Finite Group Databasebeta

← C2×C3×#64283 #13181489

Subgroups — C2×C3×#64283 #13181489

Order 1728.

1176 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3792C212yesmaximal, complemented
3C3×#64283C218yesmaximal, direct product factor
4C2×C3×#3788C212yesmaximal, complemented
5C2×C3×#3787C212yesmaximal, complemented
6C2×#64283C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C411yescharacteristic
8C22×C33×C4C411yescharacteristic
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×C3×C4×C3⋊D3C414yes—
11C2×C3×C4×C3⋊D3C411yescharacteristic
12C2×C3×C4×C3⋊D3C411yescharacteristic
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2214yes—
16C3×#3792C2218yescomplemented
17C2×C3×#622C2214yescomplemented
18C3×#3788C2218yescomplemented
19C3×#3787C2218yescomplemented
20C22×C4×C3⋊D3C2×C311yescharacteristic
21C22×C3×C4×D3—62no—
22C2×#3792C2×C312yescomplemented
23#64283C2×C318yesdirect product factor
24C2×#3788C2×C312yescomplemented
25C2×#3787C2×C312yescomplemented
26C23×C33C2×C411yescharacteristic
27C22×C3×C3⋊D3—24no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C22×C3×C3⋊D3C2×C414yes—
30C22×C3×C3⋊D3C2×C411yescharacteristic
31C2×C33×C4C2×C414yes—
32C2×C33×C4C2×C411yescharacteristic
33C2×C33×C4C2×C411yescharacteristic
34C2×C3×C3⋊Dic3C2×C412yes—
35C2×C3×C3⋊Dic3C2×C411yescharacteristic
36C2×C3×C3⋊Dic3C2311yescharacteristic
37C2×C3×C3⋊Dic3C2312yes—
38C3×C4×C3⋊D3C2×C412yes—
39C3×C4×C3⋊D3C2×C414yes—
40C3×C4×C3⋊D3C2×C412yes—
41C3×C4×C3⋊D3C2×C414yes—
42C3×C4×C3⋊D3C2312yes—
43C3×C4×C3⋊D3C2312yes—
44C3×#622C2318yescomplemented
45C22×C3×M16—91nomaximal
46C23×C3⋊D3C3×C411yescharacteristic
47C23×C3×D3—62no—
48C22×C32×C4—22no—
49C22×C32×C4—42no—
50C22×C32×C4C3×C411yescharacteristic
51C22×C3⋊Dic3C22×C311yescharacteristic
52C2×C4×C3⋊D3C3×C414yes—
53C2×C4×C3⋊D3C3×C411yescharacteristic
54C2×C4×C3⋊D3C3×C411yescharacteristic
55C2×C4×C3⋊D3C22×C311yescharacteristic
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C4×C3⋊D3C22×C314yes—
58C22×C3×Dic3—62no—
59C2×C3×C4×D3—62no—
60C2×C3×C4×D3—68no—
61C2×C3×C4×D3—62no—
62C2×C3×C4×D3—62no—
63C2×C3×C4×D3—68no—
64C2×C3×C4×D3—62no—
65#3792C22×C318yescomplemented
66C2×#622C22×C314yescomplemented
67#3788C22×C318yescomplemented
68#3787C22×C318yescomplemented
69C22×C33—22no—
70C22×C33C22×C412yes—
71C22×C33C22×C411yescharacteristic
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3—28no—
74C2×C3×C3⋊D3M1614yescomplemented
75C2×C3×C3⋊D3C22×C412yes—
76C2×C3×C3⋊D3C22×C412yes—
77C33×C4C22×C412yes—
78C33×C4C22×C412yes—
79C3×C3⋊Dic3C22×C411yescharacteristic
80C3×C3⋊Dic3C22×C412yes—
81C3×C3⋊Dic3C2411yescharacteristic
82C23×C3×C4—91no—
83C22×C4×D3—62no—
84C2×C3×M16—92no—
85C2×C3×M16—98no—
86C2×C3×M16—92no—
87C22×C3×C8—92no—
88C23×C32—22no—
89C23×C32—42no—
90C23×C32C2×C3×C411yescharacteristic
91C22×C3⋊D3—24no—
92C22×C3⋊D3C2×C3×C411yescharacteristic
93C22×C3⋊D3C2×C3×C414yes—
94C22×C3⋊D3C2×C3×C411yescharacteristic
95C22×C3×D3—62no—
96C22×C3×D3—68no—
97C22×C3×D3—62no—
98C22×C3×D3—616no—
99C2×C32×C4—22no—
100C2×C32×C4—28no—
101C2×C32×C4—22no—
102C2×C32×C4—48no—
103C2×C32×C4—42no—
104C2×C32×C4—42no—
105C2×C32×C4C2×C3×C414yes—
106C2×C32×C4C2×C3×C411yescharacteristic
107C2×C32×C4C2×C3×C411yescharacteristic
108C2×C3⋊Dic3C2×C3×C412yes—
109C2×C3⋊Dic3C2×C3×C411yescharacteristic
110C2×C3⋊Dic3C23×C311yescharacteristic
111C2×C3⋊Dic3C23×C312yes—
112C4×C3⋊D3C2×C3×C412yes—
113C4×C3⋊D3C2×C3×C414yes—
114C4×C3⋊D3C2×C3×C412yes—
115C4×C3⋊D3C2×C3×C414yes—
116C4×C3⋊D3C23×C312yes—
117C4×C3⋊D3C23×C312yes—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—62no—
120C2×C3×Dic3—62no—
121C2×C3×Dic3—64no—
122C3×C4×D3—68no—
123C3×C4×D3—64no—
124C3×C4×D3—64no—
125C3×C4×D3—68no—
126C3×C4×D3—64no—
127C3×C4×D3—64no—
128#622C23×C318yescomplemented
129C22×M16—91noSylow
130C2×C33—22no—
131C2×C33C2×M1612yescomplemented
132C2×C33C23×C411yescharacteristic
133C3×C3⋊D3—22no—
134C3×C3⋊D3C2×M1614yescomplemented
135C24×C3—91no—
136C23×D3—62no—
137C22×C3×C4—22no—
138C22×C3×C4—42no—
139C22×C3×C4—91no—
140C22×C3×C4—91no—
141C22×C3×C4—94no—
142C22×C3×C4—91no—
143C22×C3×C4—94no—
144C22×C3×C4—91no—
145C22×C3×C4—91no—
146C22×C3×C4#15711yescharacteristic
147C22×Dic3—62no—
148C2×C4×D3—62no—
149C2×C4×D3—68no—
150C2×C4×D3—62no—
151C2×C4×D3—62no—
152C2×C4×D3—62no—
153C2×C4×D3—68no—
154C3×M16—98no—
155C3×M16—98no—
156C2×C3×C8—98no—
157C2×C3×C8—94no—
158C22×C32—22no—
159C22×C32—28no—
160C22×C32—24no—
161C22×C32—22no—
162C22×C32—48no—
163C22×C32—44no—
164C22×C32—42no—
165C22×C32C22×C3×C412yes—
166C22×C32C22×C3×C411yescharacteristic
167C2×C3⋊D3—22no—
168C2×C3⋊D3—28no—
169C2×C3⋊D3C3×M1614yescomplemented
170C2×C3⋊D3C22×C3×C412yes—
171C2×C3⋊D3C22×C3×C412yes—
172C2×C3×D3—68no—
173C2×C3×D3—64no—
174C2×C3×D3—64no—
175C2×C3×D3—632no—
176C2×C3×D3—68no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—44no—
180C32×C4—44no—
181C32×C4C22×C3×C412yes—
182C32×C4C22×C3×C412yes—
183C3⋊Dic3C22×C3×C411yescharacteristic
184C3⋊Dic3C22×C3×C412yes—
185C3⋊Dic3C24×C311yescharacteristic
186C3×Dic3—62no—
187C3×Dic3—64no—
188C3×Dic3—62no—
189C23×C4—91no—
190C2×M16—92no—
191C2×M16—98no—
192C2×M16—92no—
193C22×C8—92no—
194C33C22×M1611yescharacteristic, complemented, Sylow
195C23×C3—22no—
196C23×C3—42no—
197C23×C3—91no—
198C23×C3—94no—
199C23×C3—91no—
200C23×C3—184no—
201C23×C3C2×#15711yescharacteristic
202C22×D3—62no—
203C22×D3—68no—
204C22×D3—62no—
205C22×D3—616no—
206C2×C3×C4—28no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—48no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—91no—
213C2×C3×C4—92no—
214C2×C3×C4—92no—
215C2×C3×C4—91no—
216C2×C3×C4—92no—
217C2×C3×C4—92no—
218C2×C3×C4—94no—
219C2×C3×C4—92no—
220C2×C3×C4—92no—
221C2×C3×C4—94no—
222C2×C3×C4C2×#15711yescharacteristic
223C2×C3×C4C2×#15714yes—
224C2×C3×C4C2×#15711yescharacteristic
225C2×Dic3—64no—
226C2×Dic3—62no—
227C2×Dic3—62no—
228C2×Dic3—64no—
229C4×D3—64no—
230C4×D3—68no—
231C4×D3—64no—
232C4×D3—68no—
233C4×D3—64no—
234C4×D3—64no—
235C3×C8—98no—
236C2×C32—22no—
237C2×C32—28no—
238C2×C32—24no—
239C2×C32—22no—
240C2×C32—48no—
241C2×C32—44no—
242C2×C32—42no—
243C2×C32C2×C3×M1612yescomplemented
244C2×C32C23×C3×C411yescharacteristic
245C3⋊D3—22no—
246C3⋊D3C2×C3×M1614yescomplemented
247C3×D3—68no—
248C3×D3—68no—
249C24—91no—
250C22×C4—91no—
251C22×C4—91no—
252C22×C4—91no—
253C22×C4—94no—
254C22×C4—91no—
255C22×C4—94no—
256C22×C4—91no—
257C22×C4C3×#15711yescharacteristic
258M16—98no—
259M16—98no—
260C2×C8—98no—
261C2×C8—94no—
262C22×C3—28no—
263C22×C3—22no—
264C22×C3—24no—
265C22×C3—22no—
266C22×C3—48no—
267C22×C3—44no—
268C22×C3—42no—
269C22×C3—94no—
270C22×C3—92no—
271C22×C3—92no—
272C22×C3—188no—
273C22×C3—182no—
274C22×C3C22×#15712yes—
275C22×C3C22×#15711yescharacteristic
276C2×D3—64no—
277C2×D3—64no—
278C2×D3—68no—
279C2×D3—632no—
280C2×D3—68no—
281C3×C4—24no—
282C3×C4—24no—
283C3×C4—44no—
284C3×C4—44no—
285C3×C4—91no—
286C3×C4—91no—
287C3×C4—92no—
288C3×C4C22×#15712yes—
289C3×C4C22×#15712yes—
290Dic3—62no—
291Dic3—64no—
292Dic3—62no—
293C32—22no—
294C32—42no—
295C32C22×C3×M1611yescharacteristic, complemented
296C23—91no—
297C23—94no—
298C23—91no—
299C23—184no—
300C23C2×C3×#15711yescharacteristic
301C2×C4—91no—
302C2×C4—92no—
303C2×C4—92no—
304C2×C4—91no—
305C2×C4—92no—
306C2×C4—92no—
307C2×C4—92no—
308C2×C4—94no—
309C2×C4—92no—
310C2×C4—94no—
311C2×C4C2×C3×#15714yes—
312C2×C4C2×C3×#15711yescharacteristic
313C2×C4C2×C3×#15711yescharacteristic
314C8—98no—
315C2×C3—28no—
316C2×C3—22no—
317C2×C3—22no—
318C2×C3—24no—
319C2×C3—48no—
320C2×C3—42no—
321C2×C3—44no—
322C2×C3—94no—
323C2×C3—182no—
324C2×C3#6428312yesdirect product factor
325C2×C3C23×#15711yescharacteristic
326D3—68no—
327D3—68no—
328C22—22no—
329C22—92no—
330C22—94no—
331C22—92no—
332C22—188no—
333C22—182no—
334C22C22×C3×#15712yes—
335C22C22×C3×#15711yescharacteristic
336C4—91no—
337C4—91no—
338C4—92no—
339C4C22×C3×#15712yes—
340C4C22×C3×#15712yes—
341C3—22no—
342C3—42no—
343C3C2×#6428311yescharacteristic, direct product factor
344C2—22no—
345C2—94no—
346C2—182no—
347C2C3×#6428312yesdirect product factor
348C2C23×C3×#15711yescharacteristic
subgroupG/H
centerC2×C3×C4C2×#157
commutatorC2×C32C23×C3×C4
FrattiniC2C23×C3×#157
FittingC22×C33×C4C4
socleC22×C33C22×C4
prime radicalC23×C3×C3⋊D3C4
cosocleC3⋊Dic3C24×C3
hypercenterC22×C3×C4#157
hypocenterC32C22×C3×M16
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3792C2semidirect
3. C3×#64283C2direct (N × H)
4. C2×C3×#3788C2semidirect
5. C2×C3×#3787C2semidirect
6. C2×#64283C3direct (N × H)
16. C3×#3792C22semidirect
17. C2×C3×#622C22semidirect
18. C3×#3788C22semidirect
19. C3×#3787C22semidirect
22. C2×#3792C2×C3semidirect
23. #64283C2×C3direct (N × H)
24. C2×#3788C2×C3semidirect
25. C2×#3787C2×C3semidirect
44. C3×#622C23semidirect
65. #3792C22×C3semidirect
66. C2×#622C22×C3semidirect
67. #3788C22×C3semidirect
68. #3787C22×C3semidirect
74. C2×C3×C3⋊D3M16semidirect
128. #622C23×C3semidirect
131. C2×C33C2×M16semidirect
134. C3×C3⋊D3C2×M16semidirect
169. C2×C3⋊D3C3×M16semidirect
194. C33C22×M16semidirect
243. C2×C32C2×C3×M16semidirect
246. C3⋊D3C2×C3×M16semidirect
295. C32C22×C3×M16semidirect
324. C2×C3#64283direct (N × H)
343. C3C2×#64283direct (N × H)
347. C2C3×#64283direct (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×C3⋊D32
C2×C3×#37922
C3×#642832
C2×C3×#37882
C2×C3×#37872
C2×#642833
C22×C3×M169