Finite Group Databasebeta

← #5762

Subgroups — #5762

Order 192.

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

normal subgroups only

This group has 258 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
1#1007C211yesmaximal, characteristic, complemented
2#1004C211yesmaximal, characteristic, complemented
3C2×C3×SD16C211yesmaximal, characteristic, complemented
4#942C211yesmaximal, characteristic, complemented
5#932C211yesmaximal, characteristic, complemented
6#916C212yesmaximal, complemented
7#915C212yesmaximal, complemented
8#914C212yesmaximal, complemented
9D3×SD16C212yesmaximal, complemented
10#904C211yesmaximal, characteristic, complemented
11#901C211yesmaximal, characteristic, complemented
12#577—31nomaximal, Sylow
13C2×C3×Q8C2211yescharacteristic, complemented
14C2×C3×D4C2211yescharacteristic
15C2×#67—22no—
16#239—21no—
17#239C2212yescomplemented
18D3×Q8—21no—
19D3×Q8C2212yescomplemented
20#237—21no—
21#237C2212yes—
22D3×D4—21no—
23D3×D4C2212yes—
24#235—22no—
25#235C2211yescharacteristic
26#235C2211yescharacteristic
27#235C2211yescharacteristic, complemented
28C3×SD16C2212yescomplemented
29C3×SD16C2212yescomplemented
30C2×C3×C8C2211yescharacteristic, complemented
31#216C2212yescomplemented
32#215C2212yescomplemented
33#214C2212yescomplemented
34#213C2212yescomplemented
35#208C2211yescharacteristic, complemented
36Dic12C2211yescharacteristic, complemented
37D24C2211yescharacteristic, complemented
38#204C2211yescharacteristic, complemented
39#204C2211yescharacteristic, complemented
40ΓL(1,𝔽25)C2212yescomplemented
41D3×C8C2212yescomplemented
422−1+4—31no—
43H(5,𝔽2)—31no—
44#137—31no—
45#137—32no—
46AGL(1,ℤ/8)—31no—
47AGL(1,ℤ/8)—32no—
48#135—32no—
49#135—31no—
50C2×SD16—32no—
51C2×SD16D311yescharacteristic, complemented
52#131—31no—
53C23×C3—21no—
54C22×D3—22no—
55C3×Q8—21no—
56C3×Q8C2312yes—
57C3×D4—21no—
58C3×D4C2312yes—
59C2×C3×C4—21no—
60C2×C3×C4C2311yescharacteristic
61#67—21no—
62#67—21no—
63#67—22no—
64#67—22no—
65#67D412yes—
66C2×Dic3—22no—
67D12—22no—
68D12D411yescharacteristic
69D12C2311yescharacteristic
70D12C2311yescharacteristic
71C4×D3—22no—
72C4×D3—21no—
73C4×D3—21no—
74C4×D3—22no—
75C4×D3C2312yes—
76Dic6—22no—
77Dic6D411yescharacteristic
78Dic6C2311yescharacteristic
79Dic6C2311yescharacteristic
80C3×C8C2311yescharacteristic
81C3×C8C2311yescharacteristic
82C3⋊C8C2312yes—
83#41—31no—
84#41—32no—
85#41—32no—
86#41—31no—
87#41—31no—
88#41—62no—
89#41—61no—
90#41—61no—
91C2×Q8—32no—
92C2×Q8—61no—
93C2×Q8C2×D311yescharacteristic, complemented
94C2×D4—32no—
95C2×D4—61no—
96C2×D4—62no—
97C2×D4C2×D311yescharacteristic
98Q16—32no—
99Q16—31no—
100SD16—32no—
101SD16—32no—
102SD16—31no—
103SD16—31no—
104SD16C2×D312yescomplemented
105SD16C2×D312yescomplemented
106D8—32no—
107D8—31no—
108M16—31no—
109M16—32no—
110C2×C8—32no—
111C2×C8C2×D311yescharacteristic, complemented
112C22×C3—22no—
113C22×C3—81no—
114C22×C3C2×D411yescharacteristic
115C2×D3—21no—
116C2×D3—21no—
117C2×D3—82no—
118C2×D3C2×D412yes—
119C3×C4—22no—
120C3×C4C2×D411yescharacteristic
121C3×C4C2411yescharacteristic
122Dic3—21no—
123Dic3—21no—
124Dic3C2×D412yes—
125C23—21no—
126C23—62no—
127Q8—21no—
128Q8—31no—
129Q8—31no—
130Q8—31no—
131Q8—62no—
132Q8C22×D312yes—
133D4—21no—
134D4—32no—
135D4—31no—
136D4—31no—
137D4—31no—
138D4—62no—
139D4—62no—
140D4—62no—
141D4—61no—
142D4—61no—
143D4C22×D312yes—
144C2×C4—21no—
145C2×C4—32no—
146C2×C4—62no—
147C2×C4—62no—
148C2×C4—62no—
149C2×C4—61no—
150C2×C4—61no—
151C2×C4C22×D311yescharacteristic
152C8—32no—
153C8C22×D311yescharacteristic
154C8C22×D311yescharacteristic
155C2×C3—21no—
156C2×C3—42no—
157C2×C3C22×D411yescharacteristic
158D3—22no—
159D3—41no—
160D3—41no—
161C22—22no—
162C22—32no—
163C22—61no—
164C22—61no—
165C22—81no—
166C22—242no—
167C22D3×D411yescharacteristic
168C4—22no—
169C4—32no—
170C4—61no—
171C4—61no—
172C4D3×D411yescharacteristic
173C4C23×D311yescharacteristic
174C3#57711yescharacteristic, complemented, Sylow
175C2—21no—
176C2—42no—
177C2—62no—
178C2—121no—
179C2—121no—
180C2C2×D3×D411yescharacteristic
subgroupG/H
centerC2C2×D3×D4
commutator, cosocleC3×C4C24
FrattiniC4C23×D3
FittingC2×C3×SD16C2
socleC2×C3C22×D4
hypercenterC2×SD16D3
hypocenterC3#577
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×SD16

Complements

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

NH ≅ G/Ntype
1. #1007C2semidirect
2. #1004C2semidirect
3. C2×C3×SD16C2semidirect
4. #942C2semidirect
5. #932C2semidirect
6. #916C2semidirect
7. #915C2semidirect
8. #914C2semidirect
9. D3×SD16C2semidirect
10. #904C2semidirect
11. #901C2semidirect
13. C2×C3×Q8C22semidirect
17. #239C22semidirect
19. D3×Q8C22semidirect
27. #235C22semidirect
28. C3×SD16C22semidirect
29. C3×SD16C22semidirect
30. C2×C3×C8C22semidirect
31. #216C22semidirect
32. #215C22semidirect
33. #214C22semidirect
34. #213C22semidirect
35. #208C22semidirect
36. Dic12C22semidirect
37. D24C22semidirect
38. #204C22semidirect
39. #204C22semidirect
40. ΓL(1,𝔽25)C22semidirect
41. D3×C8C22semidirect
51. C2×SD16D3semidirect
93. C2×Q8C2×D3semidirect
104. SD16C2×D3semidirect
105. SD16C2×D3semidirect
111. C2×C8C2×D3semidirect
174. C3#577semidirect

Maximal subgroups

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

subgroupindex
#10072
#10042
C2×C3×SD162
#9422
#9322
#9162
#9152
#9142
D3×SD162
#9042
#9012
#5773