Finite Group Databasebeta

← C24×C3×D32 #13181865

Subgroups — C24×C3×D32 #13181865

Order 1728.

24978 conjugacy classes of subgroups; only the 7280 normal subgroups are shown.

normal subgroups only

This group has 24978 conjugacy classes of subgroups — too many to draw. Only its 7280 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
1C24×C3×C3⋊D3C211yesmaximal, characteristic, complemented
2C24×C32×D3C212yesmaximal, complemented
3C23×C3×D32C2160yesmaximal, direct product factor
4C25×C3×D3—32nomaximal
5C24×D32C311yesmaximal, characteristic, direct product factor
6C24×C33C2211yescharacteristic, complemented
7C23×C3×C3⋊D3C22130yescomplemented
8C23×C32×D3C22160yescomplemented
9C22×C3×D32C221560yesdirect product factor
10C25×C32—32no—
11C24×C3⋊D3C2×C311yescharacteristic, complemented
12C24×C3×D3—22no—
13C24×C3×D3—32no—
14C24×C3×D3—3120no—
15C24×C3×D3—61no—
16C24×C3×D3D312yesdirect product factor
17C24×C3×D3C2×C312yescomplemented
18C23×D32C2×C3160yesdirect product factor
19C23×C33C23115yescomplemented
20C22×C3×C3⋊D3C231140yescomplemented
21C22×C32×D3C231280yescomplemented
22C2×C3×D32C231960yesdirect product factor
23C26×C3—91no—
24C25×D3—32no—
25C24×C32—21no—
26C24×C32—22no—
27C24×C32—360no—
28C24×C32—41no—
29C24×C32C2×D312yescomplemented
30C24×C32C22×C311yescharacteristic, complemented
31C23×C3⋊D3C22×C3130yescomplemented
32C23×C3×D3—260no—
33C23×C3×D3—360no—
34C23×C3×D3—31120no—
35C23×C3×D3—630no—
36C23×C3×D3C2×D3160yesdirect product factor
37C23×C3×D3C22×C3160yescomplemented
38C22×D32C22×C31560yesdirect product factor
39C22×C33C24135yescomplemented
40C2×C3×C3⋊D3C241120yescomplemented
41C2×C32×D3C241240yescomplemented
42C3×D32C241256yesdirect product factor
43C25×C3—32no—
44C25×C3—32no—
45C25×C3—62no—
46C25×C3—960no—
47C25×C3—91no—
48C24×D3—32no—
49C24×D3—3120no—
50C24×D3—61no—
51C24×D3C3×D312yesdirect product factor
52C23×C32—215no—
53C23×C32—230no—
54C23×C32—3280no—
55C23×C32—415no—
56C23×C32C22×D3130yescomplemented
57C23×C32C23×C3115yescomplemented
58C22×C3⋊D3C23×C31140yescomplemented
59C22×C3×D3—2280no—
60C22×C3×D3—3280no—
61C22×C3×D3—31920no—
62C22×C3×D3—6140no—
63C22×C3×D3C22×D31280yesdirect product factor
64C22×C3×D3C23×C31280yescomplemented
65C2×D32C23×C31960yesdirect product factor
66C26—91noSylow
67C2×C33C25115yescomplemented
68C3×C3⋊D3C25116yescomplemented
69C32×D3C25132yescomplemented
70C24×C3—22no—
71C24×C3—21no—
72C24×C3—360no—
73C24×C3—360no—
74C24×C3—41no—
75C24×C3—660no—
76C24×C3—9560no—
77C24×C3—930no—
78C24×C3D3211yescharacteristic, direct product factor
79C24×C3C2×C3×D312yescomplemented
80C23×D3—360no—
81C23×D3—31120no—
82C23×D3—630no—
83C23×D3C2×C3×D3160yesdirect product factor
84C22×C32—235no—
85C22×C32—270no—
86C22×C32—3240no—
87C22×C32—435no—
88C22×C32C23×D3170yescomplemented
89C22×C32C24×C3135yescomplemented
90C2×C3⋊D3C24×C31120yescomplemented
91C2×C3×D3—2240no—
92C2×C3×D3—3240no—
93C2×C3×D3—3512no—
94C2×C3×D3—6120no—
95C2×C3×D3C23×D31240yesdirect product factor
96C2×C3×D3C24×C31240yescomplemented
97D32C24×C31256yesdirect product factor
98C25—32no—
99C25—960no—
100C25—91no—
101C33C2611yescharacteristic, complemented, Sylow
102C23×C3—230no—
103C23×C3—215no—
104C23×C3—3280no—
105C23×C3—3280no—
106C23×C3—415no—
107C23×C3—6280no—
108C23×C3—9960no—
109C23×C3—9140no—
110C23×C3C2×D32115yesdirect product factor
111C23×C3C22×C3×D3130yescomplemented
112C22×D3—3280no—
113C22×D3—31920no—
114C22×D3—6140no—
115C22×D3C22×C3×D31280yesdirect product factor
116C2×C32—215no—
117C2×C32—230no—
118C2×C32—332no—
119C2×C32—415no—
120C2×C32C24×D3130yescomplemented
121C2×C32C25×C3115yescomplemented
122C3⋊D3C25×C3116yescomplemented
123C3×D3—232no—
124C3×D3—332no—
125C3×D3—616no—
126C3×D3C24×D3132yesdirect product factor
127C3×D3C25×C3132yescomplemented
128C24—360no—
129C24—9560no—
130C24—930no—
131C24C3×D3211yescharacteristic, direct product factor
132C22×C3—235no—
133C22×C3—270no—
134C22×C3—3240no—
135C22×C3—3240no—
136C22×C3—435no—
137C22×C3—6240no—
138C22×C3—9256no—
139C22×C3—9120no—
140C22×C3C22×D32135yesdirect product factor
141C22×C3C23×C3×D3170yescomplemented
142C2×D3—3240no—
143C2×D3—3512no—
144C2×D3—6120no—
145C2×D3C23×C3×D31240yesdirect product factor
146C32—21no—
147C32—22no—
148C32—41no—
149C32C25×D312yescomplemented
150C32C26×C311yescharacteristic, complemented
151C23—3280no—
152C23—9960no—
153C23—9140no—
154C23C2×C3×D32115yesdirect product factor
155C2×C3—215no—
156C2×C3—230no—
157C2×C3—332no—
158C2×C3—332no—
159C2×C3—415no—
160C2×C3—632no—
161C2×C3—916no—
162C2×C3C23×D32115yesdirect product factor
163C2×C3C24×C3×D3130yescomplemented
164D3—332no—
165D3—616no—
166D3C24×C3×D3132yesdirect product factor
167C22—3240no—
168C22—9256no—
169C22—9120no—
170C22C22×C3×D32135yesdirect product factor
171C3—21no—
172C3—22no—
173C3—41no—
174C3C24×D3211yescharacteristic, direct product factor
175C3C25×C3×D312yescomplemented
176C2—332no—
177C2—916no—
178C2C23×C3×D32115yesdirect product factor
subgroupG/H
center, hypercenterC24×C3D32
commutator, cosocle, hypocenterC32C26×C3
FrattiniC1G
Fitting, socleC24×C33C22
LCSG ▷ C32
UCSC1 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C24×C3×C3⋊D3C2semidirect
2. C24×C32×D3C2semidirect
3. C23×C3×D32C2direct (N × H)
5. C24×D32C3direct (N × H)
6. C24×C33C22semidirect
7. C23×C3×C3⋊D3C22semidirect
8. C23×C32×D3C22semidirect
9. C22×C3×D32C22direct (N × H)
11. C24×C3⋊D3C2×C3semidirect
16. C24×C3×D3D3direct (N × H)
17. C24×C3×D3C2×C3semidirect
18. C23×D32C2×C3direct (N × H)
19. C23×C33C23semidirect
20. C22×C3×C3⋊D3C23semidirect
21. C22×C32×D3C23semidirect
22. C2×C3×D32C23direct (N × H)
29. C24×C32C2×D3semidirect
30. C24×C32C22×C3semidirect
31. C23×C3⋊D3C22×C3semidirect
36. C23×C3×D3C2×D3direct (N × H)
37. C23×C3×D3C22×C3semidirect
38. C22×D32C22×C3direct (N × H)
39. C22×C33C24semidirect
40. C2×C3×C3⋊D3C24semidirect
41. C2×C32×D3C24semidirect
42. C3×D32C24direct (N × H)
51. C24×D3C3×D3direct (N × H)
56. C23×C32C22×D3semidirect
57. C23×C32C23×C3semidirect
58. C22×C3⋊D3C23×C3semidirect
63. C22×C3×D3C22×D3direct (N × H)
64. C22×C3×D3C23×C3semidirect
65. C2×D32C23×C3direct (N × H)
67. C2×C33C25semidirect
68. C3×C3⋊D3C25semidirect
69. C32×D3C25semidirect
78. C24×C3D32direct (N × H)
79. C24×C3C2×C3×D3semidirect
83. C23×D3C2×C3×D3direct (N × H)
88. C22×C32C23×D3semidirect
89. C22×C32C24×C3semidirect
90. C2×C3⋊D3C24×C3semidirect
95. C2×C3×D3C23×D3direct (N × H)
96. C2×C3×D3C24×C3semidirect
97. D32C24×C3direct (N × H)
101. C33C26semidirect
110. C23×C3C2×D32direct (N × H)
111. C23×C3C22×C3×D3semidirect
115. C22×D3C22×C3×D3direct (N × H)
120. C2×C32C24×D3semidirect
121. C2×C32C25×C3semidirect
122. C3⋊D3C25×C3semidirect
126. C3×D3C24×D3direct (N × H)
127. C3×D3C25×C3semidirect
131. C24C3×D32direct (N × H)
140. C22×C3C22×D32direct (N × H)
141. C22×C3C23×C3×D3semidirect
145. C2×D3C23×C3×D3direct (N × H)
149. C32C25×D3semidirect
150. C32C26×C3semidirect
154. C23C2×C3×D32direct (N × H)
162. C2×C3C23×D32direct (N × H)
163. C2×C3C24×C3×D3semidirect
166. D3C24×C3×D3direct (N × H)
170. C22C22×C3×D32direct (N × H)
174. C3C24×D32direct (N × H)
175. C3C25×C3×D3semidirect
178. C2C23×C3×D32direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C3×C3⋊D32
C24×C32×D32
C23×C3×D322
C25×C3×D33
C24×D323