Finite Group Databasebeta

← C22×C32×D32 #13049879

Subgroups — C22×C32×D32 #13049879

Order 1296.

2356 conjugacy classes of subgroups; only the 624 normal subgroups are shown.

normal subgroups only

This group has 2356 conjugacy classes of subgroups — too many to draw. Only its 624 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
1C22×C32×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C33×D3C212yesmaximal, complemented
3C2×C32×D32C2112yesmaximal, direct product factor
4C23×C32×D3—32nomaximal
5C22×C3×D32C314yesmaximal, direct product factor
6C22×C34C2211yescharacteristic, complemented
7C2×C32×C3⋊D3C2216yescomplemented
8C2×C33×D3C22112yescomplemented
9C32×D32C22116yesdirect product factor
10C23×C33—32no—
11C22×C3×C3⋊D3C2×C314yescomplemented
12C22×C32×D3—28no—
13C22×C32×D3—32no—
14C22×C32×D3—324no—
15C22×C32×D3—61no—
16C22×C32×D3D312yesdirect product factor
17C22×C32×D3C2×C318yescomplemented
18C2×C3×D32C2×C3148yesdirect product factor
19C2×C34C2313yescomplemented
20C32×C3⋊D3C2314yescomplemented
21C33×D3C2318yescomplemented
22C24×C32—91no—
23C23×C3×D3—38no—
24C22×D32C3211yescharacteristic, direct product factor
25C22×C33—21no—
26C22×C33—28no—
27C22×C33—312no—
28C22×C33—44no—
29C22×C33C2×D312yescomplemented
30C22×C33C22×C314yescomplemented
31C2×C3×C3⋊D3C22×C3124yescomplemented
32C2×C32×D3—248no—
33C2×C32×D3—312no—
34C2×C32×D3—332no—
35C2×C32×D3—66no—
36C2×C32×D3C2×D3112yesdirect product factor
37C2×C32×D3C22×C3148yescomplemented
38C3×D32C22×C3164yesdirect product factor
39C34C2411yescharacteristic, complemented, Sylow
40C23×C32—32no—
41C23×C32—38no—
42C23×C32—68no—
43C23×C32—912no—
44C23×C32—91no—
45C22×C3⋊D3C2×C3211yescharacteristic, complemented
46C22×C3×D3—28no—
47C22×C3×D3—38no—
48C22×C3×D3—396no—
49C22×C3×D3—64no—
50C22×C3×D3C3×D318yesdirect product factor
51C22×C3×D3C2×C3212yescomplemented
52C2×D32C2×C32112yesdirect product factor
53C2×C33—23no—
54C2×C33—224no—
55C2×C33—38no—
56C2×C33—412no—
57C2×C33C22×D316yescomplemented
58C2×C33C23×C3112yescomplemented
59C3×C3⋊D3C23×C3116yescomplemented
60C32×D3—232no—
61C32×D3—38no—
62C32×D3—64no—
63C32×D3C22×D318yesdirect product factor
64C32×D3C23×C3132yescomplemented
65C24×C3—94no—
66C23×D3—32no—
67C22×C32—28no—
68C22×C32—24no—
69C22×C32—28no—
70C22×C32—312no—
71C22×C32—348no—
72C22×C32—44no—
73C22×C32—412no—
74C22×C32—44no—
75C22×C32—648no—
76C22×C32—916no—
77C22×C32—96no—
78C22×C32D3211yescharacteristic, direct product factor
79C22×C32C2×C3×D318yescomplemented
80C22×C32C22×C3211yescharacteristic, complemented
81C2×C3⋊D3C22×C3216yescomplemented
82C2×C3×D3—248no—
83C2×C3×D3—348no—
84C2×C3×D3—3128no—
85C2×C3×D3—624no—
86C2×C3×D3C2×C3×D3148yesdirect product factor
87C2×C3×D3C22×C32112yescomplemented
88D32C22×C32116yesdirect product factor
89C33—21no—
90C33—28no—
91C33—44no—
92C33C23×D312yescomplemented
93C33C24×C314yescomplemented
94C23×C3—38no—
95C23×C3—32no—
96C23×C3—68no—
97C23×C3—948no—
98C23×C3—94no—
99C22×D3—32no—
100C22×D3—324no—
101C22×D3—61no—
102C22×D3C32×D312yesdirect product factor
103C2×C32—224no—
104C2×C32—212no—
105C2×C32—224no—
106C2×C32—38no—
107C2×C32—332no—
108C2×C32—412no—
109C2×C32—436no—
110C2×C32—412no—
111C2×C32—632no—
112C2×C32—94no—
113C2×C32C2×D3213yesdirect product factor
114C2×C32C22×C3×D3124yescomplemented
115C2×C32C23×C3213yescomplemented
116C3⋊D3C23×C3214yescomplemented
117C3×D3—232no—
118C3×D3—332no—
119C3×D3—616no—
120C3×D3C22×C3×D3132yesdirect product factor
121C3×D3C23×C3218yescomplemented
122C24—91noSylow
123C22×C3—21no—
124C22×C3—28no—
125C22×C3—348no—
126C22×C3—312no—
127C22×C3—44no—
128C22×C3—648no—
129C22×C3—964no—
130C22×C3—924no—
131C22×C3C3×D3214yesdirect product factor
132C22×C3C2×C32×D312yescomplemented
133C2×D3—312no—
134C2×D3—332no—
135C2×D3—66no—
136C2×D3C2×C32×D3112yesdirect product factor
137C32—28no—
138C32—24no—
139C32—28no—
140C32—44no—
141C32—412no—
142C32—44no—
143C32C22×D3211yescharacteristic, direct product factor
144C32C23×C3×D318yescomplemented
145C32C24×C3211yescharacteristic, complemented
146C23—32no—
147C23—912no—
148C23—91no—
149C2×C3—23no—
150C2×C3—224no—
151C2×C3—332no—
152C2×C3—38no—
153C2×C3—412no—
154C2×C3—632no—
155C2×C3—916no—
156C2×C3C2×C3×D32112yesdirect product factor
157C2×C3C22×C32×D316yescomplemented
158D3—38no—
159D3—64no—
160D3C22×C32×D318yesdirect product factor
161C22—312no—
162C22—916no—
163C22—96no—
164C22C32×D3211yescharacteristic, direct product factor
165C3—21no—
166C3—28no—
167C3—44no—
168C3C22×C3×D3214yesdirect product factor
169C3C23×C32×D312yescomplemented
170C2—38no—
171C2—94no—
172C2C2×C32×D3213yesdirect product factor
subgroupG/H
center, hypercenterC22×C32D32
commutator, cosocle, hypocenterC32C24×C32
FrattiniC1G
Fitting, socleC22×C34C22
LCSG ▷ C32
UCSC1 ◁ C22×C32

Complements

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

NH ≅ G/Ntype
1. C22×C32×C3⋊D3C2semidirect
2. C22×C33×D3C2semidirect
3. C2×C32×D32C2direct (N × H)
5. C22×C3×D32C3direct (N × H)
6. C22×C34C22semidirect
7. C2×C32×C3⋊D3C22semidirect
8. C2×C33×D3C22semidirect
9. C32×D32C22direct (N × H)
11. C22×C3×C3⋊D3C2×C3semidirect
16. C22×C32×D3D3direct (N × H)
17. C22×C32×D3C2×C3semidirect
18. C2×C3×D32C2×C3direct (N × H)
19. C2×C34C23semidirect
20. C32×C3⋊D3C23semidirect
21. C33×D3C23semidirect
24. C22×D32C32direct (N × H)
29. C22×C33C2×D3semidirect
30. C22×C33C22×C3semidirect
31. C2×C3×C3⋊D3C22×C3semidirect
36. C2×C32×D3C2×D3direct (N × H)
37. C2×C32×D3C22×C3semidirect
38. C3×D32C22×C3direct (N × H)
39. C34C24semidirect
45. C22×C3⋊D3C2×C32semidirect
50. C22×C3×D3C3×D3direct (N × H)
51. C22×C3×D3C2×C32semidirect
52. C2×D32C2×C32direct (N × H)
57. C2×C33C22×D3semidirect
58. C2×C33C23×C3semidirect
59. C3×C3⋊D3C23×C3semidirect
63. C32×D3C22×D3direct (N × H)
64. C32×D3C23×C3semidirect
78. C22×C32D32direct (N × H)
79. C22×C32C2×C3×D3semidirect
80. C22×C32C22×C32semidirect
81. C2×C3⋊D3C22×C32semidirect
86. C2×C3×D3C2×C3×D3direct (N × H)
87. C2×C3×D3C22×C32semidirect
88. D32C22×C32direct (N × H)
92. C33C23×D3semidirect
93. C33C24×C3semidirect
102. C22×D3C32×D3direct (N × H)
113. C2×C32C2×D32direct (N × H)
114. C2×C32C22×C3×D3semidirect
115. C2×C32C23×C32semidirect
116. C3⋊D3C23×C32semidirect
120. C3×D3C22×C3×D3direct (N × H)
121. C3×D3C23×C32semidirect
131. C22×C3C3×D32direct (N × H)
132. C22×C3C2×C32×D3semidirect
136. C2×D3C2×C32×D3direct (N × H)
143. C32C22×D32direct (N × H)
144. C32C23×C3×D3semidirect
145. C32C24×C32semidirect
156. C2×C3C2×C3×D32direct (N × H)
157. C2×C3C22×C32×D3semidirect
160. D3C22×C32×D3direct (N × H)
164. C22C32×D32direct (N × H)
168. C3C22×C3×D32direct (N × H)
169. C3C23×C32×D3semidirect
172. C2C2×C32×D32direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C3⋊D32
C22×C33×D32
C2×C32×D322
C23×C32×D33
C22×C3×D323