Finite Group Databasebeta

← #15689764

Subgroups — #15689764

Order 3008.

334 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 107 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×#10625354C211yesmaximal, characteristic, complemented
2C22×D188C211yesmaximal, characteristic, complemented
3#13094013C211yesmaximal, characteristic, complemented
4#13094009C212yesmaximal, complemented
5#13093999C211yesmaximal, characteristic, complemented
6#13093999C212yesmaximal, complemented
7#13093994C212yesmaximal, complemented
8#13093986C211yesmaximal, characteristic, complemented
9C4×C22⋊D47C211yesmaximal, characteristic, complemented
10C22⋊C4×D47C212yesmaximal, complemented
11C22⋊Q8×C47C211yesmaximal, characteristic, complemented
12C23×D47C2212yes—
13C2×Q8×C47C2211yescharacteristic, complemented
14C22×C4×C47C2211yescharacteristic
15C2×C22⋊D47C2211yescharacteristic
16#10625354—24no—
17C2×D188—24no—
18C2×D188C2212yes—
19C2×D188C2211yescharacteristic
20C2×D188C2211yescharacteristic
21C2×D188C2212yescomplemented
22C2×D188C2212yescomplemented
23C2×C4×D47C2212yescomplemented
24C2×C4×D47C2211yescharacteristic, complemented
25C2×C4×D47C2212yescomplemented
26C4⋊C4×C47C2211yescharacteristic, complemented
27C4⋊C4×C47C2212yescomplemented
28C22⋊C4×C47C2212yes—
29C22⋊Dic47C2211yescharacteristic
30(C2×D47)⋊C4C2212yescomplemented
31(C2×D47)⋊C4C2212yescomplemented
32(C2×D47)⋊C4C2211yescharacteristic, complemented
33(C2×D47)⋊C4C2212yescomplemented
34(C2×D47)⋊C4C2212yescomplemented
35Dic47⋊C4C2211yescharacteristic, complemented
36C4×Dic47C2211yescharacteristic, complemented
37C23×C47C2311yescharacteristic
38C22×D47—24no—
39C22×D47—24no—
40C22×D47—42no—
41C22×D47C2312yes—
42C22×D47C2311yescharacteristic
43C22×D47C2312yes—
44Q8×C47—22no—
45C2×C4×C47—22no—
46C2×C4×C47C2311yescharacteristic
47C2×C4×C47C2312yes—
48C2×C4×C47C2312yes—
49C2×C4×C47C2311yescharacteristic
50C22⋊D47—22no—
51C2×Dic47C2311yescharacteristic
52C2×Dic47C2311yescharacteristic
53C2×Dic47C2311yescharacteristic
54D188—24no—
55D188—24no—
56D188—22no—
57D188—42no—
58D188D414yescomplemented
59C4×D47—24no—
60C4×D47—22no—
61C4×D47—42no—
62C22×C47—21no—
63C22×C47—21no—
64C22×C47#4112yescomplemented
65C22×C47C2411yescharacteristic
66C2×D47—22no—
67C2×D47—22no—
68C2×D47—21no—
69C2×D47—22no—
70C2×D47—21no—
71C2×D47—22no—
72C2×D47—21no—
73C2×D47—22no—
74C2×D47—48no—
75C2×D47C2×D414yes—
76C4×C47—22no—
77C4×C47—21no—
78C4×C47—22no—
79C4×C47C2×D412yes—
80Dic47—21no—
81Dic47—21no—
82Dic47—21no—
83C2×C47—22no—
84C2×C47C22×D411yescharacteristic
85C2×C47C2×#4111yescharacteristic
86C2×C47H(5,𝔽2)11yescharacteristic
87D47—24no—
88D47—41no—
89D47—42no—
90#546—471nomaximal, Sylow
91C47#54611yescharacteristic, complemented, Sylow
92C2×#41—471no—
93C22×D4—471no—
94#124—471no—
95#123—472no—
96C22⋊Q8D4711yescharacteristic, complemented
97#121—471no—
98#121—472no—
99#120—472no—
100C4×D4—471no—
101C4×D4—471no—
102C2×C22⋊C4—472no—
103C24—472no—
104#41—944no—
105C2×Q8C2×D4711yescharacteristic, complemented
106C2×D4—471no—
107C2×D4—471no—
108C2×D4—472no—
109C2×D4—471no—
110C2×D4—472no—
111C2×D4—472no—
112C2×D4—944no—
113C22×C4—472no—
114C22×C4—471no—
115C22×C4—472no—
116C22×C4C2×D4711yescharacteristic
117C4⋊C4—471no—
118C4⋊C4C2×D4711yescharacteristic, complemented
119C4⋊C4C2×D4712yescomplemented
120C22⋊C4—471no—
121C22⋊C4—472no—
122C22⋊C4—472no—
123C22⋊C4—472no—
124C22⋊C4—471no—
125C22⋊C4—472no—
126C22⋊C4C2×D4712yes—
127C42—471no—
128C23—472no—
129C23—472no—
130C23—471no—
131C23—944no—
132C23—944no—
133C23—1882no—
134C23C22×D4711yescharacteristic
135Q8—22no—
136D4—474no—
137D4—944no—
138D4—944no—
139D4—942no—
140D4—942no—
141D4—1882no—
142C2×C4—22no—
143C2×C4—471no—
144C2×C4—471no—
145C2×C4—471no—
146C2×C4—944no—
147C2×C4—942no—
148C2×C4—1882no—
149C2×C4C22×D4712yes—
150C2×C4C22×D4712yes—
151C2×C4C22×D4711yescharacteristic
152C2×C4C22×D4711yescharacteristic
153C22—21no—
154C22—21no—
155C22—474no—
156C22—942no—
157C22—942no—
158C22—941no—
159C22—942no—
160C22—941no—
161C22—942no—
162C22—941no—
163C22—942no—
164C22—1888no—
165C22#1062535412yescomplemented
166C22C23×D4711yescharacteristic
167C4—21no—
168C4—22no—
169C4—22no—
170C4—941no—
171C4—941no—
172C4—941no—
173C4D4×D4712yes—
174C2—22no—
175C2—944no—
176C2—1882no—
177C2—1881no—
178C2C2×D4×D4711yescharacteristic
179C2C2×#1062535411yescharacteristic
180C2#1309407011yescharacteristic
subgroupG/H
center, FrattiniC22C23×D47
commutator, socle, cosocleC22×C47C24
FittingC22⋊Q8×C47C2
hypercenterC22⋊Q8D47
hypocenterC47#546
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#10625354C2semidirect
2. C22×D188C2semidirect
3. #13094013C2semidirect
4. #13094009C2semidirect
5. #13093999C2semidirect
6. #13093999C2semidirect
7. #13093994C2semidirect
8. #13093986C2semidirect
9. C4×C22⋊D47C2semidirect
10. C22⋊C4×D47C2semidirect
11. C22⋊Q8×C47C2semidirect
13. C2×Q8×C47C22semidirect
21. C2×D188C22semidirect
22. C2×D188C22semidirect
23. C2×C4×D47C22semidirect
24. C2×C4×D47C22semidirect
25. C2×C4×D47C22semidirect
26. C4⋊C4×C47C22semidirect
27. C4⋊C4×C47C22semidirect
30. (C2×D47)⋊C4C22semidirect
31. (C2×D47)⋊C4C22semidirect
32. (C2×D47)⋊C4C22semidirect
33. (C2×D47)⋊C4C22semidirect
34. (C2×D47)⋊C4C22semidirect
35. Dic47⋊C4C22semidirect
36. C4×Dic47C22semidirect
58. D188D4semidirect
64. C22×C47#41semidirect
91. C47#546semidirect
96. C22⋊Q8D47semidirect
105. C2×Q8C2×D47semidirect
118. C4⋊C4C2×D47semidirect
119. C4⋊C4C2×D47semidirect
165. C22#10625354semidirect

Maximal subgroups

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

subgroupindex
C2×#106253542
C22×D1882
#130940132
#130940092
#130939992
#130939942
#130939862
C4×C22⋊D472
C22⋊C4×D472
C22⋊Q8×C472
#54647