25C2×Dic194C2×C412yes—
26C22×Dic97—21no—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29#13095490—22no—
30#13095490—22no—
31C4×Dic97—22no—
32C4×Dic97—22no—
33C4×Dic97C2×C411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97C2311yescharacteristic
37C22×C4×C97—21no—
38C22×C4×C97C2×C411yescharacteristic
39C4⋊C4×C97—21no—
40C4⋊C4×C97C2×C412yes—
41C23×C97C22⋊C411yescharacteristic
42C2×C4×C97—22no—
43C2×C4×C97—22no—
44C2×C4×C97—42no—
45C2×C4×C97C22⋊C411yescharacteristic
46C2×C4×C97C22×C411yescharacteristic
47C2×Dic97—24no—
48C2×Dic97—22no—
49C2×Dic97—22no—
50C2×Dic97—22no—
51C2×Dic97C22⋊C411yescharacteristic
52C2×Dic97C22⋊C411yescharacteristic
53C2×Dic97C2×D411yescharacteristic
54C2×Dic97C2×D411yescharacteristic
55Dic194—24no—
56Dic194—22no—
57Dic194M1614yescomplemented
58#11715897—24no—
59#11715897—42no—
60C22×C97—21no—
61C22×C97—21no—
62C22×C97#10412yes—
63C22×C97C2×C22⋊C411yescharacteristic
64C4×C97—21no—
65C4×C97—42no—
66C4×C97C2×M1612yes—
67Dic97—24no—
68Dic97—21no—
69Dic97—21no—
70Dic97—21no—
71Dic97#13712yes—
72C2×C97—22no—
73C2×C97#40711yescharacteristic
74C2×C97#41911yescharacteristic
75C2×C97C2×#10411yescharacteristic
76#1491—971nomaximal, Sylow
77C97#149111yescharacteristic, complemented, Sylow
78C2×C4×Q8—971no—
79#432—971no—
80C4×M16—971no—
81Q8⋊C8—974no—
82C22×Q8—971no—
83C2×M16—1941no—
84C4×Q8—972no—
85C4×Q8—972no—
86C4×Q8—1941no—
87C4×Q8—1941no—
88C2×C4⋊C4—1941no—
89C2×C4⋊C4#8760411yescharacteristic
90C2×C42—971no—
91C2×C42—1941no—
92C4⋊C8—972no—
93C22⋊C8—1941no—
94#97—1941no—
95#97—1941no—
96C4×C8—972no—
97C2×Q8—972no—
98C2×Q8—1944no—
99C2×Q8—1941no—
100C22×C4—21no—
101C22×C4—971no—
102C22×C4—971no—
103C22×C4—1941no—
104C22×C4C2×#8760411yescharacteristic
105M16—1944no—
106C2×C8—1942no—
107C2×C8—1942no—
108C4⋊C4—21no—
109C4⋊C4—1942no—
110C4⋊C4—1942no—
111C4⋊C4C2×#8760412yes—
112C42—971no—
113C42—971no—
114C42—971no—
115C42—971no—
116C42—1942no—
117C42—1942no—
118C23C22⋊#8760411yescharacteristic
119Q8—974no—
120Q8—1944no—
121Q8—1942no—
122C2×C4—22no—
123C2×C4—22no—
124C2×C4—42no—
125C2×C4—971no—
126C2×C4—971no—
127C2×C4—971no—
128C2×C4—971no—
129C2×C4—1942no—
130C2×C4—1944no—
131C2×C4—1942no—
132C2×C4—1942no—
133C2×C4C22⋊#8760411yescharacteristic
134C2×C4C22×#8760411yescharacteristic
135C8—3882no—
136C8—1944no—
137C22—21no—
138C22—21no—
139C22#1569087412yes—
140C22C2×C22⋊#8760411yescharacteristic
141C4—21no—
142C4—42no—
143C4—972no—
144C4—1944no—
145C4—1941no—
146C4—1941no—
147C4—1941no—
148C4D97⋊M1612yes—
149C2—22no—
150C2#2026170711yescharacteristic
151C2#2026172811yescharacteristic
152C2C2×#1569087411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC2×C4⋊C4×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C22⋊C4
cosocleC4×Dic97C23
hypercenterC2×C4⋊C4#87604
hypocenterC97#1491
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20261585C2semidirect
57. Dic194M16semidirect
77. C97#1491semidirect

Maximal subgroups

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

subgroupindex
#202617562
C4×#130955202
#202615852
C2×#156907752
#149197