Finite Group Databasebeta

← C23×C3×#638 #13181749

Subgroups — C23×C3×#638 #13181749

Order 1728.

11164 conjugacy classes of subgroups; only the 2030 normal subgroups are shown.

normal subgroups only

This group has 11164 conjugacy classes of subgroups — too many to draw. Only its 2030 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
1C25×C33C211yesmaximal, characteristic, complemented
2C24×C3×C3⋊D3C211yesmaximal, characteristic, complemented
3C22×C3×#638C2128yesmaximal, direct product factor
4C23×C3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C23×#638C311yesmaximal, characteristic, direct product factor
6C23×C3×#67—34nomaximal
7C24×C33—28no—
8C24×C33C2211yescharacteristic
9C24×C33C22114yescomplemented
10C23×C3×C3⋊D3—28no—
11C23×C3×C3⋊D3C22114yescomplemented
12C2×C3×#638C221112yesdirect product factor
13C22×C3×C3⋊Dic3C22114yescomplemented
14C25×C32—24no—
15C25×C32D314yescomplemented
16C25×C32C2×C311yescharacteristic, complemented
17C24×C3⋊D3C2×C311yescharacteristic, complemented
18C24×C3×D3—34no—
19C22×#638C2×C3128yesdirect product factor
20C23×C3⋊Dic3C2×C311yescharacteristic, complemented
21C22×C3×#67—3112no—
22C23×C3×Dic3—34no—
23C23×C33—256no—
24C23×C33D418yescomplemented
25C23×C33C2317yes—
26C23×C33C23128yescomplemented
27C22×C3×C3⋊D3—256no—
28C22×C3×C3⋊D3C23128yescomplemented
29C3×#638C23164yesdirect product factor
30C2×C3×C3⋊Dic3C23128yescomplemented
31C23×C3×D4—91no—
32C23×#67—34no—
33C24×C32—264no—
34C24×C32—24no—
35C24×C32—256no—
36C24×C32—232no—
37C24×C32—28no—
38C24×C32C2×D314yes—
39C24×C32C2×D3156yescomplemented
40C24×C32C22×C311yescharacteristic
41C24×C32C22×C3114yescomplemented
42C23×C3⋊D3—28no—
43C23×C3⋊D3C22×C3114yescomplemented
44C23×C3×D3—356no—
45C23×C3×D3—632no—
46C2×#638C22×C31112yesdirect product factor
47C22×C3⋊Dic3C22×C3114yescomplemented
48C2×C3×#67—3448no—
49C22×C3×Dic3—356no—
50C22×C33—256no—
51C22×C33C2×D4128yescomplemented
52C22×C33C2417yes—
53C22×C33C2418yescomplemented
54C2×C3×C3⋊D3—256no—
55C2×C3×C3⋊D3C2418yescomplemented
56C3×C3⋊Dic3C2418yescomplemented
57C25×C3—24no—
58C25×C3—91no—
59C25×C3C3×D314yescomplemented
60C25×C3C3⋊D311yescharacteristic, complemented
61C24×D3—34no—
62C22×C3×D4—928no—
63C23×C3×C4—91no—
64C22×#67—3112no—
65C23×Dic3—34no—
66C23×C32—2448no—
67C23×C32—2224no—
68C23×C32—256no—
69C23×C32—232no—
70C23×C32—228no—
71C23×C32—2112no—
72C23×C32#67132yescomplemented
73C23×C32C3×D418yescomplemented
74C23×C32C22×D3128yes—
75C23×C32C22×D31112yescomplemented
76C23×C32C23×C317yes—
77C23×C32C23×C3128yescomplemented
78C22×C3⋊D3—256no—
79C22×C3⋊D3C23×C3128yescomplemented
80C22×C3×D3—3112no—
81C22×C3×D3—6224no—
82#638C23×C3164yesdirect product factor
83C2×C3⋊Dic3C23×C3128yescomplemented
84C3×#67—3256no—
85C2×C3×Dic3—3112no—
86C23×D4—91noSylow
87C2×C33—28no—
88C2×C33C22×D4114yescomplemented
89C2×C33C2511yescharacteristic
90C3×C3⋊D3—28no—
91C24×C3—264no—
92C24×C3—232no—
93C24×C3—28no—
94C24×C3—24no—
95C24×C3—256no—
96C24×C3—914no—
97C24×C3—188no—
98C24×C3C2×C3×D314yes—
99C24×C3C2×C3×D3156yescomplemented
100C24×C3C2×C3⋊D311yescharacteristic
101C24×C3C2×C3⋊D3114yescomplemented
102C23×D3—356no—
103C23×D3—632no—
104C2×C3×D4—9112no—
105C22×C3×C4—914no—
106C2×#67—3448no—
107C22×Dic3—356no—
108C22×C32—2448no—
109C22×C32—2224no—
110C22×C32—256no—
111C22×C32—228no—
112C22×C32—232no—
113C22×C32—2112no—
114C22×C32C2×#671112yescomplemented
115C22×C32C2×C3×D4128yescomplemented
116C22×C32C23×D3128yes—
117C22×C32C23×D3132yescomplemented
118C22×C32C24×C317yes—
119C22×C32C24×C318yescomplemented
120C2×C3⋊D3—256no—
121C2×C3⋊D3C24×C318yescomplemented
122C2×C3×D3—332no—
123C2×C3×D3—6224no—
124C3⋊Dic3C24×C318yescomplemented
125C3×Dic3—332no—
126C25—91no—
127C25C3×C3⋊D311yescharacteristic, complemented
128C22×D4—928no—
129C23×C4—91no—
130C33C23×D411yescharacteristic, complemented, Sylow
131C23×C3—2448no—
132C23×C3—2224no—
133C23×C3—256no—
134C23×C3—232no—
135C23×C3—228no—
136C23×C3—2112no—
137C23×C3—928no—
138C23×C3—1856no—
139C23×C3C3×#67132yescomplemented
140C23×C3#63818yesdirect product factor
141C23×C3C22×C3×D3128yes—
142C23×C3C22×C3×D31112yescomplemented
143C23×C3C22×C3⋊D317yes—
144C23×C3C22×C3⋊D3128yescomplemented
145C22×D3—3112no—
146C22×D3—6224no—
147C3×D4—964no—
148C2×C3×C4—928no—
149#67—3256no—
150C2×Dic3—3112no—
151C2×C32—264no—
152C2×C32—232no—
153C2×C32—28no—
154C2×C32—256no—
155C2×C32—24no—
156C2×C32C22×#67156yescomplemented
157C2×C32C22×C3×D4114yescomplemented
158C2×C32C24×D314yes—
159C2×C32C25×C311yescharacteristic
160C3⋊D3—28no—
161C3×D3—632no—
162C24—28no—
163C24—914no—
164C24—188no—
165C24C2×C3×C3⋊D311yescharacteristic
166C24C2×C3×C3⋊D3114yescomplemented
167C2×D4—9112no—
168C22×C4—914no—
169C22×C3—2448no—
170C22×C3—2224no—
171C22×C3—256no—
172C22×C3—228no—
173C22×C3—232no—
174C22×C3—2112no—
175C22×C3—98no—
176C22×C3—1856no—
177C22×C3C2×C3×#671112yescomplemented
178C22×C3C2×#638128yesdirect product factor
179C22×C3C23×C3×D3128yes—
180C22×C3C23×C3×D3132yescomplemented
181C22×C3C23×C3⋊D317yes—
182C22×C3C23×C3⋊D318yescomplemented
183C2×D3—332no—
184C2×D3—6224no—
185C3×C4—98no—
186Dic3—332no—
187C32—24no—
188C32C23×#6714yescomplemented
189C32C23×C3×D411yescharacteristic, complemented
190C23—256no—
191C23—928no—
192C23—1856no—
193C23C3×#63818yesdirect product factor
194C23C22×C3×C3⋊D317yes—
195C23C22×C3×C3⋊D3128yescomplemented
196D4—964no—
197C2×C4—928no—
198C2×C3—264no—
199C2×C3—232no—
200C2×C3—28no—
201C2×C3—256no—
202C2×C3—24no—
203C2×C3—188no—
204C2×C3C22×C3×#67156yescomplemented
205C2×C3C22×#638114yesdirect product factor
206C2×C3C24×C3×D314yes—
207C2×C3C24×C3⋊D311yescharacteristic
208D3—632no—
209C22—256no—
210C22—98no—
211C22—1856no—
212C22C2×C3×#638128yesdirect product factor
213C22C23×C3×C3⋊D317yes—
214C22C23×C3×C3⋊D318yescomplemented
215C4—98no—
216C3—24no—
217C3C23×C3×#6714yescomplemented
218C3C23×#63811yescharacteristic, direct product factor
219C2—28no—
220C2—188no—
221C2C22×C3×#638114yesdirect product factor
222C2C24×C3×C3⋊D311yescharacteristic
subgroupG/H
centerC24×C3C2×C3⋊D3
commutator, cosocleC2×C32C25×C3
FrattiniC2C24×C3×C3⋊D3
FittingC25×C33C2
socleC24×C33C22
hypercenterC25×C3C3⋊D3
hypocenterC32C23×C3×D4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C24×C3 ◁ C25×C3

Complements

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

NH ≅ G/Ntype
1. C25×C33C2semidirect
2. C24×C3×C3⋊D3C2semidirect
3. C22×C3×#638C2direct (N × H)
4. C23×C3×C3⋊Dic3C2semidirect
5. C23×#638C3direct (N × H)
9. C24×C33C22semidirect
11. C23×C3×C3⋊D3C22semidirect
12. C2×C3×#638C22direct (N × H)
13. C22×C3×C3⋊Dic3C22semidirect
15. C25×C32D3semidirect
16. C25×C32C2×C3semidirect
17. C24×C3⋊D3C2×C3semidirect
19. C22×#638C2×C3direct (N × H)
20. C23×C3⋊Dic3C2×C3semidirect
24. C23×C33D4semidirect
26. C23×C33C23semidirect
28. C22×C3×C3⋊D3C23semidirect
29. C3×#638C23direct (N × H)
30. C2×C3×C3⋊Dic3C23semidirect
39. C24×C32C2×D3semidirect
41. C24×C32C22×C3semidirect
43. C23×C3⋊D3C22×C3semidirect
46. C2×#638C22×C3direct (N × H)
47. C22×C3⋊Dic3C22×C3semidirect
51. C22×C33C2×D4semidirect
53. C22×C33C24semidirect
55. C2×C3×C3⋊D3C24semidirect
56. C3×C3⋊Dic3C24semidirect
59. C25×C3C3×D3semidirect
60. C25×C3C3⋊D3semidirect
72. C23×C32#67semidirect
73. C23×C32C3×D4semidirect
75. C23×C32C22×D3semidirect
77. C23×C32C23×C3semidirect
79. C22×C3⋊D3C23×C3semidirect
82. #638C23×C3direct (N × H)
83. C2×C3⋊Dic3C23×C3semidirect
88. C2×C33C22×D4semidirect
99. C24×C3C2×C3×D3semidirect
101. C24×C3C2×C3⋊D3semidirect
114. C22×C32C2×#67semidirect
115. C22×C32C2×C3×D4semidirect
117. C22×C32C23×D3semidirect
119. C22×C32C24×C3semidirect
121. C2×C3⋊D3C24×C3semidirect
124. C3⋊Dic3C24×C3semidirect
127. C25C3×C3⋊D3semidirect
130. C33C23×D4semidirect
139. C23×C3C3×#67semidirect
140. C23×C3#638direct (N × H)
142. C23×C3C22×C3×D3semidirect
144. C23×C3C22×C3⋊D3semidirect
156. C2×C32C22×#67semidirect
157. C2×C32C22×C3×D4semidirect
166. C24C2×C3×C3⋊D3semidirect
177. C22×C3C2×C3×#67semidirect
178. C22×C3C2×#638direct (N × H)
180. C22×C3C23×C3×D3semidirect
182. C22×C3C23×C3⋊D3semidirect
188. C32C23×#67semidirect
189. C32C23×C3×D4semidirect
193. C23C3×#638direct (N × H)
195. C23C22×C3×C3⋊D3semidirect
204. C2×C3C22×C3×#67semidirect
205. C2×C3C22×#638direct (N × H)
212. C22C2×C3×#638direct (N × H)
214. C22C23×C3×C3⋊D3semidirect
217. C3C23×C3×#67semidirect
218. C3C23×#638direct (N × H)
221. C2C22×C3×#638direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C332
C24×C3×C3⋊D32
C22×C3×#6382
C23×C3×C3⋊Dic32
C23×#6383
C23×C3×#673