Finite Group Databasebeta

← C32×#575 #10596376

Subgroups — C32×#575 #10596376

Order 576.

786 conjugacy classes of subgroups; only the 474 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 474 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×C32×#41C212yesmaximal, complemented
2C32×#137C214yesmaximal, complemented
3C32×AGL(1,ℤ/8)C214yesmaximal, complemented
4C32×#135C214yesmaximal, complemented
5C2×C32×M16C211yesmaximal, characteristic, complemented
6C3×#575C314yesmaximal, direct product factor
7C32×#41—22no—
8C32×#41—22no—
9C32×#41C2214yes—
10C32×#41C2214yescomplemented
11C2×C32×Q8C2212yescomplemented
12C2×C32×D4—22no—
13C2×C32×D4C2212yescomplemented
14C22×C32×C4—22no—
15C22×C32×C4C2211yescharacteristic
16C32×Q16C2214yescomplemented
17C32×SD16C2218yescomplemented
18C32×D8C2214yescomplemented
19C32×M16C2214yescomplemented
20C2×C32×C8C2212yescomplemented
21C2×C3×#41C2×C318yescomplemented
22C3×#137C2×C3116yescomplemented
23C3×AGL(1,ℤ/8)C2×C3116yescomplemented
24C3×#135C2×C3116yescomplemented
25C2×C3×M16C2×C314yescomplemented
26C23×C32—22no—
27C23×C32D411yescharacteristic
28C32×Q8—22no—
29C32×Q8C2314yes—
30C32×D4—24no—
31C32×D4—24no—
32C32×D4—22no—
33C32×D4C2314yes—
34C2×C32×C4—22no—
35C2×C32×C4—24no—
36C2×C32×C4—24no—
37C2×C32×C4D411yescharacteristic
38C2×C32×C4D412yes—
39C2×C32×C4C2312yes—
40C2×C32×C4C2311yescharacteristic
41C32×C8C2314yes—
42#575C3211yescharacteristic, direct product factor, Sylow
43C3×#41—28no—
44C3×#41—28no—
45C3×#41C22×C3116yes—
46C3×#41C22×C3116yescomplemented
47C2×C3×Q8C22×C318yescomplemented
48C2×C3×D4—28no—
49C2×C3×D4C22×C318yescomplemented
50C22×C3×C4—28no—
51C22×C3×C4C22×C314yes—
52C3×Q16C22×C3116yescomplemented
53C3×SD16C22×C3132yescomplemented
54C3×D8C22×C3116yescomplemented
55C3×M16C22×C3116yescomplemented
56C2×C3×C8C22×C318yescomplemented
57C22×C32—24no—
58C22×C32—41no—
59C22×C32—44no—
60C22×C32C2×D412yes—
61C22×C32C2×D411yescharacteristic
62C32×C4—24no—
63C32×C4C2×D412yes—
64C32×C4C2×D411yescharacteristic
65C32×C4C2411yescharacteristic
66C2×#41C2×C3212yescomplemented
67#137C2×C3214yescomplemented
68AGL(1,ℤ/8)C2×C3214yescomplemented
69#135C2×C3214yescomplemented
70C2×M16C2×C3211yescharacteristic, complemented
71C23×C3—28no—
72C23×C3C3×D414yes—
73C3×Q8—28no—
74C3×Q8C23×C3116yes—
75C3×D4—216no—
76C3×D4—216no—
77C3×D4—28no—
78C3×D4C23×C3116yes—
79C2×C3×C4—28no—
80C2×C3×C4—216no—
81C2×C3×C4—216no—
82C2×C3×C4C3×D418yes—
83C2×C3×C4C3×D414yes—
84C2×C3×C4C23×C314yes—
85C2×C3×C4C23×C318yes—
86C3×C8C23×C3116yes—
87C2×C32—22no—
88C2×C32—21no—
89C2×C32—44no—
90C2×C32C22×D411yescharacteristic
91#41—22no—
92#41—22no—
93#41C22×C3214yes—
94#41C22×C3214yescomplemented
95C2×Q8C22×C3212yescomplemented
96C2×D4—22no—
97C2×D4C22×C3212yescomplemented
98C22×C4—22no—
99C22×C4C22×C3211yescharacteristic
100Q16C22×C3214yescomplemented
101SD16C22×C3218yescomplemented
102D8C22×C3214yescomplemented
103M16C22×C3214yescomplemented
104C2×C8C22×C3212yescomplemented
105C22×C3—216no—
106C22×C3—44no—
107C22×C3—416no—
108C22×C3C2×C3×D418yes—
109C22×C3C2×C3×D414yes—
110C3×C4—216no—
111C3×C4C2×C3×D414yes—
112C3×C4C2×C3×D418yes—
113C3×C4C24×C314yes—
114C32#57511yescharacteristic, direct product factor, Sylow
115C23—22no—
116C23C32×D411yescharacteristic
117Q8—22no—
118Q8C23×C3214yes—
119D4—24no—
120D4—24no—
121D4—22no—
122D4C23×C3214yes—
123C2×C4—22no—
124C2×C4—24no—
125C2×C4—24no—
126C2×C4C32×D412yes—
127C2×C4C32×D411yescharacteristic
128C2×C4C23×C3211yescharacteristic
129C2×C4C23×C3212yes—
130C8C23×C3214yes—
131C2×C3—28no—
132C2×C3—24no—
133C2×C3—416no—
134C2×C3C22×C3×D414yes—
135C22—24no—
136C22—41no—
137C22—44no—
138C22C2×C32×D412yes—
139C22C2×C32×D411yescharacteristic
140C4—24no—
141C4C2×C32×D411yescharacteristic
142C4C2×C32×D412yes—
143C4C24×C3211yescharacteristic
144C3C3×#57514yesdirect product factor
145C2—22no—
146C2—21no—
147C2—44no—
148C2C22×C32×D411yescharacteristic
subgroupG/H
centerC32×C4C2×D4
commutator, Frattini, cosocleC4C24×C32
socleC2×C32C22×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C32×C4 ◁ C22×C32×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41C2semidirect
2. C32×#137C2semidirect
3. C32×AGL(1,ℤ/8)C2semidirect
4. C32×#135C2semidirect
5. C2×C32×M16C2semidirect
6. C3×#575C3direct (N × H)
10. C32×#41C22semidirect
11. C2×C32×Q8C22semidirect
13. C2×C32×D4C22semidirect
16. C32×Q16C22semidirect
17. C32×SD16C22semidirect
18. C32×D8C22semidirect
19. C32×M16C22semidirect
20. C2×C32×C8C22semidirect
21. C2×C3×#41C2×C3semidirect
22. C3×#137C2×C3semidirect
23. C3×AGL(1,ℤ/8)C2×C3semidirect
24. C3×#135C2×C3semidirect
25. C2×C3×M16C2×C3semidirect
42. #575C32direct (N × H)
46. C3×#41C22×C3semidirect
47. C2×C3×Q8C22×C3semidirect
49. C2×C3×D4C22×C3semidirect
52. C3×Q16C22×C3semidirect
53. C3×SD16C22×C3semidirect
54. C3×D8C22×C3semidirect
55. C3×M16C22×C3semidirect
56. C2×C3×C8C22×C3semidirect
66. C2×#41C2×C32semidirect
67. #137C2×C32semidirect
68. AGL(1,ℤ/8)C2×C32semidirect
69. #135C2×C32semidirect
70. C2×M16C2×C32semidirect
94. #41C22×C32semidirect
95. C2×Q8C22×C32semidirect
97. C2×D4C22×C32semidirect
100. Q16C22×C32semidirect
101. SD16C22×C32semidirect
102. D8C22×C32semidirect
103. M16C22×C32semidirect
104. C2×C8C22×C32semidirect
114. C32#575direct (N × H)
144. C3C3×#575direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#412
C32×#1372
C32×AGL(1,ℤ/8)2
C32×#1352
C2×C32×M162
C3×#5753