Finite Group Databasebeta

← C24×C3×#67 #11919160

Subgroups — C24×C3×#67 #11919160

Order 1152.

23929 conjugacy classes of subgroups; only the 7626 normal subgroups are shown.

normal subgroups only

This group has 23929 conjugacy classes of subgroups — too many to draw. Only its 7626 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
1C26×C32C211yesmaximal, characteristic, complemented
2C25×C3×D3C211yesmaximal, characteristic, complemented
3C23×C3×#67C2160yesmaximal, direct product factor
4C24×C3×Dic3C211yesmaximal, characteristic, complemented
5C24×C3×D4—31nomaximal
6C24×#67C311yesmaximal, characteristic, direct product factor
7C25×C32—216no—
8C25×C32C2211yescharacteristic
9C25×C32C22130yescomplemented
10C24×C3×D3—216no—
11C24×C3×D3C22130yescomplemented
12C22×C3×#67C221560yesdirect product factor
13C23×C3×Dic3C22130yescomplemented
14C26×C3—21no—
15C26×C3—31no—
16C26×C3D311yescharacteristic, complemented
17C26×C3C2×C311yescharacteristic, complemented
18C25×D3C2×C311yescharacteristic, complemented
19C23×C3×D4—360no—
20C24×C3×C4—31no—
21C23×#67C2×C3160yesdirect product factor
22C24×Dic3C2×C311yescharacteristic, complemented
23C24×C32—2240no—
24C24×C32D4116yescomplemented
25C24×C32C23115yes—
26C24×C32C231140yescomplemented
27C23×C3×D3—2240no—
28C23×C3×D3C231140yescomplemented
29C2×C3×#67C231960yesdirect product factor
30C22×C3×Dic3C231140yescomplemented
31C24×D4—31noSylow
32C25×C3—232no—
33C25×C3—216no—
34C25×C3—216no—
35C25×C3—21no—
36C25×C3—230no—
37C25×C3—330no—
38C25×C3—616no—
39C25×C3C2×D311yescharacteristic
40C25×C3C2×D3130yescomplemented
41C25×C3C22×C311yescharacteristic
42C25×C3C22×C3130yescomplemented
43C24×D3—216no—
44C24×D3C22×C3130yescomplemented
45C22×C3×D4—3560no—
46C23×C3×C4—330no—
47C22×#67C22×C31560yesdirect product factor
48C23×Dic3C22×C3130yescomplemented
49C23×C32—2560no—
50C23×C32C2×D41120yescomplemented
51C23×C32C24135yes—
52C23×C32C241120yescomplemented
53C22×C3×D3—2560no—
54C22×C3×D3C241120yescomplemented
55C3×#67C241256yesdirect product factor
56C2×C3×Dic3C241120yescomplemented
57C26—31no—
58C26C3×D311yescharacteristic, complemented
59C23×D4—360no—
60C24×C4—31no—
61C24×C3—2480no—
62C24×C3—2240no—
63C24×C3—2240no—
64C24×C3—216no—
65C24×C3—215no—
66C24×C3—2140no—
67C24×C3—3140no—
68C24×C3—6240no—
69C24×C3#67116yesdirect product factor
70C24×C3C3×D4116yescomplemented
71C24×C3C22×D3115yes—
72C24×C3C22×D31140yescomplemented
73C24×C3C23×C3115yes—
74C24×C3C23×C31140yescomplemented
75C23×D3—2240no—
76C23×D3C23×C31140yescomplemented
77C2×C3×D4—3960no—
78C22×C3×C4—3140no—
79C2×#67C23×C31960yesdirect product factor
80C22×Dic3C23×C31140yescomplemented
81C22×C32—2240no—
82C22×C32C22×D41140yescomplemented
83C22×C32C25115yes—
84C22×C32C25116yescomplemented
85C2×C3×D3—2240no—
86C2×C3×D3C25116yescomplemented
87C3×Dic3C25116yescomplemented
88C25—216no—
89C25—330no—
90C25—616no—
91C25C2×C3×D311yescharacteristic
92C25C2×C3×D3130yescomplemented
93C22×D4—3560no—
94C23×C4—330no—
95C23×C3—21120no—
96C23×C3—2560no—
97C23×C3—2560no—
98C23×C3—235no—
99C23×C3—2120no—
100C23×C3—2120no—
101C23×C3—3120no—
102C23×C3—6560no—
103C23×C3C2×#671120yesdirect product factor
104C23×C3C2×C3×D41120yescomplemented
105C23×C3C23×D3135yes—
106C23×C3C23×D31120yescomplemented
107C23×C3C24×C3135yes—
108C23×C3C24×C31120yescomplemented
109C22×D3—2560no—
110C22×D3C24×C31120yescomplemented
111C3×D4—3256no—
112C2×C3×C4—3120no—
113#67C24×C31256yesdirect product factor
114C2×Dic3C24×C31120yescomplemented
115C2×C32—216no—
116C2×C32C23×D4130yescomplemented
117C2×C32C2611yescharacteristic
118C3×D3—216no—
119C24—2240no—
120C24—3140no—
121C24—6240no—
122C24C3×#67116yesdirect product factor
123C24C22×C3×D3115yes—
124C24C22×C3×D31140yescomplemented
125C2×D4—3960no—
126C22×C4—3140no—
127C22×C3—2480no—
128C22×C3—2240no—
129C22×C3—2240no—
130C22×C3—2140no—
131C22×C3—215no—
132C22×C3—216no—
133C22×C3—316no—
134C22×C3—6240no—
135C22×C3C22×#671140yesdirect product factor
136C22×C3C22×C3×D41140yescomplemented
137C22×C3C24×D3115yes—
138C22×C3C24×D3116yescomplemented
139C22×C3C25×C3115yes—
140C22×C3C25×C3116yescomplemented
141C2×D3—2240no—
142C2×D3C25×C3116yescomplemented
143C3×C4—316no—
144Dic3C25×C3116yescomplemented
145C32C24×D411yescharacteristic, complemented, Sylow
146C23—2560no—
147C23—3120no—
148C23—6560no—
149C23C2×C3×#671120yesdirect product factor
150C23C23×C3×D3135yes—
151C23C23×C3×D31120yescomplemented
152D4—3256no—
153C2×C4—3120no—
154C2×C3—232no—
155C2×C3—216no—
156C2×C3—216no—
157C2×C3—230no—
158C2×C3—21no—
159C2×C3—616no—
160C2×C3C23×#67130yesdirect product factor
161C2×C3C23×C3×D4130yescomplemented
162C2×C3C25×D311yescharacteristic
163C2×C3C26×C311yescharacteristic
164D3—216no—
165C22—2240no—
166C22—316no—
167C22—6240no—
168C22C22×C3×#671140yesdirect product factor
169C22C24×C3×D3115yes—
170C22C24×C3×D3116yescomplemented
171C4—316no—
172C3—21no—
173C3C24×#6711yescharacteristic, direct product factor
174C3C24×C3×D411yescharacteristic, complemented
175C2—216no—
176C2—616no—
177C2C23×C3×#67130yesdirect product factor
178C2C25×C3×D311yescharacteristic
subgroupG/H
centerC25×C3C2×D3
commutator, cosocleC2×C3C26×C3
FrattiniC2C25×C3×D3
FittingC26×C32C2
socleC25×C32C22
hypercenterC26×C3D3
hypocenterC3C24×C3×D4
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C25×C3 ◁ C26×C3

Complements

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

NH ≅ G/Ntype
1. C26×C32C2semidirect
2. C25×C3×D3C2semidirect
3. C23×C3×#67C2direct (N × H)
4. C24×C3×Dic3C2semidirect
6. C24×#67C3direct (N × H)
9. C25×C32C22semidirect
11. C24×C3×D3C22semidirect
12. C22×C3×#67C22direct (N × H)
13. C23×C3×Dic3C22semidirect
16. C26×C3D3semidirect
17. C26×C3C2×C3semidirect
18. C25×D3C2×C3semidirect
21. C23×#67C2×C3direct (N × H)
22. C24×Dic3C2×C3semidirect
24. C24×C32D4semidirect
26. C24×C32C23semidirect
28. C23×C3×D3C23semidirect
29. C2×C3×#67C23direct (N × H)
30. C22×C3×Dic3C23semidirect
40. C25×C3C2×D3semidirect
42. C25×C3C22×C3semidirect
44. C24×D3C22×C3semidirect
47. C22×#67C22×C3direct (N × H)
48. C23×Dic3C22×C3semidirect
50. C23×C32C2×D4semidirect
52. C23×C32C24semidirect
54. C22×C3×D3C24semidirect
55. C3×#67C24direct (N × H)
56. C2×C3×Dic3C24semidirect
58. C26C3×D3semidirect
69. C24×C3#67direct (N × H)
70. C24×C3C3×D4semidirect
72. C24×C3C22×D3semidirect
74. C24×C3C23×C3semidirect
76. C23×D3C23×C3semidirect
79. C2×#67C23×C3direct (N × H)
80. C22×Dic3C23×C3semidirect
82. C22×C32C22×D4semidirect
84. C22×C32C25semidirect
86. C2×C3×D3C25semidirect
87. C3×Dic3C25semidirect
92. C25C2×C3×D3semidirect
103. C23×C3C2×#67direct (N × H)
104. C23×C3C2×C3×D4semidirect
106. C23×C3C23×D3semidirect
108. C23×C3C24×C3semidirect
110. C22×D3C24×C3semidirect
113. #67C24×C3direct (N × H)
114. C2×Dic3C24×C3semidirect
116. C2×C32C23×D4semidirect
122. C24C3×#67direct (N × H)
124. C24C22×C3×D3semidirect
135. C22×C3C22×#67direct (N × H)
136. C22×C3C22×C3×D4semidirect
138. C22×C3C24×D3semidirect
140. C22×C3C25×C3semidirect
142. C2×D3C25×C3semidirect
144. Dic3C25×C3semidirect
145. C32C24×D4semidirect
149. C23C2×C3×#67direct (N × H)
151. C23C23×C3×D3semidirect
160. C2×C3C23×#67direct (N × H)
161. C2×C3C23×C3×D4semidirect
168. C22C22×C3×#67direct (N × H)
170. C22C24×C3×D3semidirect
173. C3C24×#67direct (N × H)
174. C3C24×C3×D4semidirect
177. C2C23×C3×#67direct (N × H)

Maximal subgroups

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

subgroupindex
C26×C322
C25×C3×D32
C23×C3×#672
C24×C3×Dic32
C24×C3×D43
C24×#673