Finite Group Databasebeta

← C2×C3×#63434 #13152280

Subgroups — C2×C3×#63434 #13152280

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D9C211yesmaximal, characteristic
2C2×C3×C42×C9C211yesmaximal, characteristic, complemented
3C2×C3×(C2×D9)⋊C4C212yesmaximal
4C2×C3×Dic9⋊C4C212yesmaximal, complemented
5C2×C3×C4×Dic9C211yesmaximal, characteristic, complemented
6C3×#63434C218yesmaximal, direct product factor
7C2×C3×#872—31nomaximal
8C2×#63434C311yesmaximal, characteristic, direct product factor
9C23×C3×D9C2211yescharacteristic
10C22×C3×C4×C9C2212yes—
11C22×C3×C4×C9C2211yescharacteristic
12C22×C3×Dic9C2212yes—
13C22×C3×Dic9C2211yescharacteristic
14C2×C3×C4×D9C418yescomplemented
15C2×C3×C4×D9C2214yes—
16C3×C42×C9C2214yescomplemented
17C3×(C2×D9)⋊C4C2218yes—
18C3×Dic9⋊C4C2218yescomplemented
19C3×C4×Dic9C2214yescomplemented
20C22×C3×C4×D3—31no—
21C2×C32×C42D311yescharacteristic
22C2×C3×(C2×D3)⋊C4—32no—
23C2×C3×Dic3⋊C4—32no—
24C2×C3×C4×Dic3—31no—
25C3×#872—38no—
26C22×C4×D9C2×C311yescharacteristic
27C2×C42×C9—21no—
28C2×C42×C9C2×C311yescharacteristic, complemented
29C2×(C2×D9)⋊C4C2×C312yes—
30C2×Dic9⋊C4C2×C312yescomplemented
31C2×C4×Dic9C2×C311yescharacteristic, complemented
32#63434C2×C318yesdirect product factor
33C23×C3×C9C2311yescharacteristic
34C22×C3×D9—24no—
35C22×C3×D9C2×C414yes—
36C22×C3×D9C2312yes—
37C2×C3×C4×C9—24no—
38C2×C3×C4×C9C2×C414yes—
39C2×C3×C4×C9C2314yes—
40C2×C3×C4×C9C2312yes—
41C2×C3×Dic9—24no—
42C2×C3×Dic9C2×C414yes—
43C2×C3×Dic9C2314yes—
44C2×C3×Dic9C2312yes—
45C3×C4×D9C2×C4116yescomplemented
46C2×C3×#117—91no—
47C2×#872—31no—
48C23×C3×D3—31no—
49C22×C32×C4C2×D312yes—
50C22×C32×C4C2×D311yescharacteristic
51C22×C3×Dic3—31no—
52C22×C3×Dic3—32no—
53C2×C3×C4×D3—38no—
54C2×C3×C4×D3—34no—
55C23×D9C22×C311yescharacteristic
56C32×C42C2×D314yes—
57C3×(C2×D3)⋊C4—38no—
58C3×Dic3⋊C4—38no—
59C3×C4×Dic3—34no—
60C22×C4×C9—22no—
61C22×C4×C9—21no—
62C22×C4×C9C22×C312yes—
63C22×C4×C9C22×C311yescharacteristic
64C22×Dic9C22×C312yes—
65C22×Dic9C22×C311yescharacteristic
66C2×C4×D9C3×C418yescomplemented
67C2×C4×D9C22×C314yes—
68C42×C9—24no—
69C42×C9C22×C314yescomplemented
70(C2×D9)⋊C4C22×C318yes—
71Dic9⋊C4C22×C318yescomplemented
72C4×Dic9C22×C314yescomplemented
73C22×C3×C9C22×C412yes—
74C22×C3×C9#4114yes—
75C22×C3×C9C2411yescharacteristic
76C2×C3×D9—28no—
77C2×C3×D9—24no—
78C2×C3×D9C22×C414yes—
79C3×C4×C9—24no—
80C3×C4×C9C22×C414yes—
81C3×Dic9—24no—
82C3×Dic9C22×C414yes—
83C23×C3×C4—91no—
84C22×C4×D3—31no—
85C3×#117—98no—
86C2×C3×C4⋊C4—92no—
87C2×C3×C22⋊C4—92no—
88C2×C3×C42—21no—
89C2×C3×C42—91no—
90C2×C3×C42D911yescharacteristic, complemented
91C2×C3×C42C3×D311yescharacteristic
92C2×(C2×D3)⋊C4—32no—
93C2×Dic3⋊C4—32no—
94C2×C4×Dic3—31no—
95#872—38no—
96C23×C32C22×D311yescharacteristic
97C22×C3×D3—34no—
98C22×C3×D3—32no—
99C22×C3×D3—64no—
100C2×C32×C4—24no—
101C2×C32×C4C4×D314yes—
102C2×C32×C4C22×D314yes—
103C2×C32×C4C22×D312yes—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—32no—
106C2×C3×Dic3—34no—
107C2×C3×Dic3—64no—
108C3×C4×D3—316no—
109C23×C9—21no—
110C23×C9C23×C311yescharacteristic
111C22×D9—24no—
112C22×D9C2×C3×C414yes—
113C22×D9C23×C312yes—
114C2×C4×C9—28no—
115C2×C4×C9—24no—
116C2×C4×C9—24no—
117C2×C4×C9—24no—
118C2×C4×C9—22no—
119C2×C4×C9C2×C3×C414yes—
120C2×C4×C9C23×C314yes—
121C2×C4×C9C23×C312yes—
122C2×Dic9—24no—
123C2×Dic9C2×C3×C414yes—
124C2×Dic9C23×C314yes—
125C2×Dic9C23×C312yes—
126C4×D9C2×C3×C4116yescomplemented
127C2×#117—91noSylow
128C2×C3×C9#11714yescomplemented
129C2×C3×C9C23×C411yescharacteristic
130C2×C3×C9C2×#4112yes—
131C3×D9—24no—
132C24×C3—91no—
133C23×D3—31no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—91no—
137C22×C3×C4—98no—
138C22×C3×C4—94no—
139C22×C3×C4—92no—
140C22×C3×C4C2×D912yes—
141C22×C3×C4C2×D911yescharacteristic
142C22×C3×C4C2×C3×D312yes—
143C22×C3×C4C2×C3×D311yescharacteristic
144C22×Dic3—31no—
145C22×Dic3—32no—
146C2×C4×D3—38no—
147C2×C4×D3—34no—
148C3×C4⋊C4—98no—
149C3×C22⋊C4—98no—
150C3×C42—24no—
151C3×C42—94no—
152C3×C42C2×D914yescomplemented
153C3×C42C2×C3×D314yes—
154(C2×D3)⋊C4—38no—
155Dic3⋊C4—38no—
156C4×Dic3—34no—
157C22×C32C2×C4×D312yes—
158C22×C32#23514yes—
159C22×C32C23×D311yescharacteristic
160C2×C3×D3—34no—
161C2×C3×D3—68no—
162C2×C3×D3—64no—
163C32×C4—24no—
164C32×C4C2×C4×D314yes—
165C3×Dic3—34no—
166C3×Dic3—64no—
167C22×C9—21no—
168C22×C9—22no—
169C22×C9—24no—
170C22×C9C22×C3×C412yes—
171C22×C9C3×#4114yes—
172C22×C9C24×C311yescharacteristic
173C2×D9—28no—
174C2×D9—24no—
175C2×D9C22×C3×C414yes—
176C4×C9—28no—
177C4×C9—24no—
178C4×C9—24no—
179C4×C9C22×C3×C414yes—
180Dic9—24no—
181Dic9C22×C3×C414yes—
182C23×C4—91no—
183#117—98no—
184C2×C4⋊C4—92no—
185C2×C22⋊C4—92no—
186C2×C42—91no—
187C2×C42C3×D911yescharacteristic, complemented
188C3×C9C2×#11711yescharacteristic, complemented, Sylow
189C23×C3—21no—
190C23×C3—94no—
191C23×C3—92no—
192C23×C3—184no—
193C23×C3C22×D911yescharacteristic
194C23×C3C22×C3×D311yescharacteristic
195C22×D3—32no—
196C22×D3—34no—
197C22×D3—64no—
198C2×C3×C4—28no—
199C2×C3×C4—24no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—24no—
203C2×C3×C4—22no—
204C2×C3×C4—94no—
205C2×C3×C4—92no—
206C2×C3×C4—916no—
207C2×C3×C4—94no—
208C2×C3×C4—184no—
209C2×C3×C4C4×D914yes—
210C2×C3×C4C22×D914yes—
211C2×C3×C4C22×D912yes—
212C2×C3×C4C3×C4×D314yes—
213C2×C3×C4C22×C3×D314yes—
214C2×C3×C4C22×C3×D312yes—
215C2×Dic3—32no—
216C2×Dic3—34no—
217C2×Dic3—34no—
218C2×Dic3—64no—
219C4×D3—316no—
220C2×C32#87214yes—
221C2×C32C22×C4×D311yescharacteristic
222C2×C32C2×#23512yes—
223C3×D3—64no—
224C2×C9—21no—
225C2×C9—22no—
226C2×C9—24no—
227C2×C9C3×#11714yescomplemented
228C2×C9C23×C3×C411yescharacteristic
229C2×C9C2×C3×#4112yes—
230D9—24no—
231C24—91no—
232C22×C4—91no—
233C22×C4—98no—
234C22×C4—94no—
235C22×C4—92no—
236C22×C4C2×C3×D912yes—
237C22×C4C2×C3×D911yescharacteristic
238C4⋊C4—98no—
239C22⋊C4—98no—
240C42—94no—
241C42C2×C3×D914yescomplemented
242C22×C3—21no—
243C22×C3—22no—
244C22×C3—24no—
245C22×C3—94no—
246C22×C3—188no—
247C22×C3—184no—
248C22×C3C2×C4×D912yes—
249C22×C3#369714yes—
250C22×C3C23×D911yescharacteristic
251C22×C3C2×C3×C4×D312yes—
252C22×C3C3×#23514yes—
253C22×C3C23×C3×D311yescharacteristic
254C2×D3—34no—
255C2×D3—64no—
256C2×D3—68no—
257C3×C4—28no—
258C3×C4—24no—
259C3×C4—24no—
260C3×C4—24no—
261C3×C4—94no—
262C3×C4—184no—
263C3×C4C2×C4×D914yes—
264C3×C4C2×C3×C4×D314yes—
265Dic3—34no—
266Dic3—64no—
267C32C2×#87211yescharacteristic
268C9—21no—
269C9C2×C3×#11711yescharacteristic, complemented
270C23—92no—
271C23—94no—
272C23—184no—
273C23C22×C3×D911yescharacteristic
274C2×C4—24no—
275C2×C4—92no—
276C2×C4—94no—
277C2×C4—916no—
278C2×C4—94no—
279C2×C4—184no—
280C2×C4C3×C4×D914yes—
281C2×C4C22×C3×D914yes—
282C2×C4C22×C3×D912yes—
283C2×C3—21no—
284C2×C3—22no—
285C2×C3—24no—
286C2×C3—184no—
287C2×C3#6343414yesdirect product factor
288C2×C3C22×C4×D911yescharacteristic
289C2×C3C2×#369712yes—
290C2×C3C3×#87214yes—
291C2×C3C22×C3×C4×D311yescharacteristic
292C2×C3C2×C3×#23512yes—
293D3—64no—
294C22—94no—
295C22—184no—
296C22—188no—
297C22C2×C3×C4×D912yes—
298C22C3×#369714yes—
299C22C23×C3×D911yescharacteristic
300C4—24no—
301C4—94no—
302C4—184no—
303C4C2×C3×C4×D914yes—
304C3—21no—
305C3C2×#6343411yescharacteristic, direct product factor
306C3C2×C3×#87211yescharacteristic
307C2—184no—
308C2C3×#6343414yesdirect product factor
309C2C22×C3×C4×D911yescharacteristic
310C2C2×C3×#369712yes—
subgroupG/H
centerC22×C3×C4C2×D9
commutatorC2×C9C23×C3×C4
FrattiniC22×C3C23×C3×D3
FittingC2×C3×C42×C9C2
socleC23×C32C22×D3
prime radicalC23×C3×D9C22
cosocleC22×C9C24×C3
hypercenterC2×C3×C42D9
hypocenterC9C2×C3×#117
LCSG ▷ C2×C9 ▷ C9
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
2. C2×C3×C42×C9C2semidirect
4. C2×C3×Dic9⋊C4C2semidirect
5. C2×C3×C4×Dic9C2semidirect
6. C3×#63434C2direct (N × H)
8. C2×#63434C3direct (N × H)
14. C2×C3×C4×D9C4semidirect
16. C3×C42×C9C22semidirect
18. C3×Dic9⋊C4C22semidirect
19. C3×C4×Dic9C22semidirect
28. C2×C42×C9C2×C3semidirect
30. C2×Dic9⋊C4C2×C3semidirect
31. C2×C4×Dic9C2×C3semidirect
32. #63434C2×C3direct (N × H)
45. C3×C4×D9C2×C4semidirect
66. C2×C4×D9C3×C4semidirect
69. C42×C9C22×C3semidirect
71. Dic9⋊C4C22×C3semidirect
72. C4×Dic9C22×C3semidirect
90. C2×C3×C42D9semidirect
126. C4×D9C2×C3×C4semidirect
128. C2×C3×C9#117semidirect
152. C3×C42C2×D9semidirect
187. C2×C42C3×D9semidirect
188. C3×C9C2×#117semidirect
227. C2×C9C3×#117semidirect
241. C42C2×C3×D9semidirect
269. C9C2×C3×#117semidirect
287. C2×C3#63434direct (N × H)
305. C3C2×#63434direct (N × H)
308. C2C3×#63434direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×D92
C2×C3×C42×C92
C2×C3×(C2×D9)⋊C42
C2×C3×Dic9⋊C42
C2×C3×C4×Dic92
C3×#634342
C2×C3×#8723
C2×#634343