Finite Group Databasebeta

← H(3,𝔽4)×D89 #26034371

Subgroups — H(3,𝔽4)×D89 #26034371

Order 11392.

1408 conjugacy classes of subgroups; only the 459 normal subgroups are shown.

normal subgroups only

This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 459 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#19152341C216yesmaximal, complemented
2#19152340C219yesmaximal, complemented
3#124×D89C219yesmaximal, complemented
4#120×D89C216yesmaximal, complemented
5H(3,𝔽4)×C89C211yesmaximal, characteristic, complemented
6C24×D89C2212yescomplemented
7C2×Q8×D89C2213yescomplemented
8C2×D4×D89C2219yes—
9#15668001C2219yescomplemented
10#15668000C2219yescomplemented
11#15667999C2219yescomplemented
12#15667998C22136yescomplemented
13#15667982C22118yescomplemented
14#15667981C2216yescomplemented
15#15667980C22118yescomplemented
16C22⋊C4×D89C22118yescomplemented
17C42×D89C2213yescomplemented
18#124×C89C2219yescomplemented
19#120×C89C2216yescomplemented
20C23×D89—412no—
21C23×D89C2316yes—
22C23×D89C2316yescomplemented
23C2×Dic178C2319yescomplemented
24Q8×D89—46no—
25C2×D356C2319yes—
26C2×C22⋊D89C23118yes—
27D4×D89—418no—
28C2×C4×D89C2319yes—
29C22⋊Dic89C23118yescomplemented
30#13083551C23136yescomplemented
31C4×Dic89C2319yescomplemented
32C24×C89C2312yescomplemented
33C2×Q8×C89C2313yescomplemented
34C2×D4×C89C2319yes—
35C22⋊C4×C89C23118yescomplemented
36C42×C89C2313yescomplemented
37C23×C89—46no—
38C23×C89C2416yes—
39C22×D89—236no—
40C22×D89—418no—
41C22×D89—432no—
42C22×D89C2416yes—
43C22×D89C2411yescharacteristic
44Q8×C89—43no—
45D4×C89—49no—
46C2×C4×C89C2419yes—
47C22⋊D89—418no—
48C2×Dic89C2419yes—
49D356—49no—
50C4×D89—218no—
51Dic178—49no—
52C22×C89—218no—
53C22×C89—48no—
54C22×C89C2511yescharacteristic
55C2×D89—218no—
56C2×D89—424no—
57C2×D89—424no—
58C2×D89H(5,𝔽2)16yes—
59C4×C89—29no—
60Dic89—29no—
61C2×C89—46no—
62C2×C89C2×H(5,𝔽2)13yes—
63D89—46no—
64D89H(3,𝔽4)14yesdirect product factor
65C2×H(3,𝔽4)—891nomaximal, Sylow
66C89C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
67H(3,𝔽4)—899no—
68H(3,𝔽4)—896no—
69H(3,𝔽4)D8911yescharacteristic, direct product factor
70C2×#124—899no—
71C2×#120—896no—
72C25—892no—
73C22×Q8—893no—
74C22×D4—899no—
75#124—899no—
76#124—899no—
77#124—8936no—
78#124—899no—
79#124C2×D8919yescomplemented
80#120—896no—
81#120—8918no—
82#120—8918no—
83#120C2×D8916yescomplemented
84C2×C22⋊C4—8918no—
85C2×C42—893no—
86C24—896no—
87C24—896no—
88C24—35612no—
89C24C22×D8912yescomplemented
90C2×Q8—899no—
91C2×Q8—3566no—
92C2×Q8C22×D8913yescomplemented
93C2×D4—8918no—
94C2×D4—899no—
95C2×D4—35618no—
96C2×D4C22×D8919yes—
97C22×C4—899no—
98C22⋊C4—8918no—
99C22⋊C4—8936no—
100C22⋊C4C22×D89118yescomplemented
101C42—899no—
102C42C22×D8913yescomplemented
103C23—46no—
104C23—891no—
105C23—896no—
106C23—35618no—
107C23—35632no—
108C23—17836no—
109C23C23×D8916yes—
110Q8—43no—
111Q8—3569no—
112D4—49no—
113D4—3569no—
114D4—35618no—
115C2×C4—899no—
116C2×C4—17818no—
117C2×C4C23×D8919yes—
118C22—218no—
119C22—48no—
120C22—896no—
121C22—35624no—
122C22—35624no—
123C22—17818no—
124C22C24×D8911yescharacteristic
125C4—29no—
126C4—1789no—
127C2—46no—
128C2—894no—
129C2—3566no—
130C2H(5,𝔽2)×D8913yes—
subgroupG/H
center, FrattiniC22C24×D89
commutator, socle, cosocleC22×C89C25
FittingH(3,𝔽4)×C89C2
hypercenterH(3,𝔽4)D89
hypocenterC89C2×H(3,𝔽4)
LCSG ▷ C22×C89 ▷ C89
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #19152341C2semidirect
2. #19152340C2semidirect
3. #124×D89C2semidirect
4. #120×D89C2semidirect
5. H(3,𝔽4)×C89C2semidirect
6. C24×D89C22semidirect
7. C2×Q8×D89C22semidirect
9. #15668001C22semidirect
10. #15668000C22semidirect
11. #15667999C22semidirect
12. #15667998C22semidirect
13. #15667982C22semidirect
14. #15667981C22semidirect
15. #15667980C22semidirect
16. C22⋊C4×D89C22semidirect
17. C42×D89C22semidirect
18. #124×C89C22semidirect
19. #120×C89C22semidirect
22. C23×D89C23semidirect
23. C2×Dic178C23semidirect
29. C22⋊Dic89C23semidirect
30. #13083551C23semidirect
31. C4×Dic89C23semidirect
32. C24×C89C23semidirect
33. C2×Q8×C89C23semidirect
35. C22⋊C4×C89C23semidirect
36. C42×C89C23semidirect
64. D89H(3,𝔽4)direct (N × H)
66. C89C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D89direct (N × H)
79. #124C2×D89semidirect
83. #120C2×D89semidirect
89. C24C22×D89semidirect
92. C2×Q8C22×D89semidirect
100. C22⋊C4C22×D89semidirect
102. C42C22×D89semidirect

Maximal subgroups

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

subgroupindex
#191523412
#191523402
#124×D892
#120×D892
H(3,𝔽4)×C892
C2×H(3,𝔽4)89