Finite Group Databasebeta

← H(3,𝔽4)×D251 #44564484

Subgroups — H(3,𝔽4)×D251 #44564484

Order 32128.

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#30632430C216yesmaximal, complemented
2#30632429C219yesmaximal, complemented
3#124×D251C219yesmaximal, complemented
4#120×D251C216yesmaximal, complemented
5H(3,𝔽4)×C251C211yesmaximal, characteristic, complemented
6C24×D251C2212yescomplemented
7C2×Q8×D251C2213yescomplemented
8C2×D4×D251C2219yes—
9#22548704C2219yescomplemented
10#22548703C2219yescomplemented
11#22548702C2219yescomplemented
12#22548701C22136yescomplemented
13#22548685C22118yescomplemented
14#22548684C2216yescomplemented
15#22548683C22118yescomplemented
16C22⋊C4×D251C22118yescomplemented
17C42×D251C2213yescomplemented
18#124×C251C2219yescomplemented
19#120×C251C2216yescomplemented
20C23×D251—412no—
21C23×D251C2316yes—
22C23×D251C2316yescomplemented
23C2×Dic502C2319yescomplemented
24Q8×D251—46no—
25C2×D1004C2319yes—
26C2×C22⋊D251C23118yes—
27D4×D251—418no—
28C2×C4×D251C2319yes—
29C22⋊Dic251C23118yescomplemented
30(C2×D251)⋊C4C23136yescomplemented
31C4×Dic251C2319yescomplemented
32C24×C251C2312yescomplemented
33C2×Q8×C251C2313yescomplemented
34C2×D4×C251C2319yes—
35C22⋊C4×C251C23118yescomplemented
36C42×C251C2313yescomplemented
37C22×D251—236no—
38C22×D251—418no—
39C22×D251—432no—
40C22×D251C2416yes—
41C22×D251C2411yescharacteristic
42Dic502—49no—
43D1004—49no—
44C22⋊D251—418no—
45C2×Dic251C2419yes—
46C4×D251—218no—
47C23×C251—46no—
48C23×C251C2416yes—
49Q8×C251—43no—
50D4×C251—49no—
51C2×C4×C251C2419yes—
52C22×C251—218no—
53C22×C251—48no—
54C22×C251C2511yescharacteristic
55C2×D251—218no—
56C2×D251—424no—
57C2×D251—424no—
58C2×D251H(5,𝔽2)16yes—
59C4×C251—29no—
60Dic251—29no—
61C2×C251—46no—
62C2×C251C2×H(5,𝔽2)13yes—
63D251—46no—
64D251H(3,𝔽4)14yesdirect product factor
65C251C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
66C2×H(3,𝔽4)—2511nomaximal, Sylow
67H(3,𝔽4)—2519no—
68H(3,𝔽4)—2516no—
69H(3,𝔽4)D25111yescharacteristic, direct product factor
70C2×#124—2519no—
71C2×#120—2516no—
72C25—2512no—
73C22×Q8—2513no—
74C22×D4—2519no—
75#124—2519no—
76#124—2519no—
77#124—25136no—
78#124—2519no—
79#124C2×D25119yescomplemented
80#120—2516no—
81#120—25118no—
82#120—25118no—
83#120C2×D25116yescomplemented
84C2×C22⋊C4—25118no—
85C2×C42—2513no—
86C24—100412no—
87C24—2516no—
88C24—2516no—
89C24C22×D25112yescomplemented
90C2×Q8—10046no—
91C2×Q8—2519no—
92C2×Q8C22×D25113yescomplemented
93C2×D4—100418no—
94C2×D4—25118no—
95C2×D4—2519no—
96C2×D4C22×D25119yes—
97C22×C4—2519no—
98C22⋊C4—25118no—
99C22⋊C4—25136no—
100C22⋊C4C22×D251118yescomplemented
101C42—2519no—
102C42C22×D25113yescomplemented
103C23—46no—
104C23—100418no—
105C23—100432no—
106C23—50236no—
107C23—2516no—
108C23—2511no—
109C23C23×D25116yes—
110Q8—43no—
111Q8—10049no—
112D4—49no—
113D4—10049no—
114D4—100418no—
115C2×C4—50218no—
116C2×C4—2519no—
117C2×C4C23×D25119yes—
118C22—218no—
119C22—48no—
120C22—100424no—
121C22—100424no—
122C22—50218no—
123C22—2516no—
124C22C24×D25111yescharacteristic
125C4—29no—
126C4—5029no—
127C2—46no—
128C2—10046no—
129C2—2514no—
130C2H(5,𝔽2)×D25113yes—
subgroupG/H
center, FrattiniC22C24×D251
commutator, socle, cosocleC22×C251C25
FittingH(3,𝔽4)×C251C2
hypercenterH(3,𝔽4)D251
hypocenterC251C2×H(3,𝔽4)
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #30632430C2semidirect
2. #30632429C2semidirect
3. #124×D251C2semidirect
4. #120×D251C2semidirect
5. H(3,𝔽4)×C251C2semidirect
6. C24×D251C22semidirect
7. C2×Q8×D251C22semidirect
9. #22548704C22semidirect
10. #22548703C22semidirect
11. #22548702C22semidirect
12. #22548701C22semidirect
13. #22548685C22semidirect
14. #22548684C22semidirect
15. #22548683C22semidirect
16. C22⋊C4×D251C22semidirect
17. C42×D251C22semidirect
18. #124×C251C22semidirect
19. #120×C251C22semidirect
22. C23×D251C23semidirect
23. C2×Dic502C23semidirect
29. C22⋊Dic251C23semidirect
30. (C2×D251)⋊C4C23semidirect
31. C4×Dic251C23semidirect
32. C24×C251C23semidirect
33. C2×Q8×C251C23semidirect
35. C22⋊C4×C251C23semidirect
36. C42×C251C23semidirect
64. D251H(3,𝔽4)direct (N × H)
65. C251C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D251direct (N × H)
79. #124C2×D251semidirect
83. #120C2×D251semidirect
89. C24C22×D251semidirect
92. C2×Q8C22×D251semidirect
100. C22⋊C4C22×D251semidirect
102. C42C22×D251semidirect

Maximal subgroups

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

subgroupindex
#306324302
#306324292
#124×D2512
#120×D2512
H(3,𝔽4)×C2512
C2×H(3,𝔽4)251