Finite Group Databasebeta

← H(3,𝔽4)×D271 #46850483

Subgroups — H(3,𝔽4)×D271 #46850483

Order 34688.

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#31774640C216yesmaximal, complemented
2#31774639C219yesmaximal, complemented
3#124×D271C219yesmaximal, complemented
4#120×D271C216yesmaximal, complemented
5H(3,𝔽4)×C271C211yesmaximal, characteristic, complemented
6C24×D271C2212yescomplemented
7C2×Q8×D271C2213yescomplemented
8C2×D4×D271C2219yes—
9#22576149C2219yescomplemented
10#22576148C2219yescomplemented
11#22576147C2219yescomplemented
12#22576146C22136yescomplemented
13#22576130C22118yescomplemented
14#22576129C2216yescomplemented
15#22576128C22118yescomplemented
16C22⋊C4×D271C22118yescomplemented
17C42×D271C2213yescomplemented
18#124×C271C2219yescomplemented
19#120×C271C2216yescomplemented
20C23×D271—412no—
21C23×D271C2316yes—
22C23×D271C2316yescomplemented
23C2×Dic542C2319yescomplemented
24Q8×D271—46no—
25C2×D1084C2319yes—
26C2×C22⋊D271C23118yes—
27D4×D271—418no—
28C2×C4×D271C2319yes—
29C22⋊Dic271C23118yescomplemented
30(C2×D271)⋊C4C23136yescomplemented
31C4×Dic271C2319yescomplemented
32C24×C271C2312yescomplemented
33C2×Q8×C271C2313yescomplemented
34C2×D4×C271C2319yes—
35C22⋊C4×C271C23118yescomplemented
36C42×C271C2313yescomplemented
37C22×D271—236no—
38C22×D271—418no—
39C22×D271—432no—
40C22×D271C2416yes—
41C22×D271C2411yescharacteristic
42Dic542—49no—
43D1084—49no—
44C22⋊D271—418no—
45C2×Dic271C2419yes—
46C4×D271—218no—
47C23×C271—46no—
48C23×C271C2416yes—
49Q8×C271—43no—
50D4×C271—49no—
51C2×C4×C271C2419yes—
52C22×C271—218no—
53C22×C271—48no—
54C22×C271C2511yescharacteristic
55C2×D271—218no—
56C2×D271—424no—
57C2×D271—424no—
58C2×D271H(5,𝔽2)16yes—
59C4×C271—29no—
60Dic271—29no—
61C2×C271—46no—
62C2×C271C2×H(5,𝔽2)13yes—
63D271—46no—
64D271H(3,𝔽4)14yesdirect product factor
65C271C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
66C2×H(3,𝔽4)—2711nomaximal, Sylow
67H(3,𝔽4)—2719no—
68H(3,𝔽4)—2716no—
69H(3,𝔽4)D27111yescharacteristic, direct product factor
70C2×#124—2719no—
71C2×#120—2716no—
72C25—2712no—
73C22×Q8—2713no—
74C22×D4—2719no—
75#124—2719no—
76#124—2719no—
77#124—27136no—
78#124—2719no—
79#124C2×D27119yescomplemented
80#120—2716no—
81#120—27118no—
82#120—27118no—
83#120C2×D27116yescomplemented
84C2×C22⋊C4—27118no—
85C2×C42—2713no—
86C24—2716no—
87C24—2716no—
88C24—108412no—
89C24C22×D27112yescomplemented
90C2×Q8—2719no—
91C2×Q8—10846no—
92C2×Q8C22×D27113yescomplemented
93C2×D4—27118no—
94C2×D4—2719no—
95C2×D4—108418no—
96C2×D4C22×D27119yes—
97C22×C4—2719no—
98C22⋊C4—27118no—
99C22⋊C4—27136no—
100C22⋊C4C22×D271118yescomplemented
101C42—2719no—
102C42C22×D27113yescomplemented
103C23—46no—
104C23—2711no—
105C23—2716no—
106C23—54236no—
107C23—108418no—
108C23—108432no—
109C23C23×D27116yes—
110Q8—43no—
111Q8—10849no—
112D4—49no—
113D4—10849no—
114D4—108418no—
115C2×C4—2719no—
116C2×C4—54218no—
117C2×C4C23×D27119yes—
118C22—218no—
119C22—48no—
120C22—2716no—
121C22—54218no—
122C22—108424no—
123C22—108424no—
124C22C24×D27111yescharacteristic
125C4—29no—
126C4—5429no—
127C2—46no—
128C2—2714no—
129C2—10846no—
130C2H(5,𝔽2)×D27113yes—
subgroupG/H
center, FrattiniC22C24×D271
commutator, socle, cosocleC22×C271C25
FittingH(3,𝔽4)×C271C2
hypercenterH(3,𝔽4)D271
hypocenterC271C2×H(3,𝔽4)
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #31774640C2semidirect
2. #31774639C2semidirect
3. #124×D271C2semidirect
4. #120×D271C2semidirect
5. H(3,𝔽4)×C271C2semidirect
6. C24×D271C22semidirect
7. C2×Q8×D271C22semidirect
9. #22576149C22semidirect
10. #22576148C22semidirect
11. #22576147C22semidirect
12. #22576146C22semidirect
13. #22576130C22semidirect
14. #22576129C22semidirect
15. #22576128C22semidirect
16. C22⋊C4×D271C22semidirect
17. C42×D271C22semidirect
18. #124×C271C22semidirect
19. #120×C271C22semidirect
22. C23×D271C23semidirect
23. C2×Dic542C23semidirect
29. C22⋊Dic271C23semidirect
30. (C2×D271)⋊C4C23semidirect
31. C4×Dic271C23semidirect
32. C24×C271C23semidirect
33. C2×Q8×C271C23semidirect
35. C22⋊C4×C271C23semidirect
36. C42×C271C23semidirect
64. D271H(3,𝔽4)direct (N × H)
65. C271C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D271direct (N × H)
79. #124C2×D271semidirect
83. #120C2×D271semidirect
89. C24C22×D271semidirect
92. C2×Q8C22×D271semidirect
100. C22⋊C4C22×D271semidirect
102. C42C22×D271semidirect

Maximal subgroups

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

subgroupindex
#317746402
#317746392
#124×D2712
#120×D2712
H(3,𝔽4)×C2712
C2×H(3,𝔽4)271