Finite Group Databasebeta

← C23×#6290 #13181656

Subgroups — C23×#6290 #13181656

Order 1728.

9116 conjugacy classes of subgroups; only the 1468 normal subgroups are shown.

normal subgroups only

This group has 9116 conjugacy classes of subgroups — too many to draw. Only its 1468 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×D3C211yesmaximal, characteristic, complemented
3C23×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4C22×#6290C2128yesmaximal, direct product factor
5C23×#638—31nomaximal
6C23×#625—34nomaximal
7C24×C33C2211yescharacteristic
8C23×C3×C3⋊D3—28no—
9C23×C3×C3⋊D3C22114yescomplemented
10C23×C32×D3—28no—
11C23×C32×D3C22114yescomplemented
12C22×C3⋊(C3⋊Dic3)C22114yescomplemented
13C2×#6290C221112yesdirect product factor
14C25×C32—31no—
15C24×C3⋊D3D311yescharacteristic, complemented
16C24×C3×D3—34no—
17C24×C3×D3D314yescomplemented
18C22×#638—328no—
19C23×C3⋊Dic3—31no—
20C23×C3⋊Dic3—34no—
21C23×C3⋊Dic3—64no—
22C22×#625—3112no—
23C23×C33D418yescomplemented
24C23×C33C2317yes—
25C22×C3×C3⋊D3—256no—
26C22×C3×C3⋊D3C23128yescomplemented
27C22×C32×D3—256no—
28C22×C32×D3C23128yescomplemented
29C2×C3⋊(C3⋊Dic3)C23128yescomplemented
30#6290C23164yesdirect product factor
31C23×#67—94no—
32C23×#67—91no—
33C24×C32—24no—
34C24×C32—314no—
35C24×C32—68no—
36C24×C32C2×D314yes—
37C24×C32C2×D311yescharacteristic
38C23×C3⋊D3—28no—
39C23×C3⋊D3C2×D3114yescomplemented
40C23×C3×D3—232no—
41C23×C3×D3—356no—
42C23×C3×D3—632no—
43C23×C3×D3C2×D3156yescomplemented
44C2×#638—3112no—
45C22×C3⋊Dic3—314no—
46C22×C3⋊Dic3—356no—
47C22×C3⋊Dic3—656no—
48C2×#625—3448no—
49C22×C33C2×D4128yescomplemented
50C22×C33C2417yes—
51C2×C3×C3⋊D3—256no—
52C2×C3×C3⋊D3C2418yescomplemented
53C2×C32×D3—256no—
54C2×C32×D3C2418yescomplemented
55C3⋊(C3⋊Dic3)C2418yescomplemented
56C25×C3—34no—
57C25×C3—91no—
58C24×D3—34no—
59C24×D3C3⋊D311yescharacteristic, complemented
60C22×#67—9112no—
61C22×#67—928no—
62C23×Dic3—94no—
63C23×Dic3—91no—
64C23×Dic3—184no—
65C23×C32—232no—
66C23×C32—228no—
67C23×C32—328no—
68C23×C32—656no—
69C23×C32#67132yescomplemented
70C23×C32#6718yescomplemented
71C23×C32C22×D3128yes—
72C23×C32C22×D317yes—
73C22×C3⋊D3—256no—
74C22×C3⋊D3C22×D3128yescomplemented
75C22×C3×D3—2224no—
76C22×C3×D3—3112no—
77C22×C3×D3—6224no—
78C22×C3×D3C22×D31112yescomplemented
79#638—364no—
80C2×C3⋊Dic3—328no—
81C2×C3⋊Dic3—3112no—
82C2×C3⋊Dic3—6112no—
83#625—3256no—
84C23×D4—271noSylow
85C2×C33C22×D4114yescomplemented
86C2×C33C2511yescharacteristic
87C3×C3⋊D3—28no—
88C32×D3—28no—
89C24×C3—24no—
90C24×C3—356no—
91C24×C3—632no—
92C24×C3—914no—
93C24×C3—188no—
94C24×C3D3214yes—
95C24×C3C2×C3⋊D311yescharacteristic
96C23×D3—28no—
97C23×D3—356no—
98C23×D3—632no—
99C23×D3C2×C3⋊D3114yescomplemented
100C2×#67—9448no—
101C2×#67—9112no—
102C22×Dic3—956no—
103C22×Dic3—914no—
104C22×Dic3—1856no—
105C22×C32—228no—
106C22×C32—2112no—
107C22×C32—38no—
108C22×C32—656no—
109C22×C32C2×#671112yescomplemented
110C22×C32C2×#67128yescomplemented
111C22×C32C23×D3128yes—
112C22×C32C23×D317yes—
113C2×C3⋊D3—256no—
114C2×C3⋊D3C23×D318yescomplemented
115C2×C3×D3—2224no—
116C2×C3×D3—332no—
117C2×C3×D3—6224no—
118C2×C3×D3C23×D3132yescomplemented
119C3⋊Dic3—38no—
120C3⋊Dic3—332no—
121C3⋊Dic3—632no—
122C25—31no—
123C25—91no—
124C22×D4—2728no—
125C23×C4—271no—
126C33C23×D411yescharacteristic, complemented, Sylow
127C23×C3—232no—
128C23×C3—228no—
129C23×C3—3112no—
130C23×C3—6224no—
131C23×C3—928no—
132C23×C3—1856no—
133C23×C3#625132yescomplemented
134C23×C3#63818yescomplemented
135C23×C3C2×D32128yes—
136C23×C3C22×C3⋊D317yes—
137C22×D3—256no—
138C22×D3—3112no—
139C22×D3—6224no—
140C22×D3C22×C3⋊D3128yescomplemented
141#67—9256no—
142#67—964no—
143C2×Dic3—9112no—
144C2×Dic3—928no—
145C2×Dic3—18112no—
146C2×C32—256no—
147C2×C32—24no—
148C2×C32—68no—
149C2×C32C22×#67156yescomplemented
150C2×C32C22×#67114yescomplemented
151C2×C32C24×D314yes—
152C2×C32C24×D311yescharacteristic
153C3⋊D3—28no—
154C3×D3—232no—
155C3×D3—632no—
156C24—314no—
157C24—68no—
158C24—914no—
159C24—188no—
160C24D3×C3⋊D311yescharacteristic
161C2×D4—27112no—
162C22×C4—2714no—
163C22×C3—228no—
164C22×C3—2112no—
165C22×C3—332no—
166C22×C3—6224no—
167C22×C3—98no—
168C22×C3—1856no—
169C22×C3C2×#6251112yescomplemented
170C22×C3C2×#638128yescomplemented
171C22×C3C22×D32128yes—
172C22×C3C23×C3⋊D317yes—
173C2×D3—256no—
174C2×D3—332no—
175C2×D3—6224no—
176C2×D3C23×C3⋊D318yescomplemented
177Dic3—932no—
178Dic3—98no—
179Dic3—1832no—
180C32—24no—
181C32C23×#6714yescomplemented
182C32C23×#6711yescharacteristic, complemented
183C23—328no—
184C23—656no—
185C23—928no—
186C23—1856no—
187C23#629018yesdirect product factor
188C23C2×D3×C3⋊D317yes—
189D4—2764no—
190C2×C4—2728no—
191C2×C3—256no—
192C2×C3—24no—
193C2×C3—632no—
194C2×C3—188no—
195C2×C3C22×#625156yescomplemented
196C2×C3C22×#638114yescomplemented
197C2×C3C23×D3214yes—
198C2×C3C24×C3⋊D311yescharacteristic
199D3—28no—
200D3—632no—
201C22—38no—
202C22—656no—
203C22—98no—
204C22—1856no—
205C22C2×#6290128yesdirect product factor
206C22C22×D3×C3⋊D317yes—
207C4—278no—
208C3—24no—
209C3C23×#62514yescomplemented
210C3C23×#63811yescharacteristic, complemented
211C2—68no—
212C2—188no—
213C2C22×#6290114yesdirect product factor
214C2C23×D3×C3⋊D311yescharacteristic
subgroupG/H
center, hypercenterC24D3×C3⋊D3
commutator, cosocleC2×C33C25
FrattiniC2C23×D3×C3⋊D3
Fitting, socleC24×C33C22
hypocenterC33C23×D4
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C24

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⋊(C3⋊Dic3)C2semidirect
4. C22×#6290C2direct (N × H)
9. C23×C3×C3⋊D3C22semidirect
11. C23×C32×D3C22semidirect
12. C22×C3⋊(C3⋊Dic3)C22semidirect
13. C2×#6290C22direct (N × H)
15. C24×C3⋊D3D3semidirect
17. C24×C3×D3D3semidirect
23. C23×C33D4semidirect
26. C22×C3×C3⋊D3C23semidirect
28. C22×C32×D3C23semidirect
29. C2×C3⋊(C3⋊Dic3)C23semidirect
30. #6290C23direct (N × H)
39. C23×C3⋊D3C2×D3semidirect
43. C23×C3×D3C2×D3semidirect
49. C22×C33C2×D4semidirect
52. C2×C3×C3⋊D3C24semidirect
54. C2×C32×D3C24semidirect
55. C3⋊(C3⋊Dic3)C24semidirect
59. C24×D3C3⋊D3semidirect
69. C23×C32#67semidirect
70. C23×C32#67semidirect
74. C22×C3⋊D3C22×D3semidirect
78. C22×C3×D3C22×D3semidirect
85. C2×C33C22×D4semidirect
99. C23×D3C2×C3⋊D3semidirect
109. C22×C32C2×#67semidirect
110. C22×C32C2×#67semidirect
114. C2×C3⋊D3C23×D3semidirect
118. C2×C3×D3C23×D3semidirect
126. C33C23×D4semidirect
133. C23×C3#625semidirect
134. C23×C3#638semidirect
140. C22×D3C22×C3⋊D3semidirect
149. C2×C32C22×#67semidirect
150. C2×C32C22×#67semidirect
169. C22×C3C2×#625semidirect
170. C22×C3C2×#638semidirect
176. C2×D3C23×C3⋊D3semidirect
181. C32C23×#67semidirect
182. C32C23×#67semidirect
187. C23#6290direct (N × H)
195. C2×C3C22×#625semidirect
196. C2×C3C22×#638semidirect
205. C22C2×#6290direct (N × H)
209. C3C23×#625semidirect
210. C3C23×#638semidirect
213. C2C22×#6290direct (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⋊(C3⋊Dic3)2
C22×#62902
C23×#6383
C23×#6253