Finite Group Databasebeta

← C22×C8×#87604 #27158284

Subgroups — C22×C8×#87604 #27158284

Order 12416.

760 conjugacy classes of subgroups; only the 418 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 418 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
1C22×#13095519C211yesmaximal, characteristic
2C22×C4×#87604C211yesmaximal, characteristic
3C2×C8×#87604C2112yesmaximal, direct product factor
4C22×C8×D97C211yesmaximal, characteristic
5C23×#87604C412yes—
6C2×#13095519C416yescomplemented
7C2×#13095519C2216yes—
8C22×#11715897C412yescomplemented
9C2×C4×#87604C416yes—
10C2×C4×#87604C2216yes—
11C8×#87604C22116yesdirect product factor
12C22×C4×D97C2211yescharacteristic
13C22×#11715895C411yescharacteristic, complemented
14C2×C8×D97C416yescomplemented
15C2×C8×D97C2216yes—
16C22×C8×C97C411yescharacteristic, complemented
17C22×#87604C8116yesdirect product factor
18C22×#87604C2×C4112yes—
19#13095519C2×C4112yescomplemented
20#13095519C2314yes—
21C2×#11715897C2×C4112yescomplemented
22C4×#87604C2×C4112yes—
23C4×#87604C2314yes—
24C23×D97C2×C411yescharacteristic
25C22×Dic97C2×C411yescharacteristic
26C2×C4×D97C2×C413yes—
27C2×C4×D97C2×C413yes—
28C2×C4×D97C2×C413yes—
29C2×C4×D97C2313yes—
30C2×#11715895C2×C416yescomplemented
31C8×D97C2×C4112yescomplemented
32C8×D97C2314yes—
33C22×C4×C97C2×C411yescharacteristic
34C2×C8×C97C2×C416yescomplemented
35C23×C97C4211yescharacteristic
36C22×D97C4213yes—
37C22×D97C2×C818yes—
38C22×D97C22×C413yes—
39C2×#87604C2×C8148yesdirect product factor
40C2×#87604C22×C418yes—
41C2×C4×C97C4213yes—
42C2×C4×C97C22×C413yes—
43C2×Dic97C4213yes—
44C2×Dic97C22×C413yes—
45C4×D97C4216yes—
46C4×D97C22×C413yes—
47C4×D97C22×C413yes—
48C4×D97C22×C413yes—
49C4×D97C2411yescharacteristic
50#11715897C22×C418yescomplemented
51C8×C97C22×C414yescomplemented
52#11715895C22×C414yescomplemented
53C22×C97C4×C814yescomplemented
54C22×C97C2×C4213yes—
55C2×D97C4×C8112yescomplemented
56C2×D97C2×C4213yes—
57C2×D97C22×C8112yes—
58C2×D97C23×C411yescharacteristic
59#87604C22×C8116yesdirect product factor
60C4×C97C2×C4213yes—
61C4×C97C23×C411yescharacteristic
62Dic97C2×C4213yes—
63Dic97C23×C411yescharacteristic
64C2×C97C2×C4×C816yescomplemented
65C2×C97C22×C4211yescharacteristic
66D97C2×C4×C816yescomplemented
67D97C23×C812yes—
68C22×C4×C8—971nomaximal, Sylow
69C97C22×C4×C811yescharacteristic, complemented, Sylow
70C23×C8—971no—
71C23×C8—971no—
72C22×C42—971no—
73C2×C4×C8—9712no—
74C23×C4—972no—
75C23×C4—971no—
76C22×C8—972no—
77C22×C8—976no—
78C22×C8—976no—
79C22×C8—971no—
80C22×C8—976no—
81C22×C8—976no—
82C22×C8#8760411yescharacteristic, direct product factor
83C2×C42—976no—
84C2×C42—976no—
85C4×C8—9716no—
86C24—971no—
87C22×C4—9716no—
88C22×C4—9712no—
89C22×C4—971no—
90C22×C4—973no—
91C22×C4—973no—
92C22×C4—973no—
93C22×C4—973no—
94C22×C4C2×#8760411yescharacteristic
95C2×C8—9712no—
96C2×C8—9712no—
97C2×C8—974no—
98C2×C8—976no—
99C2×C8—9712no—
100C2×C8—974no—
101C2×C8C2×#8760416yesdirect product factor
102C42—9712no—
103C42—974no—
104C23—978no—
105C23—973no—
106C23—973no—
107C23C4×#8760411yescharacteristic
108C2×C4—978no—
109C2×C4—9748no—
110C2×C4—973no—
111C2×C4—973no—
112C2×C4—973no—
113C2×C4—973no—
114C2×C4—976no—
115C2×C4—971no—
116C2×C4—973no—
117C2×C4C4×#8760413yes—
118C2×C4C22×#8760413yes—
119C8—978no—
120C8—974no—
121C8C22×#8760414yesdirect product factor
122C22—9712no—
123C22—9712no—
124C22—971no—
125C22—973no—
126C22C8×#8760414yesdirect product factor
127C22C2×C4×#8760413yes—
128C4—9716no—
129C4—971no—
130C4—973no—
131C4C2×C4×#8760413yes—
132C4C23×#8760411yescharacteristic
133C2—972no—
134C2—976no—
135C2C2×C8×#8760416yesdirect product factor
136C2C22×C4×#8760411yescharacteristic
subgroupG/H
center, hypercenterC22×C8#87604
commutator, hypocenterC97C22×C4×C8
FrattiniC4C23×#87604
FittingC22×C8×C97C4
socleC23×C97C42
prime radicalC23×D97C2×C4
cosocleC4×D97C24
LCSG ▷ C97
UCSC1 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. C2×C8×#87604C2direct (N × H)
6. C2×#13095519C4semidirect
8. C22×#11715897C4semidirect
11. C8×#87604C22direct (N × H)
13. C22×#11715895C4semidirect
14. C2×C8×D97C4semidirect
16. C22×C8×C97C4semidirect
17. C22×#87604C8direct (N × H)
19. #13095519C2×C4semidirect
21. C2×#11715897C2×C4semidirect
30. C2×#11715895C2×C4semidirect
31. C8×D97C2×C4semidirect
34. C2×C8×C97C2×C4semidirect
39. C2×#87604C2×C8direct (N × H)
50. #11715897C22×C4semidirect
51. C8×C97C22×C4semidirect
52. #11715895C22×C4semidirect
53. C22×C97C4×C8semidirect
55. C2×D97C4×C8semidirect
59. #87604C22×C8direct (N × H)
64. C2×C97C2×C4×C8semidirect
66. D97C2×C4×C8semidirect
69. C97C22×C4×C8semidirect
82. C22×C8#87604direct (N × H)
101. C2×C8C2×#87604direct (N × H)
121. C8C22×#87604direct (N × H)
126. C22C8×#87604direct (N × H)
135. C2C2×C8×#87604direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955192
C22×C4×#876042
C2×C8×#876042
C22×C8×D972
C22×C4×C897