Finite Group Databasebeta

← C2×C3×#64231 #13181028

Subgroups — C2×C3×#64231 #13181028

Order 1728.

1576 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 226 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
1C22×C3×GO+(4,𝔽2)C214yesmaximal, complemented
2C2×C3×#3802C212yesmaximal, complemented
3C2×C3×C4×#157C211yesmaximal, characteristic, complemented
4C3×#64231C218yesmaximal, direct product factor
5C2×#64231C311yesmaximal, characteristic, direct product factor
6C22×C3×D32C2214yes—
7C22×C3×#157C2212yes—
8C2×C3×GO+(4,𝔽2)—216no—
9C2×C3×GO+(4,𝔽2)C22116yescomplemented
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×D12—22no—
12C2×C3×#625—22no—
13C3×#3802—24no—
14C3×#3802C2218yescomplemented
15C3×C4×#157C2214yescomplemented
16C22×GO+(4,𝔽2)C2×C314yescomplemented
17C2×C3×D3×D4—62no—
18C2×#3802C2×C312yescomplemented
19C2×C4×#157C2×C311yescharacteristic, complemented
20#64231C2×C318yesdirect product factor
21C22×C3×C3⋊D3C2311yescharacteristic
22C22×C32×D3—24no—
23C2×C3×D32—216no—
24C2×C3×D32—24no—
25C2×C3×D32C2318yes—
26C2×C3×#157D418yescomplemented
27C2×C3×#157C2314yes—
28C3×GO+(4,𝔽2)—232no—
29C2×C33×C4D411yescharacteristic, complemented
30C2×C3×C3⋊Dic3D411yescharacteristic, complemented
31C3×C4×C3⋊D3D412yescomplemented
32C3×C4×C3⋊D3C2312yes—
33C32×D12—28no—
34C3×#625—28no—
35C2×C3×#127—91nomaximal
36C23×C3×D3—64no—
37C22×D32C22×C314yes—
38C22×#157C22×C312yes—
39C2×GO+(4,𝔽2)—216no—
40C2×GO+(4,𝔽2)C22×C3116yescomplemented
41C2×C32×D4—62no—
42C2×C4×C3⋊D3C22×C311yescharacteristic
43C2×C3×#67—64no—
44C3×D3×D4—68no—
45C3×D3×D4—68no—
46C2×C3×D12—22no—
47C2×C3×D12—22no—
48C2×C3×D12—42no—
49C2×C3×C4×D3—62no—
50C2×#625—22no—
51#3802—24no—
52#3802C22×C318yescomplemented
53C4×#157C22×C314yescomplemented
54C22×C33C2×D411yescharacteristic
55C2×C3×C3⋊D3C2×D411yescharacteristic
56C2×C3×C3⋊D3C2×D414yes—
57C2×C3×C3⋊D3C2411yescharacteristic
58C2×C32×D3—28no—
59C2×C32×D3—48no—
60C3×D32—216no—
61C3×D32—48no—
62C3×#157C2×D418yescomplemented
63C33×C4C2×D412yescomplemented
64C3×C3⋊Dic3C2×D412yescomplemented
65C22×C3×D4—92no—
66C22×C3×D4—94no—
67C2×D3×D4—62no—
68C3×#127—98no—
69C2×C3×C42—91no—
70C23×C32—64no—
71C22×C3⋊D3C23×C311yescharacteristic
72C22×C3×D3—24no—
73C22×C3×D3—24no—
74C22×C3×D3—44no—
75C22×C3×D3—68no—
76C22×C3×D3—62no—
77C22×C3×D3—68no—
78C22×C3×D3—1216no—
79C2×D32—24no—
80C2×D32—216no—
81C2×D32C23×C318yes—
82C2×#157C3×D418yescomplemented
83C2×#157C23×C314yes—
84GO+(4,𝔽2)—232no—
85C32×D4—68no—
86C2×C32×C4—22no—
87C2×C32×C4—42no—
88C2×C32×C4C3×D411yescharacteristic, complemented
89C2×C3⋊Dic3C3×D411yescharacteristic, complemented
90C4×C3⋊D3C3×D412yescomplemented
91C4×C3⋊D3C23×C312yes—
92C3×#67—616no—
93C2×C3×Dic3—62no—
94C3×D12—28no—
95C3×D12—28no—
96C3×D12—48no—
97C3×C4×D3—64no—
98C3×C4×D3—64no—
99#625—28no—
100C2×#127—91noSylow
101C2×C33#12712yescomplemented
102C2×C33C22×D411yescharacteristic
103C3×C3⋊D3#12712yescomplemented
104C3×C3⋊D3C22×D412yes—
105C32×D3—48no—
106C24×C3—94no—
107C23×D3—64no—
108C2×C3×D4—62no—
109C2×C3×D4—62no—
110C2×C3×D4—98no—
111C2×C3×D4—916no—
112C2×C3×D4—122no—
113C2×C3×D4—184no—
114C2×C3×D4—182no—
115C2×C3×D4—1816no—
116C22×C3×C4—92no—
117C22×C3×C4—91no—
118C2×#67—64no—
119D3×D4—68no—
120D3×D4—68no—
121C2×D12—22no—
122C2×C4×D3—62no—
123C3×C42—94no—
124C22×C32—22no—
125C22×C32—42no—
126C22×C32—68no—
127C22×C32—128no—
128C22×C32C2×C3×D411yescharacteristic
129C2×C3⋊D3C2×C3×D411yescharacteristic
130C2×C3⋊D3C2×C3×D414yes—
131C2×C3⋊D3C24×C311yescharacteristic
132C2×C3×D3—28no—
133C2×C3×D3—28no—
134C2×C3×D3—48no—
135C2×C3×D3—48no—
136C2×C3×D3—48no—
137C2×C3×D3—62no—
138C2×C3×D3—68no—
139C2×C3×D3—62no—
140C2×C3×D3—88no—
141C2×C3×D3—1216no—
142C2×C3×D3—1216no—
143D32—216no—
144D32—48no—
145#157C2×C3×D418yescomplemented
146C32×C4—24no—
147C32×C4—44no—
148C32×C4C2×C3×D412yescomplemented
149C3⋊Dic3C2×C3×D412yescomplemented
150C3×Dic3—64no—
151C22×D4—92no—
152C22×D4—94no—
153#127—98no—
154C2×C42—91no—
155C33C2×#12711yescharacteristic, complemented, Sylow
156C23×C3—64no—
157C23×C3—64no—
158C23×C3—98no—
159C23×C3—91no—
160C23×C3—124no—
161C23×C3—184no—
162C23×C3—1816no—
163C22×D3—24no—
164C22×D3—68no—
165C22×D3—62no—
166C22×D3—68no—
167C22×D3—1216no—
168C3×D4—68no—
169C3×D4—68no—
170C3×D4—128no—
171C3×D4—188no—
172C3×D4—1832no—
173C2×C3×C4—22no—
174C2×C3×C4—42no—
175C2×C3×C4—98no—
176C2×C3×C4—94no—
177C2×C3×C4—91no—
178C2×C3×C4—92no—
179C2×C3×C4—92no—
180C2×C3×C4GO+(4,𝔽2)11yescharacteristic, complemented
181#67—616no—
182C2×Dic3—62no—
183D12—28no—
184C4×D3—64no—
185C4×D3—64no—
186C2×C32—22no—
187C2×C32—24no—
188C2×C32—44no—
189C2×C32—42no—
190C2×C32—128no—
191C2×C32C3×#12712yescomplemented
192C2×C32C22×C3×D411yescharacteristic
193C3⋊D3C3×#12712yescomplemented
194C3⋊D3C22×C3×D412yes—
195C3×D3—48no—
196C3×D3—48no—
197C3×D3—64no—
198C3×D3—64no—
199C3×D3—88no—
200C24—94no—
201C2×D4—62no—
202C2×D4—98no—
203C2×D4—916no—
204C2×D4—184no—
205C2×D4—182no—
206C2×D4—1816no—
207C22×C4—92no—
208C22×C4—91no—
209C42—94no—
210C22×C3—22no—
211C22×C3—42no—
212C22×C3—68no—
213C22×C3—68no—
214C22×C3—91no—
215C22×C3—91no—
216C22×C3—94no—
217C22×C3—128no—
218C22×C3—128no—
219C22×C3—128no—
220C22×C3—1816no—
221C22×C3—248no—
222C22×C3—368no—
223C22×C3C2×GO+(4,𝔽2)11yescharacteristic
224C2×D3—28no—
225C2×D3—48no—
226C2×D3—62no—
227C2×D3—62no—
228C2×D3—68no—
229C2×D3—1216no—
230C2×D3—1216no—
231C3×C4—24no—
232C3×C4—44no—
233C3×C4—98no—
234C3×C4—92no—
235C3×C4C2×GO+(4,𝔽2)12yescomplemented
236Dic3—64no—
237C32—22no—
238C32—42no—
239C32C2×C3×#12711yescharacteristic, complemented
240C23—64no—
241C23—98no—
242C23—91no—
243C23—184no—
244C23—1816no—
245D4—68no—
246D4—188no—
247D4—1832no—
248C2×C4—94no—
249C2×C4—98no—
250C2×C4—91no—
251C2×C4—92no—
252C2×C4—92no—
253C2×C4C3×GO+(4,𝔽2)11yescharacteristic, complemented
254C2×C3—22no—
255C2×C3—24no—
256C2×C3—42no—
257C2×C3—44no—
258C2×C3—92no—
259C2×C3—92no—
260C2×C3—128no—
261C2×C3—128no—
262C2×C3—248no—
263C2×C3#6423112yesdirect product factor
264C2×C3C22×GO+(4,𝔽2)11yescharacteristic
265D3—48no—
266D3—64no—
267D3—64no—
268C22—68no—
269C22—91no—
270C22—91no—
271C22—94no—
272C22—128no—
273C22—1816no—
274C22—368no—
275C22C2×C3×GO+(4,𝔽2)11yescharacteristic
276C4—98no—
277C4—92no—
278C4C2×C3×GO+(4,𝔽2)12yescomplemented
279C3—22no—
280C3—42no—
281C3C2×#6423111yescharacteristic, direct product factor
282C2—92no—
283C2—92no—
284C2—128no—
285C2C3×#6423112yesdirect product factor
286C2C22×C3×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C2×GO+(4,𝔽2)
commutator, cosocleC2×C3⋊D3C24×C3
FrattiniC2C22×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
hypercenterC2×C3×C4GO+(4,𝔽2)
hypocenterC32C2×C3×#127
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×GO+(4,𝔽2)C2semidirect
2. C2×C3×#3802C2semidirect
3. C2×C3×C4×#157C2semidirect
4. C3×#64231C2direct (N × H)
5. C2×#64231C3direct (N × H)
9. C2×C3×GO+(4,𝔽2)C22semidirect
14. C3×#3802C22semidirect
15. C3×C4×#157C22semidirect
16. C22×GO+(4,𝔽2)C2×C3semidirect
18. C2×#3802C2×C3semidirect
19. C2×C4×#157C2×C3semidirect
20. #64231C2×C3direct (N × H)
26. C2×C3×#157D4semidirect
29. C2×C33×C4D4semidirect
30. C2×C3×C3⋊Dic3D4semidirect
31. C3×C4×C3⋊D3D4semidirect
40. C2×GO+(4,𝔽2)C22×C3semidirect
52. #3802C22×C3semidirect
53. C4×#157C22×C3semidirect
62. C3×#157C2×D4semidirect
63. C33×C4C2×D4semidirect
64. C3×C3⋊Dic3C2×D4semidirect
82. C2×#157C3×D4semidirect
88. C2×C32×C4C3×D4semidirect
89. C2×C3⋊Dic3C3×D4semidirect
90. C4×C3⋊D3C3×D4semidirect
101. C2×C33#127semidirect
103. C3×C3⋊D3#127semidirect
145. #157C2×C3×D4semidirect
148. C32×C4C2×C3×D4semidirect
149. C3⋊Dic3C2×C3×D4semidirect
155. C33C2×#127semidirect
180. C2×C3×C4GO+(4,𝔽2)semidirect
191. C2×C32C3×#127semidirect
193. C3⋊D3C3×#127semidirect
235. C3×C4C2×GO+(4,𝔽2)semidirect
239. C32C2×C3×#127semidirect
253. C2×C4C3×GO+(4,𝔽2)semidirect
263. C2×C3#64231direct (N × H)
278. C4C2×C3×GO+(4,𝔽2)semidirect
281. C3C2×#64231direct (N × H)
285. C2C3×#64231direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×GO+(4,𝔽2)2
C2×C3×#38022
C2×C3×C4×#1572
C3×#642312
C2×#642313
C2×C3×#1279