Finite Group Databasebeta

← H(3,𝔽4)×D269 #46825004

Subgroups — H(3,𝔽4)×D269 #46825004

Order 34432.

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#31771348C216yesmaximal, complemented
2#31771347C219yesmaximal, complemented
3#124×D269C219yesmaximal, complemented
4#120×D269C216yesmaximal, complemented
5H(3,𝔽4)×C269C211yesmaximal, characteristic, complemented
6C24×D269C2212yescomplemented
7C2×Q8×D269C2213yescomplemented
8C2×D4×D269C2219yes—
9#22575442C2219yescomplemented
10#22575441C2219yescomplemented
11#22575440C2219yescomplemented
12#22575439C22136yescomplemented
13#22575423C22118yescomplemented
14#22575422C2216yescomplemented
15#22575421C22118yescomplemented
16C22⋊C4×D269C22118yescomplemented
17C42×D269C2213yescomplemented
18#124×C269C2219yescomplemented
19#120×C269C2216yescomplemented
20C23×D269—412no—
21C23×D269C2316yes—
22C23×D269C2316yescomplemented
23C2×Dic538C2319yescomplemented
24Q8×D269—46no—
25C2×D1076C2319yes—
26C2×C22⋊D269C23118yes—
27D4×D269—418no—
28C2×C4×D269C2319yes—
29C22⋊Dic269C23118yescomplemented
30#16864379C23136yescomplemented
31C4×Dic269C2319yescomplemented
32C24×C269C2312yescomplemented
33C2×Q8×C269C2313yescomplemented
34C2×D4×C269C2319yes—
35C22⋊C4×C269C23118yescomplemented
36C42×C269C2313yescomplemented
37C22×D269—236no—
38C22×D269—418no—
39C22×D269—432no—
40C22×D269C2416yes—
41C22×D269C2411yescharacteristic
42Dic538—49no—
43D1076—49no—
44C22⋊D269—418no—
45C2×Dic269C2419yes—
46C4×D269—218no—
47C23×C269—46no—
48C23×C269C2416yes—
49Q8×C269—43no—
50D4×C269—49no—
51C2×C4×C269C2419yes—
52C22×C269—218no—
53C22×C269—48no—
54C22×C269C2511yescharacteristic
55C2×D269—218no—
56C2×D269—424no—
57C2×D269—424no—
58C2×D269H(5,𝔽2)16yes—
59C4×C269—29no—
60Dic269—29no—
61C2×C269—46no—
62C2×C269C2×H(5,𝔽2)13yes—
63D269—46no—
64D269H(3,𝔽4)14yesdirect product factor
65C269C2×H(3,𝔽4)11yescharacteristic, complemented, Sylow
66C2×H(3,𝔽4)—2691nomaximal, Sylow
67H(3,𝔽4)—2699no—
68H(3,𝔽4)—2696no—
69H(3,𝔽4)D26911yescharacteristic, direct product factor
70C2×#124—2699no—
71C2×#120—2696no—
72C25—2692no—
73C22×Q8—2693no—
74C22×D4—2699no—
75#124—2699no—
76#124—2699no—
77#124—26936no—
78#124—2699no—
79#124C2×D26919yescomplemented
80#120—2696no—
81#120—26918no—
82#120—26918no—
83#120C2×D26916yescomplemented
84C2×C22⋊C4—26918no—
85C2×C42—2693no—
86C24—2696no—
87C24—2696no—
88C24—107612no—
89C24C22×D26912yescomplemented
90C2×Q8—2699no—
91C2×Q8—10766no—
92C2×Q8C22×D26913yescomplemented
93C2×D4—26918no—
94C2×D4—2699no—
95C2×D4—107618no—
96C2×D4C22×D26919yes—
97C22×C4—2699no—
98C22⋊C4—26918no—
99C22⋊C4—26936no—
100C22⋊C4C22×D269118yescomplemented
101C42—2699no—
102C42C22×D26913yescomplemented
103C23—46no—
104C23—2691no—
105C23—2696no—
106C23—53836no—
107C23—107618no—
108C23—107632no—
109C23C23×D26916yes—
110Q8—43no—
111Q8—10769no—
112D4—49no—
113D4—10769no—
114D4—107618no—
115C2×C4—2699no—
116C2×C4—53818no—
117C2×C4C23×D26919yes—
118C22—218no—
119C22—48no—
120C22—2696no—
121C22—53818no—
122C22—107624no—
123C22—107624no—
124C22C24×D26911yescharacteristic
125C4—29no—
126C4—5389no—
127C2—46no—
128C2—2694no—
129C2—10766no—
130C2H(5,𝔽2)×D26913yes—
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
FittingH(3,𝔽4)×C269C2
hypercenterH(3,𝔽4)D269
hypocenterC269C2×H(3,𝔽4)
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #31771348C2semidirect
2. #31771347C2semidirect
3. #124×D269C2semidirect
4. #120×D269C2semidirect
5. H(3,𝔽4)×C269C2semidirect
6. C24×D269C22semidirect
7. C2×Q8×D269C22semidirect
9. #22575442C22semidirect
10. #22575441C22semidirect
11. #22575440C22semidirect
12. #22575439C22semidirect
13. #22575423C22semidirect
14. #22575422C22semidirect
15. #22575421C22semidirect
16. C22⋊C4×D269C22semidirect
17. C42×D269C22semidirect
18. #124×C269C22semidirect
19. #120×C269C22semidirect
22. C23×D269C23semidirect
23. C2×Dic538C23semidirect
29. C22⋊Dic269C23semidirect
30. #16864379C23semidirect
31. C4×Dic269C23semidirect
32. C24×C269C23semidirect
33. C2×Q8×C269C23semidirect
35. C22⋊C4×C269C23semidirect
36. C42×C269C23semidirect
64. D269H(3,𝔽4)direct (N × H)
65. C269C2×H(3,𝔽4)semidirect
69. H(3,𝔽4)D269direct (N × H)
79. #124C2×D269semidirect
83. #120C2×D269semidirect
89. C24C22×D269semidirect
92. C2×Q8C22×D269semidirect
100. C22⋊C4C22×D269semidirect
102. C42C22×D269semidirect

Maximal subgroups

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

subgroupindex
#317713482
#317713472
#124×D2692
#120×D2692
H(3,𝔽4)×C2692
C2×H(3,𝔽4)269