Finite Group Databasebeta

← C23×C3×#611 #13164965

Subgroups — C23×C3×#611 #13164965

Order 1728.

4834 conjugacy classes of subgroups; only the 1322 normal subgroups are shown.

normal subgroups only

This group has 4834 conjugacy classes of subgroups — too many to draw. Only its 1322 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×C3×C9C211yesmaximal, characteristic, complemented
2C24×C3×D9C211yesmaximal, characteristic, complemented
3C22×C3×#611C2128yesmaximal, direct product factor
4C23×C3×Dic9C211yesmaximal, characteristic, complemented
5C23×C3×#67—31nomaximal
6C23×#611C311yesmaximal, characteristic, direct product factor
7C24×C3×C9—28no—
8C24×C3×C9C2211yescharacteristic
9C24×C3×C9C22114yescomplemented
10C23×C3×D9—28no—
11C23×C3×D9C22114yescomplemented
12C2×C3×#611C221112yesdirect product factor
13C22×C3×Dic9C22114yescomplemented
14C25×C32D311yescharacteristic
15C24×C3×D3—31no—
16C22×C3×#67—328no—
17C23×C3×Dic3—31no—
18C25×C9—21no—
19C25×C9C2×C311yescharacteristic, complemented
20C24×D9C2×C311yescharacteristic, complemented
21C22×#611C2×C3128yesdirect product factor
22C23×Dic9C2×C311yescharacteristic, complemented
23C23×C3×C9—256no—
24C23×C3×C9D418yescomplemented
25C23×C3×C9C2317yes—
26C23×C3×C9C23128yescomplemented
27C22×C3×D9—256no—
28C22×C3×D9C23128yescomplemented
29C3×#611C23164yesdirect product factor
30C2×C3×Dic9C23128yescomplemented
31C23×C3×D4—91no—
32C23×#67—31no—
33C24×C32—28no—
34C24×C32C2×D3114yes—
35C24×C32C2×D311yescharacteristic
36C23×C3×D3—314no—
37C23×C3×D3—68no—
38C2×C3×#67—3112no—
39C22×C3×Dic3—314no—
40C24×C9—216no—
41C24×C9—21no—
42C24×C9—214no—
43C24×C9—28no—
44C24×C9C22×C311yescharacteristic
45C24×C9C22×C3114yescomplemented
46C23×D9—28no—
47C23×D9C22×C3114yescomplemented
48C2×#611C22×C31112yesdirect product factor
49C22×Dic9C22×C3114yescomplemented
50C22×C3×C9—256no—
51C22×C3×C9C2×D4128yescomplemented
52C22×C3×C9C2417yes—
53C22×C3×C9C2418yescomplemented
54C2×C3×D9—256no—
55C2×C3×D9C2418yescomplemented
56C3×Dic9C2418yescomplemented
57C25×C3—21no—
58C25×C3—91no—
59C25×C3D911yescharacteristic, complemented
60C25×C3C3×D311yescharacteristic
61C24×D3—31no—
62C22×C3×D4—928no—
63C23×C3×C4—91no—
64C22×#67—328no—
65C23×Dic3—31no—
66C23×C32—256no—
67C23×C32#6718yes—
68C23×C32C22×D317yes—
69C23×C32C22×D3128yes—
70C22×C3×D3—328no—
71C22×C3×D3—656no—
72C3×#67—364no—
73C2×C3×Dic3—328no—
74C23×C9—2112no—
75C23×C9—256no—
76C23×C9—28no—
77C23×C9—27no—
78C23×C9—228no—
79C23×C9C3×D418yescomplemented
80C23×C9C23×C317yes—
81C23×C9C23×C3128yescomplemented
82C22×D9—256no—
83C22×D9C23×C3128yescomplemented
84#611C23×C3164yesdirect product factor
85C2×Dic9C23×C3128yescomplemented
86C23×D4—91noSylow
87C2×C3×C9—28no—
88C2×C3×C9C22×D4114yescomplemented
89C2×C3×C9C2511yescharacteristic
90C3×D9—28no—
91C24×C3—216no—
92C24×C3—28no—
93C24×C3—28no—
94C24×C3—21no—
95C24×C3—214no—
96C24×C3—914no—
97C24×C3—188no—
98C24×C3C2×D911yescharacteristic
99C24×C3C2×D9114yescomplemented
100C24×C3C2×C3×D311yescharacteristic
101C24×C3C2×C3×D3114yes—
102C23×D3—314no—
103C23×D3—68no—
104C2×C3×D4—9112no—
105C22×C3×C4—914no—
106C2×#67—3112no—
107C22×Dic3—314no—
108C22×C32—256no—
109C22×C32C2×#67128yes—
110C22×C32C23×D317yes—
111C22×C32C23×D318yes—
112C2×C3×D3—38no—
113C2×C3×D3—656no—
114C3×Dic3—38no—
115C22×C9—2112no—
116C22×C9—256no—
117C22×C9—27no—
118C22×C9—28no—
119C22×C9—228no—
120C22×C9C2×C3×D4128yescomplemented
121C22×C9C24×C317yes—
122C22×C9C24×C318yescomplemented
123C2×D9—256no—
124C2×D9C24×C318yescomplemented
125Dic9C24×C318yescomplemented
126C25—91no—
127C25C3×D911yescharacteristic, complemented
128C22×D4—928no—
129C23×C4—91no—
130C3×C9C23×D411yescharacteristic, complemented, Sylow
131C23×C3—2112no—
132C23×C3—256no—
133C23×C3—256no—
134C23×C3—28no—
135C23×C3—27no—
136C23×C3—228no—
137C23×C3—928no—
138C23×C3—1856no—
139C23×C3#61118yesdirect product factor
140C23×C3C22×D917yes—
141C23×C3C22×D9128yescomplemented
142C23×C3C3×#6718yes—
143C23×C3C22×C3×D317yes—
144C23×C3C22×C3×D3128yes—
145C22×D3—328no—
146C22×D3—656no—
147C3×D4—964no—
148C2×C3×C4—928no—
149#67—364no—
150C2×Dic3—328no—
151C2×C32—28no—
152C2×C32C22×#67114yes—
153C2×C32C24×D311yescharacteristic
154C3×D3—68no—
155C2×C9—216no—
156C2×C9—28no—
157C2×C9—214no—
158C2×C9—21no—
159C2×C9C22×C3×D4114yescomplemented
160C2×C9C25×C311yescharacteristic
161D9—28no—
162C24—28no—
163C24—914no—
164C24—188no—
165C24C2×C3×D911yescharacteristic
166C24C2×C3×D9114yescomplemented
167C2×D4—9112no—
168C22×C4—914no—
169C22×C3—2112no—
170C22×C3—256no—
171C22×C3—256no—
172C22×C3—27no—
173C22×C3—28no—
174C22×C3—228no—
175C22×C3—98no—
176C22×C3—1856no—
177C22×C3C2×#611128yesdirect product factor
178C22×C3C23×D917yes—
179C22×C3C23×D918yescomplemented
180C22×C3C2×C3×#67128yes—
181C22×C3C23×C3×D317yes—
182C22×C3C23×C3×D318yes—
183C2×D3—38no—
184C2×D3—656no—
185C3×C4—98no—
186Dic3—38no—
187C32C23×#6711yescharacteristic
188C9—21no—
189C9C23×C3×D411yescharacteristic, complemented
190C23—256no—
191C23—928no—
192C23—1856no—
193C23C3×#61118yesdirect product factor
194C23C22×C3×D917yes—
195C23C22×C3×D9128yescomplemented
196D4—964no—
197C2×C4—928no—
198C2×C3—216no—
199C2×C3—28no—
200C2×C3—28no—
201C2×C3—21no—
202C2×C3—214no—
203C2×C3—188no—
204C2×C3C22×#611114yesdirect product factor
205C2×C3C24×D911yescharacteristic
206C2×C3C22×C3×#67114yes—
207C2×C3C24×C3×D311yescharacteristic
208D3—68no—
209C22—256no—
210C22—98no—
211C22—1856no—
212C22C2×C3×#611128yesdirect product factor
213C22C23×C3×D917yes—
214C22C23×C3×D918yescomplemented
215C4—98no—
216C3—21no—
217C3C23×#61111yescharacteristic, direct product factor
218C3C23×C3×#6711yescharacteristic
219C2—28no—
220C2—188no—
221C2C22×C3×#611114yesdirect product factor
222C2C24×C3×D911yescharacteristic
subgroupG/H
centerC24×C3C2×D9
commutator, cosocleC2×C9C25×C3
FrattiniC2×C3C24×C3×D3
FittingC25×C3×C9C2
socleC24×C32C2×D3
hypercenterC25×C3D9
hypocenterC9C23×C3×D4
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C24×C3 ◁ C25×C3

Complements

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

NH ≅ G/Ntype
1. C25×C3×C9C2semidirect
2. C24×C3×D9C2semidirect
3. C22×C3×#611C2direct (N × H)
4. C23×C3×Dic9C2semidirect
6. C23×#611C3direct (N × H)
9. C24×C3×C9C22semidirect
11. C23×C3×D9C22semidirect
12. C2×C3×#611C22direct (N × H)
13. C22×C3×Dic9C22semidirect
19. C25×C9C2×C3semidirect
20. C24×D9C2×C3semidirect
21. C22×#611C2×C3direct (N × H)
22. C23×Dic9C2×C3semidirect
24. C23×C3×C9D4semidirect
26. C23×C3×C9C23semidirect
28. C22×C3×D9C23semidirect
29. C3×#611C23direct (N × H)
30. C2×C3×Dic9C23semidirect
45. C24×C9C22×C3semidirect
47. C23×D9C22×C3semidirect
48. C2×#611C22×C3direct (N × H)
49. C22×Dic9C22×C3semidirect
51. C22×C3×C9C2×D4semidirect
53. C22×C3×C9C24semidirect
55. C2×C3×D9C24semidirect
56. C3×Dic9C24semidirect
59. C25×C3D9semidirect
79. C23×C9C3×D4semidirect
81. C23×C9C23×C3semidirect
83. C22×D9C23×C3semidirect
84. #611C23×C3direct (N × H)
85. C2×Dic9C23×C3semidirect
88. C2×C3×C9C22×D4semidirect
99. C24×C3C2×D9semidirect
120. C22×C9C2×C3×D4semidirect
122. C22×C9C24×C3semidirect
124. C2×D9C24×C3semidirect
125. Dic9C24×C3semidirect
127. C25C3×D9semidirect
130. C3×C9C23×D4semidirect
139. C23×C3#611direct (N × H)
141. C23×C3C22×D9semidirect
159. C2×C9C22×C3×D4semidirect
166. C24C2×C3×D9semidirect
177. C22×C3C2×#611direct (N × H)
179. C22×C3C23×D9semidirect
189. C9C23×C3×D4semidirect
193. C23C3×#611direct (N × H)
195. C23C22×C3×D9semidirect
204. C2×C3C22×#611direct (N × H)
212. C22C2×C3×#611direct (N × H)
214. C22C23×C3×D9semidirect
217. C3C23×#611direct (N × H)
221. C2C22×C3×#611direct (N × H)

Maximal subgroups

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

subgroupindex
C25×C3×C92
C24×C3×D92
C22×C3×#6112
C23×C3×Dic92
C23×C3×#673
C23×#6113