Finite Group Databasebeta

← C23×#6292 #13181657

Subgroups — C23×#6292 #13181657

Order 1728.

11164 conjugacy classes of subgroups; only the 1468 normal subgroups are shown.

normal subgroups only

This group has 11164 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⋊D3)C211yesmaximal, characteristic, complemented
2C24×C3×C3⋊D3C211yesmaximal, characteristic, complemented
3C23×C32×Dic3C211yesmaximal, characteristic, complemented
4C22×#6292C2128yesmaximal, direct product factor
5C23×C4⋊(C3⋊D3)—31nomaximal
6C23×#626—34nomaximal
7C24×C33C2211yescharacteristic
8C23×C3⋊(C3⋊D3)—28no—
9C23×C3⋊(C3⋊D3)C22114yescomplemented
10C23×C3×C3⋊D3—28no—
11C23×C3×C3⋊D3C22114yescomplemented
12C22×C32×Dic3C22114yescomplemented
13C2×#6292C221112yesdirect product factor
14C24×C3⋊D3—31no—
15C24×C3⋊D3—34no—
16C24×C3⋊D3—64no—
17C24×C3⋊D3D311yescharacteristic, complemented
18C24×C3×D3—34no—
19C23×C32×C4—31no—
20C22×C4⋊(C3⋊D3)—328no—
21C23×C3×Dic3D314yescomplemented
22C22×#626—3112no—
23C23×C33D418yescomplemented
24C23×C33C2317yes—
25C22×C3⋊(C3⋊D3)—256no—
26C22×C3⋊(C3⋊D3)C23128yescomplemented
27C22×C3×C3⋊D3—256no—
28C22×C3×C3⋊D3C23128yescomplemented
29C2×C32×Dic3C23128yescomplemented
30#6292C23164yesdirect product factor
31C23×#67—91no—
32C23×D12—94no—
33C24×C32—24no—
34C24×C32C2×D314yes—
35C24×C32C2×D311yescharacteristic
36C23×C3⋊D3—28no—
37C23×C3⋊D3—314no—
38C23×C3⋊D3—356no—
39C23×C3⋊D3—656no—
40C23×C3⋊D3—664no—
41C23×C3⋊D3—632no—
42C23×C3⋊D3—68no—
43C23×C3⋊D3C2×D3114yescomplemented
44C23×C3×D3—356no—
45C23×C3×D3—632no—
46C22×C32×C4—314no—
47C2×C4⋊(C3⋊D3)—3112no—
48C22×C3×Dic3C2×D3156yescomplemented
49C2×#626—3448no—
50C22×C33C2×D4128yescomplemented
51C22×C33C2417yes—
52C2×C3⋊(C3⋊D3)—256no—
53C2×C3⋊(C3⋊D3)C2418yescomplemented
54C2×C3×C3⋊D3—256no—
55C2×C3×C3⋊D3C2418yescomplemented
56C32×Dic3C2418yescomplemented
57C25×C3—91no—
58C24×D3—34no—
59C24×D3—94no—
60C24×D3—91no—
61C24×D3—184no—
62C23×C3×C4—34no—
63C22×#67—928no—
64C23×Dic3C3⋊D311yescharacteristic, complemented
65C22×D12—9112no—
66C23×C32—232no—
67C23×C32—228no—
68C23×C32D12132yescomplemented
69C23×C32#6718yescomplemented
70C23×C32C22×D3128yes—
71C23×C32C22×D317yes—
72C22×C3⋊D3—256no—
73C22×C3⋊D3—328no—
74C22×C3⋊D3—3112no—
75C22×C3⋊D3—6112no—
76C22×C3⋊D3—6448no—
77C22×C3⋊D3—6224no—
78C22×C3⋊D3—656no—
79C22×C3⋊D3C22×D3128yescomplemented
80C22×C3×D3—3112no—
81C22×C3×D3—6224no—
82C2×C32×C4—328no—
83C4⋊(C3⋊D3)—364no—
84C2×C3×Dic3C22×D31112yescomplemented
85#626—3256no—
86C23×D4—271noSylow
87C2×C33C22×D4114yescomplemented
88C2×C33C2511yescharacteristic
89C3⋊(C3⋊D3)—28no—
90C3×C3⋊D3—28no—
91C24×C3—24no—
92C24×C3—914no—
93C24×C3—188no—
94C24×C3D3214yes—
95C24×C3C2×C3⋊D311yescharacteristic
96C23×D3—356no—
97C23×D3—632no—
98C23×D3—956no—
99C23×D3—914no—
100C23×D3—1856no—
101C23×D3—1864no—
102C23×D3—1832no—
103C23×D3—188no—
104C22×C3×C4—356no—
105C2×#67—9112no—
106C22×Dic3C2×C3⋊D3114yescomplemented
107C2×D12—9448no—
108C22×C32—228no—
109C22×C32—2112no—
110C22×C32C2×D121112yescomplemented
111C22×C32C2×#67128yescomplemented
112C22×C32C23×D3128yes—
113C22×C32C23×D317yes—
114C2×C3⋊D3—256no—
115C2×C3⋊D3—38no—
116C2×C3⋊D3—332no—
117C2×C3⋊D3—632no—
118C2×C3⋊D3—6448no—
119C2×C3⋊D3—6224no—
120C2×C3⋊D3—656no—
121C2×C3⋊D3C23×D318yescomplemented
122C2×C3×D3—332no—
123C2×C3×D3—6224no—
124C32×C4—38no—
125C3×Dic3C23×D3132yescomplemented
126C25—91no—
127C25—271no—
128C22×D4—2728no—
129C23×C4—31no—
130C33C23×D411yescharacteristic, complemented, Sylow
131C23×C3—232no—
132C23×C3—228no—
133C23×C3—928no—
134C23×C3—1856no—
135C23×C3#626132yescomplemented
136C23×C3C4⋊(C3⋊D3)18yescomplemented
137C23×C3C2×D32128yes—
138C23×C3C22×C3⋊D317yes—
139C22×D3—3112no—
140C22×D3—6224no—
141C22×D3—9112no—
142C22×D3—928no—
143C22×D3—18112no—
144C22×D3—18448no—
145C22×D3—18224no—
146C22×D3—1856no—
147C2×C3×C4—3112no—
148#67—964no—
149C2×Dic3C22×C3⋊D3128yescomplemented
150D12—9256no—
151C2×C32—256no—
152C2×C32—24no—
153C2×C32C22×D12156yescomplemented
154C2×C32C22×#67114yescomplemented
155C2×C32C24×D314yes—
156C2×C32C24×D311yescharacteristic
157C3⋊D3—28no—
158C3⋊D3—664no—
159C3⋊D3—632no—
160C3⋊D3—68no—
161C3×D3—632no—
162C24—914no—
163C24—188no—
164C24—2714no—
165C24—548no—
166C24D3×C3⋊D311yescharacteristic
167C2×D4—27112no—
168C22×C4—314no—
169C22×C3—228no—
170C22×C3—2112no—
171C22×C3—98no—
172C22×C3—1856no—
173C22×C3C2×#6261112yescomplemented
174C22×C3C2×C4⋊(C3⋊D3)128yescomplemented
175C22×C3C22×D32128yes—
176C22×C3C23×C3⋊D317yes—
177C2×D3—332no—
178C2×D3—6224no—
179C2×D3—932no—
180C2×D3—98no—
181C2×D3—1832no—
182C2×D3—18448no—
183C2×D3—18224no—
184C2×D3—1856no—
185C3×C4—332no—
186Dic3C23×C3⋊D318yescomplemented
187C32—24no—
188C32C23×D1214yescomplemented
189C32C23×#6711yescharacteristic, complemented
190C23—928no—
191C23—1856no—
192C23—2728no—
193C23—5456no—
194C23#629218yesdirect product factor
195C23C2×D3×C3⋊D317yes—
196D4—2764no—
197C2×C4—328no—
198C2×C3—256no—
199C2×C3—24no—
200C2×C3—188no—
201C2×C3C22×#626156yescomplemented
202C2×C3C22×C4⋊(C3⋊D3)114yescomplemented
203C2×C3C23×D3214yes—
204C2×C3C24×C3⋊D311yescharacteristic
205D3—632no—
206D3—1864no—
207D3—1832no—
208D3—188no—
209C22—98no—
210C22—1856no—
211C22—278no—
212C22—5456no—
213C22C2×#6292128yesdirect product factor
214C22C22×D3×C3⋊D317yes—
215C4—38no—
216C3—24no—
217C3C23×#62614yescomplemented
218C3C23×C4⋊(C3⋊D3)11yescharacteristic, complemented
219C2—188no—
220C2—548no—
221C2C22×#6292114yesdirect product factor
222C2C23×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⋊D3)C2semidirect
2. C24×C3×C3⋊D3C2semidirect
3. C23×C32×Dic3C2semidirect
4. C22×#6292C2direct (N × H)
9. C23×C3⋊(C3⋊D3)C22semidirect
11. C23×C3×C3⋊D3C22semidirect
12. C22×C32×Dic3C22semidirect
13. C2×#6292C22direct (N × H)
17. C24×C3⋊D3D3semidirect
21. C23×C3×Dic3D3semidirect
23. C23×C33D4semidirect
26. C22×C3⋊(C3⋊D3)C23semidirect
28. C22×C3×C3⋊D3C23semidirect
29. C2×C32×Dic3C23semidirect
30. #6292C23direct (N × H)
43. C23×C3⋊D3C2×D3semidirect
48. C22×C3×Dic3C2×D3semidirect
50. C22×C33C2×D4semidirect
53. C2×C3⋊(C3⋊D3)C24semidirect
55. C2×C3×C3⋊D3C24semidirect
56. C32×Dic3C24semidirect
64. C23×Dic3C3⋊D3semidirect
68. C23×C32D12semidirect
69. C23×C32#67semidirect
79. C22×C3⋊D3C22×D3semidirect
84. C2×C3×Dic3C22×D3semidirect
87. C2×C33C22×D4semidirect
106. C22×Dic3C2×C3⋊D3semidirect
110. C22×C32C2×D12semidirect
111. C22×C32C2×#67semidirect
121. C2×C3⋊D3C23×D3semidirect
125. C3×Dic3C23×D3semidirect
130. C33C23×D4semidirect
135. C23×C3#626semidirect
136. C23×C3C4⋊(C3⋊D3)semidirect
149. C2×Dic3C22×C3⋊D3semidirect
153. C2×C32C22×D12semidirect
154. C2×C32C22×#67semidirect
173. C22×C3C2×#626semidirect
174. C22×C3C2×C4⋊(C3⋊D3)semidirect
186. Dic3C23×C3⋊D3semidirect
188. C32C23×D12semidirect
189. C32C23×#67semidirect
194. C23#6292direct (N × H)
201. C2×C3C22×#626semidirect
202. C2×C3C22×C4⋊(C3⋊D3)semidirect
213. C22C2×#6292direct (N × H)
217. C3C23×#626semidirect
218. C3C23×C4⋊(C3⋊D3)semidirect
221. C2C22×#6292direct (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⋊D3)2
C24×C3×C3⋊D32
C23×C32×Dic32
C22×#62922
C23×C4⋊(C3⋊D3)3
C23×#6263