Finite Group Databasebeta

← C3×#525 #5855

Subgroups — C3×#525 #5855

Order 192.

330 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×C3×#41C211yesmaximal, characteristic, complemented
2C23×C3×C4C211yesmaximal, characteristic, complemented
3C3×#123C212yesmaximal, complemented
4C3×C22⋊Q8C212yesmaximal, complemented
5C3×#121C212yesmaximal, complemented
6C3×#120C212yesmaximal, complemented
7C3×C4×D4C214yesmaximal, complemented
8C3×#117C211yesmaximal, characteristic, complemented
9#525C311yesmaximal, characteristic, direct product factor, Sylow
10C24×C3C2211yescharacteristic
11C3×#41—24no—
12C2×C3×Q8C2211yescharacteristic, complemented
13C2×C3×D4C2214yes—
14C2×C3×D4C2212yes—
15C2×C3×D4C2211yescharacteristic, complemented
16C22×C3×C4—24no—
17C22×C3×C4C2212yes—
18C22×C3×C4C2211yescharacteristic
19C22×C3×C4C2212yes—
20C22×C3×C4C2211yescharacteristic
21C22×C3×C4C2212yescomplemented
22C3×C4⋊C4C2212yescomplemented
23C3×C4⋊C4C2214yescomplemented
24C3×C22⋊C4C2212yes—
25C3×C22⋊C4C2214yes—
26C3×C22⋊C4C2214yescomplemented
27C3×C42C2212yescomplemented
28C2×#41C2×C311yescharacteristic, complemented
29C23×C4C2×C311yescharacteristic, complemented
30#123C2×C312yescomplemented
31C22⋊Q8C2×C312yescomplemented
32#121C2×C312yescomplemented
33#120C2×C312yescomplemented
34C4×D4C2×C314yescomplemented
35#117C2×C311yescharacteristic, complemented
36C23×C3—22no—
37C23×C3—24no—
38C23×C3C2312yes—
39C23×C3C2311yescharacteristic
40C23×C3C2312yes—
41C3×Q8—22no—
42C3×D4—24no—
43C3×D4—28no—
44C3×D4—22no—
45C2×C3×C4—22no—
46C2×C3×C4—24no—
47C2×C3×C4—24no—
48C2×C3×C4—24no—
49C2×C3×C4D414yescomplemented
50C2×C3×C4C2314yes—
51C2×C3×C4C2312yes—
52C2×C3×C4C2312yes—
53C2×C3×C4C2311yescharacteristic
54C2×C3×C4C2311yescharacteristic
55C24C22×C311yescharacteristic
56#41—24no—
57C2×Q8C22×C311yescharacteristic, complemented
58C2×D4C22×C314yes—
59C2×D4C22×C312yes—
60C2×D4C22×C311yescharacteristic, complemented
61C22×C4—24no—
62C22×C4C22×C312yes—
63C22×C4C22×C312yes—
64C22×C4C22×C311yescharacteristic
65C22×C4C22×C311yescharacteristic
66C22×C4C22×C312yescomplemented
67C4⋊C4C22×C312yescomplemented
68C4⋊C4C22×C314yescomplemented
69C22⋊C4C22×C312yes—
70C22⋊C4C22×C314yes—
71C22⋊C4C22×C314yescomplemented
72C42C22×C312yescomplemented
73C22×C3—28no—
74C22×C3—22no—
75C22×C3—22no—
76C22×C3—22no—
77C22×C3—22no—
78C22×C3—24no—
79C22×C3C2×D412yes—
80C22×C3#4114yescomplemented
81C22×C3C2411yescharacteristic
82C3×C4—22no—
83C3×C4—22no—
84C3×C4—24no—
85C3×C4C2×D412yes—
86C3×C4C2×D412yes—
87C23—22no—
88C23—24no—
89C23C23×C312yes—
90C23C23×C311yescharacteristic
91C23C23×C312yes—
92Q8—22no—
93D4—24no—
94D4—28no—
95D4—22no—
96C2×C4—22no—
97C2×C4—24no—
98C2×C4—24no—
99C2×C4—24no—
100C2×C4C3×D414yescomplemented
101C2×C4C23×C314yes—
102C2×C4C23×C312yes—
103C2×C4C23×C312yes—
104C2×C4C23×C311yescharacteristic
105C2×C4C23×C311yescharacteristic
106C2×C3—22no—
107C2×C3—24no—
108C2×C3—42no—
109C2×C3C22×D411yescharacteristic
110C2×C3C2×#4112yes—
111C22—28no—
112C22—22no—
113C22—22no—
114C22—22no—
115C22—22no—
116C22—24no—
117C22C2×C3×D412yes—
118C22C3×#4114yescomplemented
119C22C24×C311yescharacteristic
120C4—22no—
121C4—22no—
122C4—24no—
123C4C2×C3×D412yes—
124C4C2×C3×D412yes—
125C3#52511yescharacteristic, direct product factor, Sylow
126C2—22no—
127C2—24no—
128C2—42no—
129C2C22×C3×D411yescharacteristic
130C2C2×C3×#4112yes—
subgroupG/H
centerC2×C3×C4C23
commutator, Frattini, cosocleC22C24×C3
socleC22×C3C24
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41C2semidirect
2. C23×C3×C4C2semidirect
3. C3×#123C2semidirect
4. C3×C22⋊Q8C2semidirect
5. C3×#121C2semidirect
6. C3×#120C2semidirect
7. C3×C4×D4C2semidirect
8. C3×#117C2semidirect
9. #525C3direct (N × H)
12. C2×C3×Q8C22semidirect
15. C2×C3×D4C22semidirect
21. C22×C3×C4C22semidirect
22. C3×C4⋊C4C22semidirect
23. C3×C4⋊C4C22semidirect
26. C3×C22⋊C4C22semidirect
27. C3×C42C22semidirect
28. C2×#41C2×C3semidirect
29. C23×C4C2×C3semidirect
30. #123C2×C3semidirect
31. C22⋊Q8C2×C3semidirect
32. #121C2×C3semidirect
33. #120C2×C3semidirect
34. C4×D4C2×C3semidirect
35. #117C2×C3semidirect
49. C2×C3×C4D4semidirect
57. C2×Q8C22×C3semidirect
60. C2×D4C22×C3semidirect
66. C22×C4C22×C3semidirect
67. C4⋊C4C22×C3semidirect
68. C4⋊C4C22×C3semidirect
71. C22⋊C4C22×C3semidirect
72. C42C22×C3semidirect
80. C22×C3#41semidirect
100. C2×C4C3×D4semidirect
118. C22C3×#41semidirect
125. C3#525direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#412
C23×C3×C42
C3×#1232
C3×C22⋊Q82
C3×#1212
C3×#1202
C3×C4×D42
C3×#1172
#5253