Finite Group Databasebeta

← #27157195

Subgroups — #27157195

Order 12416.

264 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 62 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, complemented
2C4×#13095520C211yesmaximal, characteristic, complemented
3#20261590C214yesmaximal, complemented
4C2×#15690797C211yesmaximal, characteristic
5C2×#13095520—21no—
6C4⋊#11715897—21no—
7C22⋊#11715897—21no—
8#15690865C2212yescomplemented
9#15690865C2212yescomplemented
10C4×#11715897—21no—
11C22×Dic194C411yescharacteristic
12#15690797—21no—
13#15690797—21no—
14#15690797C412yes—
15#15690797C2212yes—
16C2×C22⋊Dic97—21no—
17C2×C22⋊Dic97—21no—
18C2×C4×Dic97C2211yescharacteristic
19C22×D4×C97C411yescharacteristic
20#13095520—24no—
21C2×#11715897—22no—
22C2×#11715897—22no—
23C2×Dic194—21no—
24C2×Dic194—42no—
25C2×Dic194C2×C412yes—
26C22×Dic97—21no—
27C22×Dic97D411yescharacteristic
28C22×Dic97D411yescharacteristic
29C22⋊Dic97—22no—
30C22⋊Dic97—22no—
31C22⋊Dic97—22no—
32C22⋊Dic97—22no—
33C4×Dic97C2×C411yescharacteristic
34C4×Dic97D411yescharacteristic
35C4×Dic97D411yescharacteristic
36C4×Dic97C2311yescharacteristic
37C24×C97—21no—
38C2×D4×C97—21no—
39C2×D4×C97—42no—
40C2×D4×C97C2×C412yes—
41C22×C4×C97C2×C411yescharacteristic
42C23×C97—22no—
43C23×C97—42no—
44C23×C97—42no—
45C23×C97—42no—
46C23×C97C22⋊C411yescharacteristic
47D4×C97—42no—
48D4×C97—42no—
49C2×C4×C97—22no—
50C2×C4×C97C22⋊C411yescharacteristic
51C2×C4×C97C22×C411yescharacteristic
52C2×Dic97—22no—
53C2×Dic97—22no—
54C2×Dic97—41no—
55C2×Dic97—42no—
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C2×D411yescharacteristic
59C2×Dic97C2×D411yescharacteristic
60Dic194—42no—
61Dic194—42no—
62#11715897—24no—
63#11715897—42no—
64C22×C97—21no—
65C22×C97—21no—
66C22×C97—42no—
67C22×C97—42no—
68C22×C97—42no—
69C22×C97—84no—
70C22×C97#10412yes—
71C22×C97C2×C22⋊C411yescharacteristic
72C4×C97—21no—
73C4×C97#10012yes—
74Dic97—21no—
75Dic97—21no—
76Dic97—21no—
77Dic97—21no—
78Dic97—42no—
79C2×C97—22no—
80C2×C97—82no—
81C2×C97C2×#10011yescharacteristic
82C2×C97C2×#10411yescharacteristic
83C2×C97#42111yescharacteristic
84#1527—971nomaximal, Sylow
85C97#152711yescharacteristic, complemented, Sylow
86C2×#124—971no—
87#432—971no—
88C4×M16—971no—
89#329—974no—
90C22×Q8—971no—
91C22×D4#8760411yescharacteristic
92C2×M16—1941no—
93#124—972no—
94#124—972no—
95#124—1941no—
96#124—1941no—
97C2×C22⋊C4—1941no—
98C2×C22⋊C4—1941no—
99C2×C42—971no—
100C4⋊C8—1941no—
101C22⋊C8—1941no—
102#97—972no—
103#97—972no—
104C4×C8—1941no—
105C24—21no—
106C2×Q8—972no—
107C2×Q8—3882no—
108C2×Q8—1941no—
109C2×D4—21no—
110C2×D4—42no—
111C2×D4C2×#8760412yes—
112C22×C4—971no—
113C22×C4—971no—
114C22×C4—1941no—
115C22×C4C2×#8760411yescharacteristic
116M16—1944no—
117C2×C8—1942no—
118C2×C8—1942no—
119C22⋊C4—1942no—
120C22⋊C4—1942no—
121C22⋊C4—1942no—
122C22⋊C4—1942no—
123C42—971no—
124C42—971no—
125C42—971no—
126C42—971no—
127C23—22no—
128C23—42no—
129C23—42no—
130C23—42no—
131C23C22⋊#8760411yescharacteristic
132Q8—3882no—
133Q8—3882no—
134D4—42no—
135D4—42no—
136C2×C4—22no—
137C2×C4—971no—
138C2×C4—971no—
139C2×C4—971no—
140C2×C4—971no—
141C2×C4—3882no—
142C2×C4—3881no—
143C2×C4—1942no—
144C2×C4—1942no—
145C2×C4C22⋊#8760411yescharacteristic
146C2×C4C22×#8760411yescharacteristic
147C8—3882no—
148C8—1944no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22—84no—
155C22#1569087412yes—
156C22C2×C22⋊#8760411yescharacteristic
157C4—21no—
158C4—3882no—
159C4—1941no—
160C4—1941no—
161C4—1941no—
162C4—1941no—
163C4#1569087012yes—
164C2—22no—
165C2—82no—
166C2C2×#1569087011yescharacteristic
167C2C2×#1569087411yescharacteristic
168C2#2026173311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting, prime radicalC22×D4×C97C4
socleC22×C97C2×C22⋊C4
cosocleC4×Dic97C23
hypercenterC22×D4#87604
hypocenterC97#1527
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20261756C2semidirect
2. C4×#13095520C2semidirect
3. #20261590C2semidirect
8. #15690865C22semidirect
9. #15690865C22semidirect
85. C97#1527semidirect

Maximal subgroups

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

subgroupindex
#202617562
C4×#130955202
#202615902
C2×#156907972
#152797