Finite Group Databasebeta

← C22×#1009 #87564

Subgroups — C22×#1009 #87564

Order 384.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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×C3×#41C211yesmaximal, characteristic, complemented
2C2×#1009C2148yesmaximal, direct product factor
3C22×#239C212yesmaximal, complemented
4C22×D3×D4C216yesmaximal, complemented
5C22×#235C213yesmaximal, complemented
6C23×D12C213yesmaximal, complemented
7C22×H(5,𝔽2)—31nomaximal, Sylow
8C24×D3C2216yescomplemented
9C2×C3×#41C22124yescomplemented
10C22×C3×Q8C2211yescharacteristic, complemented
11C22×C3×D4C2213yescomplemented
12C23×C3×C4C2213yescomplemented
13C22×#67C2216yescomplemented
14#1009C221256yesdirect product factor
15C2×#239C22148yescomplemented
16C2×D3×D4C221144yescomplemented
17C2×#235C22172yescomplemented
18C22×D12C2213yescomplemented
19C22×D12C2216yescomplemented
20C22×D12C22172yescomplemented
21C22×C4×D3C2216yescomplemented
22C22×Dic6C2211yescharacteristic, complemented
23C2×H(5,𝔽2)—348no—
24C22×#41—32no—
25C22×#41—33no—
26C22×#41D311yescharacteristic, complemented
27C23×D4—36no—
28C23×D4—33no—
29C24×C3C2313yes—
30C23×D3—424no—
31C23×D3C2316yes—
32C23×D3C23172yescomplemented
33C3×#41C23164yescomplemented
34C2×C3×Q8C23112yescomplemented
35C2×C3×D4C23136yescomplemented
36C22×C3×C4C2311yescharacteristic
37C22×C3×C4C2313yes—
38C22×C3×C4C23136yescomplemented
39C2×#67C23172yescomplemented
40C22×Dic3C2312yes—
41#239C231128yescomplemented
42D3×D4C231384yescomplemented
43#235C231192yescomplemented
44C2×D12C23136yescomplemented
45C2×D12C23172yescomplemented
46C2×D12C231192yescomplemented
47C2×C4×D3C23172yescomplemented
48C2×Dic6C23112yescomplemented
49C25—36no—
50H(5,𝔽2)—3256no—
51C2×#41—348no—
52C2×#41—372no—
53C2×#41C2×D3124yescomplemented
54C22×Q8—31no—
55C22×Q8C2×D311yescharacteristic, complemented
56C22×D4—36no—
57C22×D4—3144no—
58C22×D4—36no—
59C22×D4—33no—
60C22×D4—372no—
61C22×D4C2×D313yescomplemented
62C23×C4—36no—
63C23×C4C2×D313yescomplemented
64C23×C3—212no—
65C23×C3C24118yes—
66C23×C3C2411yescharacteristic
67C22×D3—224no—
68C22×D3—4144no—
69C22×D3C24136yes—
70C22×D3C24196yescomplemented
71C3×Q8C24116yescomplemented
72C3×D4C24148yescomplemented
73C2×C3×C4C2416yes—
74C2×C3×C4C24118yes—
75C2×C3×C4C24148yescomplemented
76#67C24196yescomplemented
77C2×Dic3C24112yes—
78D12C24148yescomplemented
79D12C24196yescomplemented
80C4×D3C24196yescomplemented
81Dic6C24116yescomplemented
82C24—36no—
83C24—372no—
84C24—1224no—
85C24C22×D313yes—
86#41—3128no—
87#41—3192no—
88#41C22×D3164yescomplemented
89C2×Q8—312no—
90C2×Q8C22×D3112yescomplemented
91C2×D4—372no—
92C2×D4—3384no—
93C2×D4—372no—
94C2×D4—336no—
95C2×D4—3192no—
96C2×D4C22×D3136yescomplemented
97C22×C4—32no—
98C22×C4—372no—
99C22×C4C22×D313yes—
100C22×C4C22×D311yescharacteristic
101C22×C4C22×D3136yescomplemented
102C22×C3—236no—
103C22×C3H(5,𝔽2)14yescomplemented
104C22×C3C25112yes—
105C22×C3C2513yes—
106C2×D3—272no—
107C2×D3—496no—
108C2×D3C25124yes—
109C3×C4C2514yes—
110C3×C4C25112yes—
111Dic3C2518yes—
112C23—212no—
113C23—336no—
114C23—396no—
115C23—624no—
116C23—12144no—
117C23C23×D3118yes—
118C23C23×D311yescharacteristic
119Q8—316no—
120Q8C23×D3116yescomplemented
121D4—396no—
122D4—396no—
123D4—348no—
124D4C23×D3148yescomplemented
125C2×C4—312no—
126C2×C4—396no—
127C2×C4C23×D3118yes—
128C2×C4C23×D316yes—
129C2×C4C23×D3148yescomplemented
130C2×C3—212no—
131C2×C3C2×H(5,𝔽2)16yescomplemented
132C2×C3C2611yescharacteristic
133D3—224no—
134C22—236no—
135C22—324no—
136C22—672no—
137C22—1296no—
138C22#100914yesdirect product factor
139C22C24×D3112yes—
140C22C24×D313yes—
141C4—38no—
142C4C24×D3112yes—
143C4C24×D314yes—
144C3C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
145C2—212no—
146C2—624no—
147C2C2×#100916yesdirect product factor
148C2C25×D311yescharacteristic
subgroupG/H
centerC23C23×D3
commutator, cosocleC2×C3C26
FrattiniC2C25×D3
FittingC22×C3×#41C2
socleC23×C3C24
hypercenterC22×#41D3
hypocenterC3C22×H(5,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×C3×#41C2semidirect
2. C2×#1009C2direct (N × H)
3. C22×#239C2semidirect
4. C22×D3×D4C2semidirect
5. C22×#235C2semidirect
6. C23×D12C2semidirect
8. C24×D3C22semidirect
9. C2×C3×#41C22semidirect
10. C22×C3×Q8C22semidirect
11. C22×C3×D4C22semidirect
12. C23×C3×C4C22semidirect
13. C22×#67C22semidirect
14. #1009C22direct (N × H)
15. C2×#239C22semidirect
16. C2×D3×D4C22semidirect
17. C2×#235C22semidirect
18. C22×D12C22semidirect
19. C22×D12C22semidirect
20. C22×D12C22semidirect
21. C22×C4×D3C22semidirect
22. C22×Dic6C22semidirect
26. C22×#41D3semidirect
32. C23×D3C23semidirect
33. C3×#41C23semidirect
34. C2×C3×Q8C23semidirect
35. C2×C3×D4C23semidirect
38. C22×C3×C4C23semidirect
39. C2×#67C23semidirect
41. #239C23semidirect
42. D3×D4C23semidirect
43. #235C23semidirect
44. C2×D12C23semidirect
45. C2×D12C23semidirect
46. C2×D12C23semidirect
47. C2×C4×D3C23semidirect
48. C2×Dic6C23semidirect
53. C2×#41C2×D3semidirect
55. C22×Q8C2×D3semidirect
61. C22×D4C2×D3semidirect
63. C23×C4C2×D3semidirect
70. C22×D3C24semidirect
71. C3×Q8C24semidirect
72. C3×D4C24semidirect
75. C2×C3×C4C24semidirect
76. #67C24semidirect
78. D12C24semidirect
79. D12C24semidirect
80. C4×D3C24semidirect
81. Dic6C24semidirect
88. #41C22×D3semidirect
90. C2×Q8C22×D3semidirect
96. C2×D4C22×D3semidirect
101. C22×C4C22×D3semidirect
103. C22×C3H(5,𝔽2)semidirect
120. Q8C23×D3semidirect
124. D4C23×D3semidirect
129. C2×C4C23×D3semidirect
131. C2×C3C2×H(5,𝔽2)semidirect
138. C22#1009direct (N × H)
144. C3C22×H(5,𝔽2)semidirect
147. C2C2×#1009direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#412
C2×#10092
C22×#2392
C22×D3×D42
C22×#2352
C23×D122
C22×H(5,𝔽2)3