Finite Group Databasebeta

← #5829

Subgroups — #5829

Order 192.

334 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 113 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
2C2×D3×D4C211yesmaximal, characteristic, complemented
3#947C211yesmaximal, characteristic, complemented
4#946C211yesmaximal, characteristic, complemented
5#939C212yesmaximal, complemented
6#937C212yesmaximal, complemented
7D4×Dic3C211yesmaximal, characteristic, complemented
8#930C211yesmaximal, characteristic, complemented
9#929C212yesmaximal, complemented
10C4×#67C211yesmaximal, characteristic, complemented
11C2×(C2×D3)⋊C4C212yesmaximal, complemented
12#546—31nomaximal, Sylow
13C23×D3C2212yes—
14C3×#41—24no—
15C2×C3×Q8C2211yescharacteristic, complemented
16C2×C3×D4C2211yescharacteristic
17C2×C3×D4C2212yescomplemented
18C22×C3×C4C2212yescomplemented
19C22×C3×C4C2211yescharacteristic, complemented
20C2×#67C2211yescharacteristic
21C2×#67C2212yes—
22C2×#67C2212yescomplemented
23C22×Dic3C2212yescomplemented
24D3×D4—24no—
25C2×D12C2211yescharacteristic
26C2×C4×D3C2211yescharacteristic
27#217C2211yescharacteristic, complemented
28#217C2212yescomplemented
29(C2×D3)⋊C4C2211yescharacteristic
30(C2×D3)⋊C4C2212yes—
31(C2×D3)⋊C4C2212yescomplemented
32(C2×D3)⋊C4C2212yescomplemented
33(C2×D3)⋊C4C2212yescomplemented
34C4⋊Dic3C2211yescharacteristic, complemented
35Dic3⋊C4C2211yescharacteristic, complemented
36Dic3⋊C4C2212yescomplemented
37C4×Dic3C2211yescharacteristic, complemented
38C2×#41D311yescharacteristic, complemented
39C22×D4—31no—
40#124—31no—
41#123—32no—
42C22⋊Q8—31no—
43#121—32no—
44#121—31no—
45#120—32no—
46C4×D4—31no—
47C4×D4—31no—
48C2×C22⋊C4—32no—
49C23×C3C2311yescharacteristic
50C23×C3C2312yes—
51C22×D3—24no—
52C22×D3—24no—
53C22×D3—42no—
54C22×D3C2312yes—
55C22×D3C2311yescharacteristic
56C3×Q8—22no—
57C3×D4—24no—
58C3×D4D414yescomplemented
59C2×C3×C4—24no—
60C2×C3×C4—22no—
61C2×C3×C4C2311yescharacteristic
62C2×C3×C4C2312yes—
63C2×C3×C4C2311yescharacteristic
64#67—24no—
65#67—22no—
66#67—42no—
67C2×Dic3—42no—
68C2×Dic3C2311yescharacteristic
69C2×Dic3C2311yescharacteristic
70C2×Dic3C2311yescharacteristic
71C2×Dic3C2312yes—
72D12—22no—
73C4×D3—22no—
74C24—32no—
75#41—24no—
76C2×Q8C2×D311yescharacteristic, complemented
77C2×D4—31no—
78C2×D4—32no—
79C2×D4—32no—
80C2×D4—31no—
81C2×D4—64no—
82C2×D4C2×D311yescharacteristic
83C2×D4C2×D312yescomplemented
84C22×C4—32no—
85C22×C4—31no—
86C22×C4C2×D312yescomplemented
87C22×C4C2×D311yescharacteristic, complemented
88C4⋊C4—31no—
89C4⋊C4—31no—
90C4⋊C4—32no—
91C22⋊C4—32no—
92C22⋊C4—31no—
93C22⋊C4—31no—
94C22⋊C4—32no—
95C22⋊C4—32no—
96C22⋊C4—32no—
97C22⋊C4—32no—
98C42—31no—
99C22×C3—22no—
100C22×C3—22no—
101C22×C3—21no—
102C22×C3—21no—
103C22×C3—21no—
104C22×C3C2×D414yes—
105C22×C3C2411yescharacteristic
106C2×D3—21no—
107C2×D3—21no—
108C2×D3—22no—
109C2×D3—22no—
110C2×D3—22no—
111C2×D3—48no—
112C2×D3#4112yescomplemented
113C3×C4—21no—
114C3×C4—22no—
115C3×C4C2×D412yes—
116Dic3—21no—
117Dic3—21no—
118Dic3—21no—
119Dic3—22no—
120C23—31no—
121C23—32no—
122C23—64no—
123C23—64no—
124C23—122no—
125C23C22×D311yescharacteristic
126C23C22×D312yes—
127Q8—22no—
128D4—24no—
129D4—62no—
130D4—64no—
131D4—62no—
132D4—122no—
133D4#6714yescomplemented
134C2×C4—24no—
135C2×C4—22no—
136C2×C4—31no—
137C2×C4—31no—
138C2×C4—32no—
139C2×C4—31no—
140C2×C4—62no—
141C2×C4—122no—
142C2×C4C22×D311yescharacteristic
143C2×C4C22×D312yes—
144C2×C4C22×D311yescharacteristic
145C2×C3—24no—
146C2×C3—41no—
147C2×C3C22×D411yescharacteristic
148C2×C3C2×#4111yescharacteristic
149C2×C3H(5,𝔽2)11yescharacteristic
150D3—22no—
151D3—42no—
152C22—22no—
153C22—22no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—32no—
158C22—61no—
159C22—61no—
160C22—62no—
161C22—62no—
162C22—62no—
163C22—128no—
164C22C2×#6714yes—
165C22C23×D311yescharacteristic
166C4—21no—
167C4—22no—
168C4—61no—
169C4—61no—
170C4—62no—
171C4—61no—
172C4C2×#6712yes—
173C3#54611yescharacteristic, complemented, Sylow
174C2—24no—
175C2—41no—
176C2—62no—
177C2—122no—
178C2D3×#4111yescharacteristic
179C2#100911yescharacteristic
180C2C22×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C23×D3
commutator, socle, cosocleC22×C3C24
FittingC2×C3×#41C2
hypercenterC2×#41D3
hypocenterC3#546
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C3×#41C2semidirect
2. C2×D3×D4C2semidirect
3. #947C2semidirect
4. #946C2semidirect
5. #939C2semidirect
6. #937C2semidirect
7. D4×Dic3C2semidirect
8. #930C2semidirect
9. #929C2semidirect
10. C4×#67C2semidirect
11. C2×(C2×D3)⋊C4C2semidirect
15. C2×C3×Q8C22semidirect
17. C2×C3×D4C22semidirect
18. C22×C3×C4C22semidirect
19. C22×C3×C4C22semidirect
22. C2×#67C22semidirect
23. C22×Dic3C22semidirect
27. #217C22semidirect
28. #217C22semidirect
31. (C2×D3)⋊C4C22semidirect
32. (C2×D3)⋊C4C22semidirect
33. (C2×D3)⋊C4C22semidirect
34. C4⋊Dic3C22semidirect
35. Dic3⋊C4C22semidirect
36. Dic3⋊C4C22semidirect
37. C4×Dic3C22semidirect
38. C2×#41D3semidirect
58. C3×D4D4semidirect
76. C2×Q8C2×D3semidirect
83. C2×D4C2×D3semidirect
86. C22×C4C2×D3semidirect
87. C22×C4C2×D3semidirect
112. C2×D3#41semidirect
133. D4#67semidirect
173. C3#546semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#412
C2×D3×D42
#9472
#9462
#9392
#9372
D4×Dic32
#9302
#9292
C4×#672
C2×(C2×D3)⋊C42
#5463