Finite Group Databasebeta

← D4×D1684 #41003476

Subgroups — D4×D1684 #41003476

Order 26944.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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×D1684C212yesmaximal, complemented
2C2×D4×D421C212yesmaximal, complemented
3#27212363C211yesmaximal, characteristic, complemented
4#27212341C212yesmaximal, complemented
5#27212340C212yesmaximal, complemented
6#27212336C214yesmaximal, complemented
7C4×D1684C211yesmaximal, characteristic, complemented
8C4×D4×C421C211yesmaximal, characteristic, complemented
9C23×D421C2214yescomplemented
10C2×D1684—28no—
11C2×D1684C2211yescharacteristic
12C2×D1684C2214yescomplemented
13C2×D1684C2212yescomplemented
14C2×D1684C2212yescomplemented
15C2×D1684C2212yescomplemented
16C2×C22⋊D421C2214yescomplemented
17D4×D421—28no—
18C2×C4×D421C2212yescomplemented
19C4⋊Dic421C2211yescharacteristic, complemented
20#20269748C2212yescomplemented
21#20269748C2214yescomplemented
22C2×D4×C421C2211yescharacteristic
23C22×C4×C421C2212yescomplemented
24C4⋊C4×C421C2211yescharacteristic, complemented
25C22⋊C4×C421C2212yescomplemented
26C42×C421C2211yescharacteristic, complemented
27C22×D421—28no—
28C22×D421—28no—
29C22×D421—44no—
30C22×D421C2314yes—
31C22×D421C2312yes—
32D1684—28no—
33D1684—24no—
34D1684—24no—
35D1684—42no—
36D1684D414yesdirect product factor
37C22⋊D421—28no—
38C2×Dic421C2312yes—
39C4×D421—24no—
40C23×C421C2312yes—
41D4×C421D414yescomplemented
42C2×C4×C421—24no—
43C2×C4×C421C2311yescharacteristic
44C2×C4×C421C2312yes—
45C2×C4×C421C2311yescharacteristic
46C2×C4×C421C2311yescharacteristic
47C22×C421—22no—
48C22×C421—22no—
49C22×C421C2×D414yescomplemented
50C22×C421C2411yescharacteristic
51C2×D421—22no—
52C2×D421—22no—
53C2×D421—24no—
54C2×D421—24no—
55C2×D421—24no—
56C2×D421—416no—
57C2×D421C2×D414yescomplemented
58C4×C421—22no—
59C4×C421—21no—
60C4×C421C2×D412yescomplemented
61C4×C421C2×D412yescomplemented
62Dic421—22no—
63C2×C421—24no—
64C2×C421C22×D411yescharacteristic
65C2×C421C22×D411yescharacteristic
66C2×C421H(5,𝔽2)11yescharacteristic
67D421—24no—
68D421—44no—
69C421D4211yescharacteristic, complemented, Sylow
70D42—4211nomaximal, Sylow
71C22×D4—4212no—
72C22×D4—4212no—
73#127—4211no—
74#121—4212no—
75#121—4212no—
76#120—4214no—
77C4×D4—4211no—
78C4×D4D42111yescharacteristic, complemented
79C24—4214no—
80C2×D4—8428no—
81C2×D4—8428no—
82C2×D4—4214no—
83C2×D4—4214no—
84C2×D4—4212no—
85C2×D4—4212no—
86C2×D4—4212no—
87C2×D4—4211no—
88C2×D4C2×D42111yescharacteristic
89C22×C4—4212no—
90C22×C4C2×D42112yescomplemented
91C4⋊C4—4211no—
92C4⋊C4C2×D42111yescharacteristic, complemented
93C22⋊C4—4212no—
94C22⋊C4—4214no—
95C22⋊C4C2×D42112yescomplemented
96C42C2×D42111yescharacteristic, complemented
97C23—8428no—
98C23—8428no—
99C23—16844no—
100C23—4214no—
101C23—4212no—
102C23C22×D42112yes—
103D4—8428no—
104D4—8428no—
105D4—8424no—
106D4—8424no—
107D4—16842no—
108D4—4214no—
109D4D168414yesdirect product factor
110C2×C4—24no—
111C2×C4—8424no—
112C2×C4—4212no—
113C2×C4C22×D42111yescharacteristic
114C2×C4C22×D42112yes—
115C2×C4C22×D42111yescharacteristic
116C2×C4C22×D42111yescharacteristic
117C22—22no—
118C22—22no—
119C22—8422no—
120C22—8422no—
121C22—8424no—
122C22—8424no—
123C22—8424no—
124C22—168416no—
125C22—4214no—
126C22C2×D168414yescomplemented
127C22C23×D42111yescharacteristic
128C4—22no—
129C4—21no—
130C4—8422no—
131C4D4×D42112yescomplemented
132C4C2×D168412yescomplemented
133C2—24no—
134C2—8424no—
135C2—16844no—
136C2C2×D4×D42111yescharacteristic
137C2C22×D168411yescharacteristic
138C2#2721241211yescharacteristic
subgroupG/H
center, FrattiniC22C23×D421
commutator, socle, cosocleC22×C421C24
FittingC4×D4×C421C2
hypercenterC4×D4D421
hypocenterC421D42
LCSG ▷ C22×C421 ▷ C421
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×D1684C2semidirect
2. C2×D4×D421C2semidirect
3. #27212363C2semidirect
4. #27212341C2semidirect
5. #27212340C2semidirect
6. #27212336C2semidirect
7. C4×D1684C2semidirect
8. C4×D4×C421C2semidirect
9. C23×D421C22semidirect
12. C2×D1684C22semidirect
13. C2×D1684C22semidirect
14. C2×D1684C22semidirect
15. C2×D1684C22semidirect
16. C2×C22⋊D421C22semidirect
18. C2×C4×D421C22semidirect
19. C4⋊Dic421C22semidirect
20. #20269748C22semidirect
21. #20269748C22semidirect
23. C22×C4×C421C22semidirect
24. C4⋊C4×C421C22semidirect
25. C22⋊C4×C421C22semidirect
26. C42×C421C22semidirect
36. D1684D4direct (N × H)
41. D4×C421D4semidirect
49. C22×C421C2×D4semidirect
57. C2×D421C2×D4semidirect
60. C4×C421C2×D4semidirect
61. C4×C421C2×D4semidirect
69. C421D42semidirect
78. C4×D4D421semidirect
90. C22×C4C2×D421semidirect
92. C4⋊C4C2×D421semidirect
95. C22⋊C4C2×D421semidirect
96. C42C2×D421semidirect
109. D4D1684direct (N × H)
126. C22C2×D1684semidirect
131. C4D4×D421semidirect
132. C4C2×D1684semidirect

Maximal subgroups

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

subgroupindex
C22×D16842
C2×D4×D4212
#272123632
#272123412
#272123402
#272123362
C4×D16842
C4×D4×C4212
D42421