Finite Group Databasebeta

← C23×Dic3×C3⋊D3 #13181650

Subgroups — C23×Dic3×C3⋊D3 #13181650

Order 1728.

9116 conjugacy classes of subgroups; only the 1980 normal subgroups are shown.

normal subgroups only

This group has 9116 conjugacy classes of subgroups — too many to draw. Only its 1980 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
2C23×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3C23×C32×Dic3C211yesmaximal, characteristic, complemented
4C22×Dic3×C3⋊D3C2128yesmaximal, direct product factor
5C23×C4×C3⋊D3—31nomaximal
6C23×D3×Dic3—34nomaximal
7C24×C33C2211yescharacteristic
8C23×C3×C3⋊D3C4116yescomplemented
9C23×C3×C3⋊D3C22114yes—
10C22×C3⋊(C3⋊Dic3)C22114yescomplemented
11C22×C32×Dic3C22114yescomplemented
12C2×Dic3×C3⋊D3C221112yesdirect product factor
13C24×C3⋊D3D311yescharacteristic
14C24×C3×D3—34no—
15C23×C32×C4—31no—
16C23×C3⋊Dic3—31no—
17C23×C3⋊Dic3—34no—
18C23×C3⋊Dic3—64no—
19C22×C4×C3⋊D3—328no—
20C23×C3×Dic3D314yescomplemented
21C22×D3×Dic3—3112no—
22C23×C33C2×C418yescomplemented
23C23×C33C2317yes—
24C22×C3×C3⋊D3C2×C41112yescomplemented
25C22×C3×C3⋊D3C23128yes—
26C2×C3⋊(C3⋊Dic3)C23128yescomplemented
27C2×C32×Dic3C23128yescomplemented
28Dic3×C3⋊D3C23164yesdirect product factor
29C24×Dic3—91no—
30C23×C4×D3—94no—
31C24×C32—24no—
32C24×C32C2×D314yes—
33C24×C32C2×D311yescharacteristic
34C23×C3⋊D3Dic3116yesdirect product factor
35C23×C3⋊D3C2×D3114yes—
36C23×C3×D3—364no—
37C23×C3×D3—356no—
38C22×C32×C4—314no—
39C22×C3⋊Dic3—314no—
40C22×C3⋊Dic3—356no—
41C22×C3⋊Dic3—656no—
42C2×C4×C3⋊D3—3112no—
43C22×C3×Dic3C2×D3156yescomplemented
44C2×D3×Dic3—3448no—
45C22×C33C22×C4128yescomplemented
46C22×C33C2417yes—
47C2×C3×C3⋊D3C22×C41112yescomplemented
48C2×C3×C3⋊D3C2418yes—
49C3⋊(C3⋊Dic3)C2418yescomplemented
50C32×Dic3C2418yescomplemented
51C25×C3—91no—
52C24×D3—34no—
53C23×C3×C4—34no—
54C23×Dic3—94no—
55C23×Dic3—91no—
56C23×Dic3—928no—
57C23×Dic3—184no—
58C23×Dic3C3⋊D311yescharacteristic, direct product factor
59C22×C4×D3—9112no—
60C23×C32—232no—
61C23×C32—228no—
62C23×C32C4×D3132yescomplemented
63C23×C32C2×Dic318yescomplemented
64C23×C32C22×D3128yes—
65C23×C32C22×D317yes—
66C22×C3⋊D3C2×Dic31112yesdirect product factor
67C22×C3⋊D3C22×D3128yes—
68C22×C3×D3—3448no—
69C22×C3×D3—3112no—
70C2×C32×C4—328no—
71C2×C3⋊Dic3—328no—
72C2×C3⋊Dic3—3112no—
73C2×C3⋊Dic3—6112no—
74C4×C3⋊D3—364no—
75C2×C3×Dic3C22×D31112yescomplemented
76D3×Dic3—3256no—
77C24×C4—271noSylow
78C2×C33C23×C4114yescomplemented
79C2×C33C2511yescharacteristic
80C3×C3⋊D3C23×C4116yescomplemented
81C24×C3—24no—
82C24×C3—916no—
83C24×C3—914no—
84C24×C3D3214yes—
85C24×C3C2×C3⋊D311yescharacteristic
86C23×D3—364no—
87C23×D3—356no—
88C22×C3×C4—356no—
89C22×Dic3—956no—
90C22×Dic3—914no—
91C22×Dic3—9112no—
92C22×Dic3—1856no—
93C22×Dic3C2×C3⋊D3114yesdirect product factor
94C2×C4×D3—9448no—
95C22×C32—228no—
96C22×C32—2112no—
97C22×C32C2×C4×D31112yescomplemented
98C22×C32C22×Dic3128yescomplemented
99C22×C32C23×D3128yes—
100C22×C32C23×D317yes—
101C2×C3⋊D3C22×Dic31112yesdirect product factor
102C2×C3⋊D3C23×D318yes—
103C2×C3×D3—3448no—
104C2×C3×D3—332no—
105C32×C4—38no—
106C3⋊Dic3—38no—
107C3⋊Dic3—332no—
108C3⋊Dic3—632no—
109C3×Dic3C23×D3132yescomplemented
110C25—91no—
111C23×C4—31no—
112C23×C4—2728no—
113C23×C4—271no—
114C33C24×C411yescharacteristic, complemented, Sylow
115C23×C3—232no—
116C23×C3—228no—
117C23×C3—928no—
118C23×C3—9112no—
119C23×C3D3×Dic3132yescomplemented
120C23×C3C4×C3⋊D318yescomplemented
121C23×C3C2×D32128yes—
122C23×C3C22×C3⋊D317yes—
123C22×D3—3112no—
124C22×D3—3448no—
125C2×C3×C4—3112no—
126C2×Dic3—9112no—
127C2×Dic3—928no—
128C2×Dic3—964no—
129C2×Dic3—18112no—
130C2×Dic3C22×C3⋊D3128yesdirect product factor
131C4×D3—9256no—
132C2×C32—256no—
133C2×C32—24no—
134C2×C32C22×C4×D3156yescomplemented
135C2×C32C23×Dic3114yescomplemented
136C2×C32C24×D314yes—
137C2×C32C24×D311yescharacteristic
138C3⋊D3C23×Dic3116yesdirect product factor
139C3×D3—364no—
140C24—916no—
141C24—914no—
142C24D3×C3⋊D311yescharacteristic
143C22×C4—314no—
144C22×C4—27112no—
145C22×C4—2714no—
146C22×C3—228no—
147C22×C3—2112no—
148C22×C3—9112no—
149C22×C3—98no—
150C22×C3C2×D3×Dic31112yescomplemented
151C22×C3C2×C4×C3⋊D3128yescomplemented
152C22×C3C22×D32128yes—
153C22×C3C23×C3⋊D317yes—
154C2×D3—3448no—
155C2×D3—332no—
156C3×C4—332no—
157Dic3—932no—
158Dic3—98no—
159Dic3—1832no—
160Dic3C23×C3⋊D318yesdirect product factor
161C32—24no—
162C32C23×C4×D314yescomplemented
163C32C24×Dic311yescharacteristic, complemented
164C23—928no—
165C23—9112no—
166C23Dic3×C3⋊D318yesdirect product factor
167C23C2×D3×C3⋊D317yes—
168C2×C4—328no—
169C2×C4—2764no—
170C2×C4—2728no—
171C2×C3—256no—
172C2×C3—24no—
173C2×C3—916no—
174C2×C3C22×D3×Dic3156yescomplemented
175C2×C3C22×C4×C3⋊D3114yescomplemented
176C2×C3C23×D3214yes—
177C2×C3C24×C3⋊D311yescharacteristic
178D3—364no—
179C22—9112no—
180C22—98no—
181C22C2×Dic3×C3⋊D3128yesdirect product factor
182C22C22×D3×C3⋊D317yes—
183C4—38no—
184C4—278no—
185C3—24no—
186C3C23×D3×Dic314yescomplemented
187C3C23×C4×C3⋊D311yescharacteristic, complemented
188C2—916no—
189C2C22×Dic3×C3⋊D3114yesdirect product factor
190C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC24D3×C3⋊D3
commutator, hypocenterC33C24×C4
FrattiniC2C23×D3×C3⋊D3
Fitting, socleC24×C33C22
prime radicalC24×C3×C3⋊D3C2
cosocleC2×C33C25
LCSG ▷ C33
UCSC1 ◁ C24

Complements

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

NH ≅ G/Ntype
2. C23×C3⋊(C3⋊Dic3)C2semidirect
3. C23×C32×Dic3C2semidirect
4. C22×Dic3×C3⋊D3C2direct (N × H)
8. C23×C3×C3⋊D3C4semidirect
10. C22×C3⋊(C3⋊Dic3)C22semidirect
11. C22×C32×Dic3C22semidirect
12. C2×Dic3×C3⋊D3C22direct (N × H)
20. C23×C3×Dic3D3semidirect
22. C23×C33C2×C4semidirect
24. C22×C3×C3⋊D3C2×C4semidirect
26. C2×C3⋊(C3⋊Dic3)C23semidirect
27. C2×C32×Dic3C23semidirect
28. Dic3×C3⋊D3C23direct (N × H)
34. C23×C3⋊D3Dic3direct (N × H)
43. C22×C3×Dic3C2×D3semidirect
45. C22×C33C22×C4semidirect
47. C2×C3×C3⋊D3C22×C4semidirect
49. C3⋊(C3⋊Dic3)C24semidirect
50. C32×Dic3C24semidirect
58. C23×Dic3C3⋊D3direct (N × H)
62. C23×C32C4×D3semidirect
63. C23×C32C2×Dic3semidirect
66. C22×C3⋊D3C2×Dic3direct (N × H)
75. C2×C3×Dic3C22×D3semidirect
78. C2×C33C23×C4semidirect
80. C3×C3⋊D3C23×C4semidirect
93. C22×Dic3C2×C3⋊D3direct (N × H)
97. C22×C32C2×C4×D3semidirect
98. C22×C32C22×Dic3semidirect
101. C2×C3⋊D3C22×Dic3direct (N × H)
109. C3×Dic3C23×D3semidirect
114. C33C24×C4semidirect
119. C23×C3D3×Dic3semidirect
120. C23×C3C4×C3⋊D3semidirect
130. C2×Dic3C22×C3⋊D3direct (N × H)
134. C2×C32C22×C4×D3semidirect
135. C2×C32C23×Dic3semidirect
138. C3⋊D3C23×Dic3direct (N × H)
150. C22×C3C2×D3×Dic3semidirect
151. C22×C3C2×C4×C3⋊D3semidirect
160. Dic3C23×C3⋊D3direct (N × H)
162. C32C23×C4×D3semidirect
163. C32C24×Dic3semidirect
166. C23Dic3×C3⋊D3direct (N × H)
174. C2×C3C22×D3×Dic3semidirect
175. C2×C3C22×C4×C3⋊D3semidirect
181. C22C2×Dic3×C3⋊D3direct (N × H)
186. C3C23×D3×Dic3semidirect
187. C3C23×C4×C3⋊D3semidirect
189. C2C22×Dic3×C3⋊D3direct (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
C23×C3⋊(C3⋊Dic3)2
C23×C32×Dic32
C22×Dic3×C3⋊D32
C23×C4×C3⋊D33
C23×D3×Dic33