Finite Group Databasebeta

← #27140179

Subgroups — #27140179

Order 12416.

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#20260792C211yesmaximal, characteristic, complemented
2C4×#13095495C211yesmaximal, characteristic, complemented
3D4⋊#11715895C214yesmaximal, complemented
4C2×C4×D4×C97C211yesmaximal, characteristic
5C2×#13095495—21no—
6C4⋊#11715895C2212yescomplemented
7C22⋊#11715895—21no—
8#15690721—21no—
9#15690721—21no—
10C4×#11715895C2212yescomplemented
11C22×D4×C97C411yescharacteristic
12C23×C4×C97—21no—
13C4×D4×C97—21no—
14C4×D4×C97—21no—
15C4×D4×C97C412yes—
16C4×D4×C97C2212yes—
17C2×C4⋊C4×C97C411yescharacteristic
18C2×C22⋊C4×C97—21no—
19C2×C42×C97C2211yescharacteristic
20#13095495—24no—
21C2×#11715895—22no—
22C2×#11715895—22no—
23C24×C97—21no—
24C2×D4×C97—24no—
25C2×D4×C97—21no—
26C2×D4×C97C2×C412yes—
27C22×C4×C97—21no—
28C22×C4×C97—22no—
29C22×C4×C97—22no—
30C22×C4×C97—44no—
31C22×C4×C97C2×C411yescharacteristic
32C22×C4×C97D411yescharacteristic
33C22×C4×C97D411yescharacteristic
34C4⋊C4×C97—21no—
35C4⋊C4×C97C2×C412yes—
36C22⋊C4×C97—22no—
37C22⋊C4×C97—22no—
38C42×C97C2×C411yescharacteristic
39C42×C97D411yescharacteristic
40C42×C97D411yescharacteristic
41C42×C97C2311yescharacteristic
42C23×C97—24no—
43C23×C97—22no—
44C23×C97—42no—
45C23×C97—42no—
46C23×C97C22⋊C411yescharacteristic
47D4×C97—24no—
48D4×C97—22no—
49D4×C97M1614yescomplemented
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—22no—
53C2×C4×C97—22no—
54C2×C4×C97—44no—
55C2×C4×C97—42no—
56C2×C4×C97—44no—
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22⋊C411yescharacteristic
59C2×C4×C97C22⋊C411yescharacteristic
60C2×C4×C97C22×C411yescharacteristic
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63#11715895—24no—
64#11715895—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97—24no—
68C22×C97—48no—
69C22×C97—42no—
70C22×C97—42no—
71C22×C97#10412yes—
72C22×C97C2×C22⋊C411yescharacteristic
73C4×C97—21no—
74C4×C97—21no—
75C4×C97—21no—
76C4×C97—21no—
77C4×C97—42no—
78C4×C97C2×M1612yes—
79C4×C97AGL(1,ℤ/8)12yes—
80C2×C97—22no—
81C2×C97—44no—
82C2×C97#40711yescharacteristic
83C2×C97#41811yescharacteristic
84C2×C97C2×#10411yescharacteristic
85#1490—971nomaximal, Sylow
86C97#149011yescharacteristic, complemented, Sylow
87C2×C4×D4D9711yescharacteristic
88#432—971no—
89C4×M16—971no—
90D4⋊C8—974no—
91C22×D4Dic9711yescharacteristic
92C23×C4—21no—
93C2×M16—1941no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4Dic9712yes—
97C4×D4C2×D9712yes—
98C2×C4⋊C4Dic9711yescharacteristic
99C2×C22⋊C4—21no—
100C2×C42C2×D9711yescharacteristic
101C4⋊C8—972no—
102C22⋊C8—1941no—
103#97—1941no—
104#97—1941no—
105C4×C8—972no—
106C24—21no—
107C2×D4—24no—
108C2×D4—21no—
109C2×D4C2×Dic9712yes—
110C22×C4—21no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—44no—
114C22×C4C2×Dic9711yescharacteristic
115C22×C4C22⋊D9711yescharacteristic
116C22×C4C22⋊D9711yescharacteristic
117M16—1944no—
118C2×C8—1942no—
119C2×C8—1942no—
120C4⋊C4—21no—
121C4⋊C4C2×Dic9712yes—
122C22⋊C4—22no—
123C22⋊C4—22no—
124C42C2×Dic9711yescharacteristic
125C42C22⋊D9711yescharacteristic
126C42C22⋊D9711yescharacteristic
127C42C22×D9711yescharacteristic
128C23—24no—
129C23—22no—
130C23—42no—
131C23—42no—
132C23C22⋊Dic9711yescharacteristic
133D4—24no—
134D4—22no—
135D4#1309549514yescomplemented
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—44no—
141C2×C4—42no—
142C2×C4—44no—
143C2×C4C22⋊Dic9711yescharacteristic
144C2×C4C22⋊Dic9711yescharacteristic
145C2×C4C22⋊Dic9711yescharacteristic
146C2×C4C22×Dic9711yescharacteristic
147C2×C4C2×C22⋊D9711yescharacteristic
148C2×C4C2×C22⋊D9711yescharacteristic
149C8—3882no—
150C8—1944no—
151C22—21no—
152C22—21no—
153C22—24no—
154C22—48no—
155C22—42no—
156C22—42no—
157C22#1569073712yes—
158C22C2×C22⋊Dic9711yescharacteristic
159C4—21no—
160C4—21no—
161C4—21no—
162C4—21no—
163C4—42no—
164C4C2×#1309549512yes—
165C4#1569083612yes—
166C2—22no—
167C2—44no—
168C2#2026069911yescharacteristic
169C2#2026074011yescharacteristic
170C2C2×#1569073711yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
FittingC2×C4×D4×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D4×C97C4
cosocleC42×C97C23
hypercenterC2×C4×D4D97
hypocenterC97#1490
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #20260792C2semidirect
2. C4×#13095495C2semidirect
3. D4⋊#11715895C2semidirect
6. C4⋊#11715895C22semidirect
10. C4×#11715895C22semidirect
49. D4×C97M16semidirect
86. C97#1490semidirect
135. D4#13095495semidirect

Maximal subgroups

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

subgroupindex
#202607922
C4×#130954952
D4⋊#117158952
C2×C4×D4×C972
#149097