Finite Group Databasebeta

← C23×D3×D21 #72246733

Subgroups — C23×D3×D21 #72246733

Order 2016.

3126 conjugacy classes of subgroups; only the 690 normal subgroups are shown.

normal subgroups only

This group has 3126 conjugacy classes of subgroups — too many to draw. Only its 690 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
1C23×C3⋊D21C211yesmaximal, characteristic, complemented
2C23×C3×D21C211yesmaximal, characteristic, complemented
3C23×C3×D3×C7C211yesmaximal, characteristic, complemented
4C22×D3×D21C2128yesmaximal, direct product factor
5C24×D21—31nomaximal
6C23×D3×D7—31nomaximal
7C23×C32×C7C2211yescharacteristic, complemented
8C22×C3⋊D21C22114yescomplemented
9C22×C3×D21C22114yescomplemented
10C22×C3×D3×C7C22114yescomplemented
11C2×D3×D21C221112yesdirect product factor
12C24×C3×C7—31no—
13C23×D21—31no—
14C23×D21—31no—
15C23×D21—328no—
16C23×D21—61no—
17C23×D21D311yescharacteristic, direct product factor
18C23×D3×C7D311yescharacteristic, complemented
19C23×C3×D7—31no—
20C22×D3×D7—328no—
21C23×D32—71nomaximal, Hall
22C22×C32×C7C2317yescomplemented
23C2×C3⋊D21C23128yescomplemented
24C2×C3×D21C23128yescomplemented
25C2×C3×D3×C7C23128yescomplemented
26D3×D21C23164yesdirect product factor
27C24×D7—91noHall
28C23×C3×C7—21no—
29C23×C3×C7—314no—
30C23×C3×C7C2×D311yescharacteristic, complemented
31C23×C3×C7C2×D311yescharacteristic, complemented
32C22×D21—314no—
33C22×D21—314no—
34C22×D21—3112no—
35C22×D21—614no—
36C22×D21C2×D3114yesdirect product factor
37C22×D3×C7C2×D3114yescomplemented
38C22×C3×D7—314no—
39C2×D3×D7—3112no—
40C23×C3⋊D3—71no—
41C23×C3×D3—71no—
42C23×C3×D3D711yescharacteristic, complemented
43C22×D32—728no—
44C2×C32×C7C2417yescomplemented
45C3⋊D21C2418yescomplemented
46C3×D21C2418yescomplemented
47C3×D3×C7C2418yescomplemented
48C24×C7—31no—
49C23×D7—31no—
50C23×D7—928no—
51C23×D7—91no—
52C24×D3—211no—
53C24×D3—211no—
54C22×C3×C7—27no—
55C22×C3×C7—328no—
56C22×C3×C7C22×D317yescomplemented
57C22×C3×C7C22×D317yescomplemented
58C2×D21—328no—
59C2×D21—328no—
60C2×D21—364no—
61C2×D21—628no—
62C2×D21C22×D3128yesdirect product factor
63C2×D3×C7C22×D3128yescomplemented
64C2×C3×D7—328no—
65D3×D7—364no—
66C23×C32C2×D711yescharacteristic, complemented
67C22×C3⋊D3—714no—
68C22×C3×D3—714no—
69C22×C3×D3C2×D7114yescomplemented
70C2×D32—7112no—
71C32×C7C2511yescharacteristic, complemented, Hall
72C23×C7—314no—
73C23×C7D3211yescharacteristic, complemented
74C22×D7—314no—
75C22×D7—9112no—
76C22×D7—914no—
77C24×C3—31no—
78C24×C3—211no—
79C23×D3—71no—
80C23×D3—2128no—
81C23×D3—211no—
82C23×D3—211no—
83C23×D3—2128no—
84C23×D3—421no—
85C23×D3D2111yescharacteristic, direct product factor
86C2×C3×C7—27no—
87C2×C3×C7—38no—
88C2×C3×C7C23×D317yescomplemented
89C2×C3×C7C23×D317yescomplemented
90D21—38no—
91D21—38no—
92D21—68no—
93D21C23×D318yesdirect product factor
94C3×D7—38no—
95D3×C7C23×D318yescomplemented
96C22×C32C22×D717yescomplemented
97C2×C3⋊D3—728no—
98C2×C3×D3—728no—
99C2×C3×D3C22×D7128yescomplemented
100D32—764no—
101C25—631noSylow
102C22×C7—328no—
103C22×C7C2×D3217yescomplemented
104C2×D7—328no—
105C2×D7—964no—
106C2×D7—928no—
107C23×C3—21no—
108C23×C3—314no—
109C23×C3—2114no—
110C23×C3D3×D711yescharacteristic, complemented
111C23×C3C2×D2111yescharacteristic, complemented
112C22×D3—714no—
113C22×D3—21112no—
114C22×D3—2114no—
115C22×D3—2114no—
116C22×D3—21112no—
117C22×D3—4214no—
118C22×D3C2×D21114yesdirect product factor
119C3×C7—21no—
120C3×C7C24×D311yescharacteristic, complemented
121C3×C7C24×D311yescharacteristic, complemented
122C2×C32C23×D717yescomplemented
123C3⋊D3—78no—
124C3×D3—78no—
125C3×D3C23×D718yescomplemented
126C24—31no—
127C24—211no—
128C24—6328no—
129C24—631no—
130C2×C7—38no—
131C2×C7C22×D3217yescomplemented
132D7—38no—
133D7—98no—
134C22×C3—27no—
135C22×C3—328no—
136C22×C3—2128no—
137C22×C3C2×D3×D717yescomplemented
138C22×C3C22×D2117yescomplemented
139C2×D3—728no—
140C2×D3—2164no—
141C2×D3—2128no—
142C2×D3—2128no—
143C2×D3—2164no—
144C2×D3—4228no—
145C2×D3C22×D21128yesdirect product factor
146C32C24×D711yescharacteristic, complemented, Sylow
147C23—314no—
148C23—2114no—
149C23—63112no—
150C23—6314no—
151C23D3×D2111yescharacteristic, direct product factor
152C7C23×D3211yescharacteristic, complemented, Sylow
153C2×C3—27no—
154C2×C3—38no—
155C2×C3—218no—
156C2×C3C22×D3×D717yescomplemented
157C2×C3C23×D2117yescomplemented
158D3—78no—
159D3—218no—
160D3—218no—
161D3—428no—
162D3C23×D2118yesdirect product factor
163C22—328no—
164C22—2128no—
165C22—6364no—
166C22—6328no—
167C22C2×D3×D2117yesdirect product factor
168C3—21no—
169C3C23×D3×D711yescharacteristic, complemented
170C3C24×D2111yescharacteristic, complemented
171C2—38no—
172C2—218no—
173C2—638no—
174C2C22×D3×D2117yesdirect product factor
subgroupG/H
center, hypercenterC23D3×D21
commutator, cosocle, hypocenterC32×C7C25
FrattiniC1G
Fitting, socleC23×C32×C7C22
LCSG ▷ C32×C7
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×C3⋊D21C2semidirect
2. C23×C3×D21C2semidirect
3. C23×C3×D3×C7C2semidirect
4. C22×D3×D21C2direct (N × H)
7. C23×C32×C7C22semidirect
8. C22×C3⋊D21C22semidirect
9. C22×C3×D21C22semidirect
10. C22×C3×D3×C7C22semidirect
11. C2×D3×D21C22direct (N × H)
17. C23×D21D3direct (N × H)
18. C23×D3×C7D3semidirect
22. C22×C32×C7C23semidirect
23. C2×C3⋊D21C23semidirect
24. C2×C3×D21C23semidirect
25. C2×C3×D3×C7C23semidirect
26. D3×D21C23direct (N × H)
30. C23×C3×C7C2×D3semidirect
31. C23×C3×C7C2×D3semidirect
36. C22×D21C2×D3direct (N × H)
37. C22×D3×C7C2×D3semidirect
42. C23×C3×D3D7semidirect
44. C2×C32×C7C24semidirect
45. C3⋊D21C24semidirect
46. C3×D21C24semidirect
47. C3×D3×C7C24semidirect
56. C22×C3×C7C22×D3semidirect
57. C22×C3×C7C22×D3semidirect
62. C2×D21C22×D3direct (N × H)
63. C2×D3×C7C22×D3semidirect
66. C23×C32C2×D7semidirect
69. C22×C3×D3C2×D7semidirect
71. C32×C7C25semidirect
73. C23×C7D32semidirect
85. C23×D3D21direct (N × H)
88. C2×C3×C7C23×D3semidirect
89. C2×C3×C7C23×D3semidirect
93. D21C23×D3direct (N × H)
95. D3×C7C23×D3semidirect
96. C22×C32C22×D7semidirect
99. C2×C3×D3C22×D7semidirect
103. C22×C7C2×D32semidirect
110. C23×C3D3×D7semidirect
111. C23×C3C2×D21semidirect
118. C22×D3C2×D21direct (N × H)
120. C3×C7C24×D3semidirect
121. C3×C7C24×D3semidirect
122. C2×C32C23×D7semidirect
125. C3×D3C23×D7semidirect
131. C2×C7C22×D32semidirect
137. C22×C3C2×D3×D7semidirect
138. C22×C3C22×D21semidirect
145. C2×D3C22×D21direct (N × H)
146. C32C24×D7semidirect
151. C23D3×D21direct (N × H)
152. C7C23×D32semidirect
156. C2×C3C22×D3×D7semidirect
157. C2×C3C23×D21semidirect
162. D3C23×D21direct (N × H)
167. C22C2×D3×D21direct (N × H)
169. C3C23×D3×D7semidirect
170. C3C24×D21semidirect
174. C2C22×D3×D21direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3⋊D212
C23×C3×D212
C23×C3×D3×C72
C22×D3×D212
C24×D213
C23×D3×D73
C23×D327