Finite Group Databasebeta

← #5768

Subgroups — #5768

Order 192.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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
1D3×#41C212yesmaximal, complemented
2C3×#135C211yesmaximal, characteristic, complemented
3#950C212yesmaximal, complemented
4#918C212yesmaximal, complemented
5#915C212yesmaximal, complemented
6#914C212yesmaximal, complemented
7#911C212yesmaximal, complemented
8#904C211yesmaximal, characteristic, complemented
9C2×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
10#575—31nomaximal, Sylow
11C3×#41C2212yescomplemented
12#239C2212yes—
13D3×Q8C2212yescomplemented
14#237—22no—
15#237C2212yes—
16D3×D4—22no—
17D3×D4C2212yescomplemented
18#235—22no—
19#235C2212yescomplemented
20C2×C4×D3—22no—
21C2×C4×D3C2211yescharacteristic
22C3×Q16C2211yescharacteristic, complemented
23C3×SD16C2212yescomplemented
24C3×D8C2211yescharacteristic, complemented
25C2×C3×C8C2211yescharacteristic, complemented
26#216C2212yescomplemented
27#215C2212yescomplemented
28#214C2212yescomplemented
29#213C2212yescomplemented
30C2×C3⋊C8C2211yescharacteristic, complemented
31Dic12C2211yescharacteristic, complemented
32D24C2211yescharacteristic, complemented
33#204C2212yescomplemented
34ΓL(1,𝔽25)C2212yescomplemented
35ΓL(1,𝔽25)C2212yescomplemented
36C2×#41—32no—
37#137—32no—
38#137—32no—
39AGL(1,ℤ/8)—32no—
40AGL(1,ℤ/8)—32no—
41#135—32no—
42#135—31no—
43#135D311yescharacteristic, complemented
44C2×M16—31no—
45C22×D3—22no—
46C22×D3D411yescharacteristic
47C3×Q8C2312yes—
48C3×D4—22no—
49C3×D4C2312yes—
50C2×C3×C4—22no—
51C2×C3×C4C2311yescharacteristic
52#67—22no—
53#67—22no—
54#67—22no—
55C2×Dic3—22no—
56C2×Dic3D411yescharacteristic
57D12—22no—
58D12C2312yes—
59C4×D3—22no—
60C4×D3—22no—
61C4×D3—22no—
62C4×D3D412yes—
63C4×D3C2312yes—
64Dic6—22no—
65Dic6C2312yes—
66C3×C8C2311yescharacteristic
67C3×C8C2311yescharacteristic
68C3⋊C8C2312yes—
69#41—32no—
70#41—32no—
71#41—32no—
72#41—62no—
73#41—62no—
74#41C2×D312yescomplemented
75C2×Q8—32no—
76C2×D4—32no—
77C2×D4—62no—
78C22×C4—31no—
79C22×C4—62no—
80Q16—32no—
81Q16—31no—
82Q16C2×D311yescharacteristic, complemented
83SD16—32no—
84SD16—32no—
85SD16—32no—
86SD16C2×D312yescomplemented
87D8—32no—
88D8—31no—
89D8C2×D311yescharacteristic, complemented
90M16—32no—
91M16—32no—
92C2×C8—31no—
93C2×C8C2×D311yescharacteristic, complemented
94C22×C3—22no—
95C22×C3C2×D411yescharacteristic
96C2×D3—22no—
97C2×D3—41no—
98C2×D3—44no—
99C2×D3C2×D412yes—
100C3×C4—22no—
101C3×C4C2×D411yescharacteristic
102C3×C4C2411yescharacteristic
103Dic3—22no—
104Dic3C2×D412yes—
105C23—31no—
106C23—62no—
107Q8—32no—
108Q8—62no—
109Q8C22×D312yes—
110D4—22no—
111D4—32no—
112D4—62no—
113D4—62no—
114D4—62no—
115D4—62no—
116D4C22×D312yes—
117C2×C4—22no—
118C2×C4—31no—
119C2×C4—32no—
120C2×C4—32no—
121C2×C4—62no—
122C2×C4—62no—
123C2×C4—62no—
124C2×C4—62no—
125C2×C4C22×D311yescharacteristic
126C8—32no—
127C8C22×D311yescharacteristic
128C8C22×D311yescharacteristic
129C2×C3—21no—
130C2×C3—42no—
131C2×C3C22×D411yescharacteristic
132D3—22no—
133D3—42no—
134C22—22no—
135C22—32no—
136C22—62no—
137C22—121no—
138C22—124no—
139C22D3×D411yescharacteristic
140C4—22no—
141C4—32no—
142C4—62no—
143C4D3×D411yescharacteristic
144C4C23×D311yescharacteristic
145C3#57511yescharacteristic, complemented, Sylow
146C2—21no—
147C2—42no—
148C2—62no—
149C2—122no—
150C2C2×D3×D411yescharacteristic
subgroupG/H
centerC4D3×D4
commutator, cosocleC3×C4C24
FrattiniC4C23×D3
FittingC3×#135C2
socleC2×C3C22×D4
hypercenter#135D3
hypocenterC3#575
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. D3×#41C2semidirect
2. C3×#135C2semidirect
3. #950C2semidirect
4. #918C2semidirect
5. #915C2semidirect
6. #914C2semidirect
7. #911C2semidirect
8. #904C2semidirect
9. C2×ΓL(1,𝔽25)C2semidirect
11. C3×#41C22semidirect
13. D3×Q8C22semidirect
17. D3×D4C22semidirect
19. #235C22semidirect
22. C3×Q16C22semidirect
23. C3×SD16C22semidirect
24. C3×D8C22semidirect
25. C2×C3×C8C22semidirect
26. #216C22semidirect
27. #215C22semidirect
28. #214C22semidirect
29. #213C22semidirect
30. C2×C3⋊C8C22semidirect
31. Dic12C22semidirect
32. D24C22semidirect
33. #204C22semidirect
34. ΓL(1,𝔽25)C22semidirect
35. ΓL(1,𝔽25)C22semidirect
43. #135D3semidirect
74. #41C2×D3semidirect
82. Q16C2×D3semidirect
86. SD16C2×D3semidirect
89. D8C2×D3semidirect
93. C2×C8C2×D3semidirect
145. C3#575semidirect

Maximal subgroups

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

subgroupindex
D3×#412
C3×#1352
#9502
#9182
#9152
#9142
#9112
#9042
C2×ΓL(1,𝔽25)2
#5753