Finite Group Databasebeta

← #38741155

Subgroups — #38741155

Order 25216.

316 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 93 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#27161136C211yesmaximal, characteristic, complemented
2C4×#15691342C211yesmaximal, characteristic, complemented
3D4⋊#13096802C214yesmaximal, complemented
4C2×C4×D4×C197C211yesmaximal, characteristic
5C2×#15691342—21no—
6C4⋊#13096802C2212yescomplemented
7C22⋊#13096802—21no—
8#20263237—21no—
9#20263237—21no—
10C4×#13096802C2212yescomplemented
11C22×D4×C197C411yescharacteristic
12C23×C4×C197—21no—
13C4×D4×C197—21no—
14C4×D4×C197—21no—
15C4×D4×C197C412yes—
16C4×D4×C197C2212yes—
17C2×C4⋊C4×C197C411yescharacteristic
18C2×C22⋊C4×C197—21no—
19C2×C42×C197C2211yescharacteristic
20#15691342—24no—
21C2×#13096802—22no—
22C2×#13096802—22no—
23C24×C197—21no—
24C2×D4×C197—24no—
25C2×D4×C197—21no—
26C2×D4×C197C2×C412yes—
27C22×C4×C197—21no—
28C22×C4×C197—22no—
29C22×C4×C197—22no—
30C22×C4×C197—44no—
31C22×C4×C197C2×C411yescharacteristic
32C22×C4×C197D411yescharacteristic
33C22×C4×C197D411yescharacteristic
34C4⋊C4×C197—21no—
35C4⋊C4×C197C2×C412yes—
36C22⋊C4×C197—22no—
37C22⋊C4×C197—22no—
38C42×C197C2×C411yescharacteristic
39C42×C197D411yescharacteristic
40C42×C197D411yescharacteristic
41C42×C197C2311yescharacteristic
42#13096802—24no—
43#13096802—42no—
44C23×C197—24no—
45C23×C197—22no—
46C23×C197—42no—
47C23×C197—42no—
48C23×C197C22⋊C411yescharacteristic
49D4×C197—24no—
50D4×C197—22no—
51D4×C197M1614yescomplemented
52C2×C4×C197—22no—
53C2×C4×C197—22no—
54C2×C4×C197—22no—
55C2×C4×C197—22no—
56C2×C4×C197—44no—
57C2×C4×C197—42no—
58C2×C4×C197—44no—
59C2×C4×C197C22⋊C411yescharacteristic
60C2×C4×C197C22⋊C411yescharacteristic
61C2×C4×C197C22⋊C411yescharacteristic
62C2×C4×C197C22×C411yescharacteristic
63C2×C4×C197C2×D411yescharacteristic
64C2×C4×C197C2×D411yescharacteristic
65C22×C197—21no—
66C22×C197—21no—
67C22×C197—24no—
68C22×C197—48no—
69C22×C197—42no—
70C22×C197—42no—
71C22×C197#10412yes—
72C22×C197C2×C22⋊C411yescharacteristic
73C4×C197—21no—
74C4×C197—21no—
75C4×C197—21no—
76C4×C197—21no—
77C4×C197—42no—
78C4×C197C2×M1612yes—
79C4×C197AGL(1,ℤ/8)12yes—
80C2×C197—22no—
81C2×C197—44no—
82C2×C197#40711yescharacteristic
83C2×C197#41811yescharacteristic
84C2×C197C2×#10411yescharacteristic
85C197#149011yescharacteristic, complemented, Sylow
86#1490—1971nomaximal, Sylow
87C2×C4×D4D19711yescharacteristic
88#432—1971no—
89C4×M16—1971no—
90D4⋊C8—1974no—
91C22×D4Dic19711yescharacteristic
92C23×C4—21no—
93C2×M16—3941no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4Dic19712yes—
97C4×D4C2×D19712yes—
98C2×C4⋊C4Dic19711yescharacteristic
99C2×C22⋊C4—21no—
100C2×C42C2×D19711yescharacteristic
101C4⋊C8—1972no—
102C22⋊C8—3941no—
103#97—3941no—
104#97—3941no—
105C4×C8—1972no—
106C24—21no—
107C2×D4—24no—
108C2×D4—21no—
109C2×D4C2×Dic19712yes—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—44no—
114C22×C4C2×Dic19711yescharacteristic
115C22×C4C22⋊D19711yescharacteristic
116C22×C4C22⋊D19711yescharacteristic
117M16—3944no—
118C2×C8—3942no—
119C2×C8—3942no—
120C4⋊C4—21no—
121C4⋊C4C2×Dic19712yes—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C42C2×Dic19711yescharacteristic
125C42C22⋊D19711yescharacteristic
126C42C22⋊D19711yescharacteristic
127C42C22×D19711yescharacteristic
128C23—24no—
129C23—22no—
130C23—42no—
131C23—42no—
132C23C22⋊Dic19711yescharacteristic
133D4—24no—
134D4—22no—
135D4#1569134214yescomplemented
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—44no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4C22⋊Dic19711yescharacteristic
144C2×C4C22⋊Dic19711yescharacteristic
145C2×C4C22⋊Dic19711yescharacteristic
146C2×C4C22×Dic19711yescharacteristic
147C2×C4C2×C22⋊D19711yescharacteristic
148C2×C4C2×C22⋊D19711yescharacteristic
149C8—7882no—
150C8—3944no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—48no—
155C22—42no—
156C22—42no—
157C22#2026325312yes—
158C22C2×C22⋊Dic19711yescharacteristic
159C4—21no—
160C4—21no—
161C4—21no—
162C4—21no—
163C4—42no—
164C4C2×#1569134212yes—
165C4#2026335212yes—
166C2—22no—
167C2—44no—
168C2#2716104311yescharacteristic
169C2#2716108411yescharacteristic
170C2C2×#2026325311yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D197
commutatorC2×C4×C197C22×C4
FrattiniC42C22×D197
FittingC2×C4×D4×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC22×D4×C197C4
cosocleC42×C197C23
hypercenterC2×C4×D4D197
hypocenterC197#1490
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #27161136C2semidirect
2. C4×#15691342C2semidirect
3. D4⋊#13096802C2semidirect
6. C4⋊#13096802C22semidirect
10. C4×#13096802C22semidirect
51. D4×C197M16semidirect
85. C197#1490semidirect
135. D4#15691342semidirect

Maximal subgroups

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

subgroupindex
#271611362
C4×#156913422
D4⋊#130968022
C2×C4×D4×C1972
#1490197