Finite Group Databasebeta

← C2×C3×#64286 #13181492

Subgroups — C2×C3×#64286 #13181492

Order 1728.

1240 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×C22⋊#157C212yesmaximal
3C3×#64286C218yesmaximal, direct product factor
4C2×C3×C4⋊#157C212yesmaximal, complemented
5C2×C3×C4×#157C212yesmaximal, complemented
6C2×#64286C311yesmaximal, characteristic, direct product factor
7C23×C3×C3⋊D3C2211yescharacteristic
8C22×C3×#157C2214yes—
9C22×C33×C4C411yescharacteristic, complemented
10C22×C3×C3⋊Dic3C411yescharacteristic, complemented
11C2×C3×C4×C3⋊D3C414yescomplemented
12C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
13C2×C3×C4×C3⋊D3C411yescharacteristic, complemented
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2214yes—
17C3×C22⋊#157C2218yes—
18C3×C4⋊#157C2218yescomplemented
19C3×C4×#157C2218yescomplemented
20C22×C4×C3⋊D3C2×C311yescharacteristic
21C22×C3×C4×D3—62no—
22C2×C22⋊#157C2×C312yes—
23#64286C2×C318yesdirect product factor
24C2×C4⋊#157C2×C312yescomplemented
25C2×C4×#157C2×C312yescomplemented
26C23×C33C2×C411yescharacteristic
27C22×C3×C3⋊D3—24no—
28C22×C3×C3⋊D3C2×C411yescharacteristic
29C22×C3×C3⋊D3C2×C412yes—
30C22×C3×C3⋊D3C2312yes—
31C22×C3×C3⋊D3C2311yescharacteristic
32C2×C3×#157—28no—
33C2×C3×#157C2318yes—
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4C2×C411yescharacteristic
36C2×C33×C4C2×C414yescomplemented
37C2×C3×C3⋊Dic3C2×C411yescharacteristic
38C2×C3×C3⋊Dic3C2×C411yescharacteristic
39C2×C3×C3⋊Dic3C2×C414yescomplemented
40C3×C4×C3⋊D3C2×C412yes—
41C3×C4×C3⋊D3C2×C412yes—
42C3×C4×C3⋊D3C2×C414yescomplemented
43C3×C4×C3⋊D3C2×C414yescomplemented
44C3×C4×C3⋊D3C2312yes—
45C3×C4×C3⋊D3C2312yes—
46C2×C3×#117—91nomaximal
47C23×C3⋊D3C22×C311yescharacteristic
48C23×C3×D3—62no—
49C22×#157C22×C314yes—
50C22×C32×C4—22no—
51C22×C32×C4—42no—
52C22×C32×C4C3×C411yescharacteristic, complemented
53C22×C3⋊Dic3C3×C411yescharacteristic, complemented
54C2×C4×C3⋊D3C3×C414yescomplemented
55C2×C4×C3⋊D3C3×C411yescharacteristic, complemented
56C2×C4×C3⋊D3C3×C411yescharacteristic, complemented
57C2×C4×C3⋊D3C22×C311yescharacteristic
58C2×C4×C3⋊D3C22×C311yescharacteristic
59C2×C4×C3⋊D3C22×C314yes—
60C22×C3×Dic3—62no—
61C2×C3×C4×D3—68no—
62C2×C3×C4×D3—62no—
63C2×C3×C4×D3—62no—
64C2×C3×C4×D3—62no—
65C2×C3×C4×D3—62no—
66C2×C3×C4×D3—68no—
67C22⋊#157C22×C318yes—
68C4⋊#157C22×C318yescomplemented
69C4×#157C22×C318yescomplemented
70C22×C33—22no—
71C22×C33C22×C412yes—
72C22×C33C22×C411yescharacteristic
73C2×C3×C3⋊D3—22no—
74C2×C3×C3⋊D3—24no—
75C2×C3×C3⋊D3—24no—
76C2×C3×C3⋊D3C22×C412yes—
77C2×C3×C3⋊D3C22×C411yescharacteristic
78C2×C3×C3⋊D3#4114yes—
79C2×C3×C3⋊D3C2411yescharacteristic
80C3×#157—28no—
81C33×C4C22×C412yes—
82C33×C4C22×C412yes—
83C3×C3⋊Dic3C22×C412yes—
84C3×C3⋊Dic3C22×C412yes—
85C23×C3×C4—91no—
86C22×C4×D3—62no—
87C3×#117—98no—
88C2×C3×C4⋊C4—92no—
89C2×C3×C22⋊C4—92no—
90C2×C3×C42—92no—
91C23×C32—22no—
92C23×C32—42no—
93C23×C32C2×C3×C411yescharacteristic
94C22×C3⋊D3—24no—
95C22×C3⋊D3C2×C3×C411yescharacteristic
96C22×C3⋊D3C2×C3×C412yes—
97C22×C3⋊D3C23×C312yes—
98C22×C3⋊D3C23×C311yescharacteristic
99C22×C3×D3—64no—
100C22×C3×D3—62no—
101C22×C3×D3—62no—
102C22×C3×D3—64no—
103C22×C3×D3—616no—
104C2×#157—28no—
105C2×#157C23×C318yes—
106C2×C32×C4—22no—
107C2×C32×C4—22no—
108C2×C32×C4—28no—
109C2×C32×C4—42no—
110C2×C32×C4—42no—
111C2×C32×C4—48no—
112C2×C32×C4C2×C3×C411yescharacteristic
113C2×C32×C4C2×C3×C411yescharacteristic
114C2×C32×C4C2×C3×C414yescomplemented
115C2×C3⋊Dic3C2×C3×C411yescharacteristic
116C2×C3⋊Dic3C2×C3×C411yescharacteristic
117C2×C3⋊Dic3C2×C3×C414yescomplemented
118C4×C3⋊D3C2×C3×C412yes—
119C4×C3⋊D3C2×C3×C412yes—
120C4×C3⋊D3C2×C3×C414yescomplemented
121C4×C3⋊D3C2×C3×C414yescomplemented
122C4×C3⋊D3C23×C312yes—
123C4×C3⋊D3C23×C312yes—
124C2×C3×Dic3—62no—
125C2×C3×Dic3—62no—
126C2×C3×Dic3—68no—
127C3×C4×D3—64no—
128C3×C4×D3—64no—
129C3×C4×D3—68no—
130C3×C4×D3—68no—
131C3×C4×D3—64no—
132C3×C4×D3—64no—
133C2×#117—91noSylow
134C2×C33—22no—
135C2×C33#11712yescomplemented
136C2×C33C23×C411yescharacteristic
137C3×C3⋊D3—22no—
138C3×C3⋊D3#11712yescomplemented
139C3×C3⋊D3C2×#4112yes—
140C24×C3—91no—
141C23×D3—62no—
142C22×C3×C4—22no—
143C22×C3×C4—42no—
144C22×C3×C4—91no—
145C22×C3×C4—91no—
146C22×C3×C4—91no—
147C22×C3×C4—91no—
148C22×C3×C4—91no—
149C22×C3×C4—94no—
150C22×C3×C4—94no—
151C22×C3×C4—94no—
152C22×C3×C4#15711yescharacteristic, complemented
153C22×Dic3—62no—
154C2×C4×D3—62no—
155C2×C4×D3—62no—
156C2×C4×D3—68no—
157C2×C4×D3—62no—
158C2×C4×D3—62no—
159C2×C4×D3—68no—
160C3×C4⋊C4—98no—
161C3×C22⋊C4—98no—
162C3×C42—98no—
163C22×C32—22no—
164C22×C32—28no—
165C22×C32—24no—
166C22×C32—22no—
167C22×C32—48no—
168C22×C32—44no—
169C22×C32—42no—
170C22×C32C22×C3×C412yes—
171C22×C32C22×C3×C411yescharacteristic
172C2×C3⋊D3—22no—
173C2×C3⋊D3—24no—
174C2×C3⋊D3—24no—
175C2×C3⋊D3C22×C3×C412yes—
176C2×C3⋊D3C22×C3×C411yescharacteristic
177C2×C3⋊D3C3×#4114yes—
178C2×C3⋊D3C24×C311yescharacteristic
179C2×C3×D3—62no—
180C2×C3×D3—64no—
181C2×C3×D3—68no—
182C2×C3×D3—62no—
183C2×C3×D3—616no—
184C2×C3×D3—616no—
185C2×C3×D3—68no—
186#157—28no—
187C32×C4—24no—
188C32×C4—24no—
189C32×C4—44no—
190C32×C4—44no—
191C32×C4C22×C3×C412yes—
192C32×C4C22×C3×C412yes—
193C3⋊Dic3C22×C3×C412yes—
194C3⋊Dic3C22×C3×C412yes—
195C3×Dic3—64no—
196C3×Dic3—64no—
197C23×C4—91no—
198#117—98no—
199C2×C4⋊C4—92no—
200C2×C22⋊C4—92no—
201C2×C42—92no—
202C33C2×#11711yescharacteristic, complemented, Sylow
203C23×C3—22no—
204C23×C3—42no—
205C23×C3—91no—
206C23×C3—92no—
207C23×C3—92no—
208C23×C3—91no—
209C23×C3—184no—
210C23×C3C2×#15711yescharacteristic
211C22×D3—64no—
212C22×D3—62no—
213C22×D3—62no—
214C22×D3—64no—
215C22×D3—616no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—28no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—48no—
222C2×C3×C4—94no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—92no—
226C2×C3×C4—92no—
227C2×C3×C4—92no—
228C2×C3×C4—94no—
229C2×C3×C4—92no—
230C2×C3×C4—94no—
231C2×C3×C4—98no—
232C2×C3×C4—188no—
233C2×C3×C4C2×#15711yescharacteristic
234C2×C3×C4C2×#15711yescharacteristic
235C2×C3×C4C2×#15714yescomplemented
236C2×Dic3—62no—
237C2×Dic3—62no—
238C2×Dic3—68no—
239C4×D3—64no—
240C4×D3—64no—
241C4×D3—68no—
242C4×D3—68no—
243C4×D3—64no—
244C4×D3—64no—
245C2×C32—22no—
246C2×C32—28no—
247C2×C32—24no—
248C2×C32—22no—
249C2×C32—48no—
250C2×C32—44no—
251C2×C32—42no—
252C2×C32C3×#11712yescomplemented
253C2×C32C23×C3×C411yescharacteristic
254C3⋊D3—22no—
255C3⋊D3C3×#11712yescomplemented
256C3⋊D3C2×C3×#4112yes—
257C3×D3—64no—
258C3×D3—64no—
259C3×D3—68no—
260C24—91no—
261C22×C4—91no—
262C22×C4—91no—
263C22×C4—91no—
264C22×C4—91no—
265C22×C4—91no—
266C22×C4—94no—
267C22×C4—94no—
268C22×C4—94no—
269C22×C4C3×#15711yescharacteristic, complemented
270C4⋊C4—98no—
271C22⋊C4—98no—
272C42—98no—
273C22×C3—28no—
274C22×C3—22no—
275C22×C3—24no—
276C22×C3—22no—
277C22×C3—48no—
278C22×C3—44no—
279C22×C3—42no—
280C22×C3—94no—
281C22×C3—91no—
282C22×C3—91no—
283C22×C3—92no—
284C22×C3—184no—
285C22×C3—184no—
286C22×C3—182no—
287C22×C3C22×#15712yes—
288C22×C3C22×#15711yescharacteristic
289C2×D3—62no—
290C2×D3—64no—
291C2×D3—62no—
292C2×D3—68no—
293C2×D3—616no—
294C2×D3—616no—
295C2×D3—68no—
296C3×C4—24no—
297C3×C4—24no—
298C3×C4—44no—
299C3×C4—44no—
300C3×C4—92no—
301C3×C4—92no—
302C3×C4—188no—
303C3×C4C22×#15712yes—
304C3×C4C22×#15712yes—
305Dic3—64no—
306Dic3—64no—
307C32—22no—
308C32—42no—
309C32C2×C3×#11711yescharacteristic, complemented
310C23—91no—
311C23—92no—
312C23—92no—
313C23—91no—
314C23—184no—
315C23C2×C3×#15711yescharacteristic
316C2×C4—94no—
317C2×C4—91no—
318C2×C4—91no—
319C2×C4—92no—
320C2×C4—92no—
321C2×C4—92no—
322C2×C4—92no—
323C2×C4—94no—
324C2×C4—94no—
325C2×C4—98no—
326C2×C4—188no—
327C2×C4C2×C3×#15711yescharacteristic
328C2×C4C2×C3×#15711yescharacteristic
329C2×C4C2×C3×#15714yescomplemented
330C2×C3—28no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—24no—
334C2×C3—48no—
335C2×C3—42no—
336C2×C3—44no—
337C2×C3—92no—
338C2×C3—92no—
339C2×C3—182no—
340C2×C3#6428612yesdirect product factor
341C2×C3C23×#15711yescharacteristic
342D3—64no—
343D3—64no—
344D3—68no—
345C22—22no—
346C22—91no—
347C22—91no—
348C22—94no—
349C22—92no—
350C22—184no—
351C22—184no—
352C22—182no—
353C22C22×C3×#15712yes—
354C22C22×C3×#15711yescharacteristic
355C4—92no—
356C4—92no—
357C4—188no—
358C4C22×C3×#15712yes—
359C4C22×C3×#15712yes—
360C3—22no—
361C3—42no—
362C3C2×#6428611yescharacteristic, direct product factor
363C2—22no—
364C2—92no—
365C2—92no—
366C2—182no—
367C2C3×#6428612yesdirect product factor
368C2C23×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⋊D3C22
cosocleC2×C3⋊D3C24×C3
hypercenterC22×C3×C4#157
hypocenterC32C2×C3×#117
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
3. C3×#64286C2direct (N × H)
4. C2×C3×C4⋊#157C2semidirect
5. C2×C3×C4×#157C2semidirect
6. C2×#64286C3direct (N × H)
9. C22×C33×C4C4semidirect
10. C22×C3×C3⋊Dic3C4semidirect
11. C2×C3×C4×C3⋊D3C4semidirect
12. C2×C3×C4×C3⋊D3C4semidirect
13. C2×C3×C4×C3⋊D3C4semidirect
18. C3×C4⋊#157C22semidirect
19. C3×C4×#157C22semidirect
23. #64286C2×C3direct (N × H)
24. C2×C4⋊#157C2×C3semidirect
25. C2×C4×#157C2×C3semidirect
36. C2×C33×C4C2×C4semidirect
39. C2×C3×C3⋊Dic3C2×C4semidirect
42. C3×C4×C3⋊D3C2×C4semidirect
43. C3×C4×C3⋊D3C2×C4semidirect
52. C22×C32×C4C3×C4semidirect
53. C22×C3⋊Dic3C3×C4semidirect
54. C2×C4×C3⋊D3C3×C4semidirect
55. C2×C4×C3⋊D3C3×C4semidirect
56. C2×C4×C3⋊D3C3×C4semidirect
68. C4⋊#157C22×C3semidirect
69. C4×#157C22×C3semidirect
114. C2×C32×C4C2×C3×C4semidirect
117. C2×C3⋊Dic3C2×C3×C4semidirect
120. C4×C3⋊D3C2×C3×C4semidirect
121. C4×C3⋊D3C2×C3×C4semidirect
135. C2×C33#117semidirect
138. C3×C3⋊D3#117semidirect
152. C22×C3×C4#157semidirect
202. C33C2×#117semidirect
235. C2×C3×C4C2×#157semidirect
252. C2×C32C3×#117semidirect
255. C3⋊D3C3×#117semidirect
269. C22×C4C3×#157semidirect
309. C32C2×C3×#117semidirect
329. C2×C4C2×C3×#157semidirect
340. C2×C3#64286direct (N × H)
362. C3C2×#64286direct (N × H)
367. C2C3×#64286direct (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×C22⋊#1572
C3×#642862
C2×C3×C4⋊#1572
C2×C3×C4×#1572
C2×#642863
C2×C3×#1179