Finite Group Databasebeta

← C3×C4×#872 #11800765

Subgroups — C3×C4×#872 #11800765

Order 1152.

1633 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#872C212yesmaximal
2C2×C3×C42×D3C211yesmaximal, characteristic
3C32×C43C211yesmaximal, characteristic, complemented
4C3×C4×(C2×D3)⋊C4C214yesmaximal
5C3×#4932C212yesmaximal, complemented
6C3×C4×Dic3⋊C4C214yesmaximal, complemented
7C3×C42×Dic3C211yesmaximal, characteristic, complemented
8C3×C4×#117—31nomaximal
9C4×#872C311yesmaximal, characteristic, direct product factor
10C22×C3×C4×D3C2212yes—
11C22×C3×C4×D3C2211yescharacteristic
12C2×C32×C42C2214yes—
13C2×C32×C42C2211yescharacteristic
14C2×C32×C42C2212yes—
15C2×C3×(C2×D3)⋊C4C2214yes—
16C2×C3×Dic3⋊C4C2214yes—
17C2×C3×C4×Dic3C2214yes—
18C2×C3×C4×Dic3C2212yes—
19C2×C3×C4×Dic3C2211yescharacteristic
20C2×C3×C4×Dic3C2212yes—
21C3×#872C4116yesdirect product factor
22C3×C42×D3C418yescomplemented
23C3×C3⋊#95C2218yes—
24C2×C3×#117—32no—
25C22×C3×C42—31no—
26C2×#872C2×C312yes—
27C2×C42×D3C2×C311yescharacteristic
28C3×C4×C4⋊C4—34no—
29C3×C4×C22⋊C4—34no—
30C3×#376—32no—
31C3×C43—21no—
32C3×C43—31no—
33C3×C43D311yescharacteristic, complemented
34C3×C43C2×C311yescharacteristic, complemented
35C4×(C2×D3)⋊C4C2×C314yes—
36#4932C2×C312yescomplemented
37C4×Dic3⋊C4C2×C314yescomplemented
38C42×Dic3C2×C311yescharacteristic, complemented
39C23×C3×D3C2311yescharacteristic
40C22×C32×C4C2314yes—
41C22×C32×C4C2311yescharacteristic
42C22×C32×C4C2312yes—
43C22×C3×Dic3C2314yes—
44C22×C3×Dic3C2312yes—
45C22×C3×Dic3C2311yescharacteristic
46C2×C3×C4×D3—28no—
47C2×C3×C4×D3C2×C418yes—
48C2×C3×C4×D3C2×C414yes—
49C2×C3×C4×D3C2×C418yes—
50C32×C42—28no—
51C32×C42C2×C414yes—
52C32×C42C2×C418yescomplemented
53C3×(C2×D3)⋊C4C2×C4116yes—
54C3×Dic3⋊C4C2×C4116yescomplemented
55C3×C4×Dic3—28no—
56C3×C4×Dic3C2×C414yes—
57C3×C4×Dic3C2×C418yes—
58C3×C4×Dic3C2×C418yescomplemented
59C4×#117—31noSylow
60C23×C3×C4—31no—
61C23×C3×C4—32no—
62C22×C4×D3C22×C312yes—
63C22×C4×D3C22×C311yescharacteristic
64C3×#117—316no—
65C2×C3×C4⋊C4—34no—
66C2×C3×C22⋊C4—34no—
67C2×C3×C42—24no—
68C2×C3×C42—21no—
69C2×C3×C42—22no—
70C2×C3×C42—32no—
71C2×C3×C42—31no—
72C2×C3×C42—32no—
73C2×C3×C42—38no—
74C2×C3×C42—34no—
75C2×C3×C42C2×D314yes—
76C2×C3×C42C2×D312yes—
77C2×C3×C42C2×D311yescharacteristic
78C2×C3×C42C22×C314yes—
79C2×C3×C42C22×C311yescharacteristic
80C2×C3×C42C22×C312yes—
81C2×(C2×D3)⋊C4C22×C314yes—
82C2×Dic3⋊C4C22×C314yes—
83C2×C4×Dic3C22×C314yes—
84C2×C4×Dic3C22×C312yes—
85C2×C4×Dic3C22×C311yescharacteristic
86C2×C4×Dic3C22×C312yes—
87#872C3×C4116yesdirect product factor
88C42×D3C3×C418yescomplemented
89C3×#95—38no—
90C3⋊#95C22×C318yes—
91C23×C32C2411yescharacteristic
92C22×C3×D3—24no—
93C22×C3×D3C22×C414yes—
94C22×C3×D3C22×C412yes—
95C2×C32×C4—28no—
96C2×C32×C4C22×C418yes—
97C2×C32×C4C22×C412yes—
98C2×C32×C4C22×C414yes—
99C2×C32×C4C22×C414yes—
100C2×C32×C4#4118yes—
101C2×C3×Dic3—28no—
102C2×C3×Dic3—24no—
103C2×C3×Dic3C22×C414yes—
104C2×C3×Dic3C22×C418yes—
105C2×C3×Dic3C22×C412yes—
106C2×C3×Dic3C22×C414yes—
107C3×C4×D3—216no—
108C3×C4×D3C42116yescomplemented
109C2×#117—32no—
110C22×C42—31no—
111C4×C4⋊C4—34no—
112C4×C22⋊C4—34no—
113#376—32no—
114C43—31no—
115C43C3×D311yescharacteristic, complemented
116C24×C3—31no—
117C23×D3C23×C311yescharacteristic
118C22×C3×C4—24no—
119C22×C3×C4—21no—
120C22×C3×C4—22no—
121C22×C3×C4—31no—
122C22×C3×C4—34no—
123C22×C3×C4—38no—
124C22×C3×C4—32no—
125C22×C3×C4—34no—
126C22×C3×C4—38no—
127C22×C3×C4—68no—
128C22×C3×C4C22×D314yes—
129C22×C3×C4C22×D311yescharacteristic
130C22×C3×C4C22×D312yes—
131C22×C3×C4C23×C314yes—
132C22×C3×C4C23×C311yescharacteristic
133C22×C3×C4C23×C312yes—
134C22×Dic3C23×C312yes—
135C22×Dic3C23×C314yes—
136C22×Dic3C23×C311yescharacteristic
137C2×C4×D3—28no—
138C2×C4×D3C2×C3×C418yes—
139C2×C4×D3C2×C3×C414yes—
140C2×C4×D3C2×C3×C418yes—
141C3×C4⋊C4—316no—
142C3×C22⋊C4—316no—
143C3×C42—216no—
144C3×C42—28no—
145C3×C42—28no—
146C3×C42—24no—
147C3×C42—28no—
148C3×C42—38no—
149C3×C42—34no—
150C3×C42—38no—
151C3×C42—68no—
152C3×C42C4×D314yes—
153C3×C42C4×D318yescomplemented
154C3×C42C2×C3×C414yes—
155C3×C42C2×C3×C418yescomplemented
156(C2×D3)⋊C4C2×C3×C4116yes—
157Dic3⋊C4C2×C3×C4116yescomplemented
158C4×Dic3—28no—
159C4×Dic3C2×C3×C414yes—
160C4×Dic3C2×C3×C418yes—
161C4×Dic3C2×C3×C418yescomplemented
162C22×C32C23×C411yescharacteristic
163C22×C32C23×C412yes—
164C22×C32C2×#4114yes—
165C2×C3×D3—24no—
166C2×C3×D3—28no—
167C2×C3×D3C2×C4214yes—
168C32×C4—28no—
169C32×C4C2×C4214yes—
170C32×C4#11718yescomplemented
171C3×Dic3—24no—
172C3×Dic3—28no—
173C3×Dic3C2×C4214yes—
174C23×C4—31no—
175C23×C4—32no—
176#117—316no—
177C2×C4⋊C4—34no—
178C2×C22⋊C4—34no—
179C2×C42—32no—
180C2×C42—31no—
181C2×C42—32no—
182C2×C42—38no—
183C2×C42—34no—
184C2×C42C2×C3×D314yes—
185C2×C42C2×C3×D311yescharacteristic
186C2×C42C2×C3×D312yes—
187#95—38no—
188C23×C3—21no—
189C23×C3—34no—
190C23×C3—32no—
191C23×C3—64no—
192C23×C3C23×D311yescharacteristic
193C23×C3C24×C311yescharacteristic
194C22×D3—24no—
195C22×D3C22×C3×C414yes—
196C22×D3C22×C3×C412yes—
197C2×C3×C4—216no—
198C2×C3×C4—28no—
199C2×C3×C4—28no—
200C2×C3×C4—28no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—24no—
204C2×C3×C4—28no—
205C2×C3×C4—32no—
206C2×C3×C4—34no—
207C2×C3×C4—316no—
208C2×C3×C4—34no—
209C2×C3×C4—38no—
210C2×C3×C4—68no—
211C2×C3×C4—64no—
212C2×C3×C4—616no—
213C2×C3×C4C2×C4×D318yes—
214C2×C3×C4C2×C4×D312yes—
215C2×C3×C4C2×C4×D314yes—
216C2×C3×C4C2×C4×D314yes—
217C2×C3×C4#23518yes—
218C2×C3×C4C22×C3×C418yes—
219C2×C3×C4C22×C3×C412yes—
220C2×C3×C4C22×C3×C414yes—
221C2×C3×C4C22×C3×C414yes—
222C2×C3×C4C3×#4118yes—
223C2×Dic3—28no—
224C2×Dic3—24no—
225C2×Dic3C22×C3×C414yes—
226C2×Dic3C22×C3×C418yes—
227C2×Dic3C22×C3×C412yes—
228C2×Dic3C22×C3×C414yes—
229C4×D3—216no—
230C4×D3C3×C42116yescomplemented
231C2×C32C22×C4211yescharacteristic
232C2×C32C2×#11712yes—
233C2×C32C4×#4114yes—
234C3×D3—24no—
235C24—31no—
236C22×C4—31no—
237C22×C4—34no—
238C22×C4—38no—
239C22×C4—32no—
240C22×C4—34no—
241C22×C4—38no—
242C22×C4—68no—
243C22×C4C22×C3×D314yes—
244C22×C4C22×C3×D311yescharacteristic
245C22×C4C22×C3×D312yes—
246C4⋊C4—316no—
247C22⋊C4—316no—
248C42—28no—
249C42—38no—
250C42—34no—
251C42—38no—
252C42—68no—
253C42C3×C4×D314yes—
254C42C3×C4×D318yescomplemented
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—24no—
258C22×C3—34no—
259C22×C3—64no—
260C22×C3—68no—
261C22×C3C22×C4×D311yescharacteristic
262C22×C3C22×C4×D312yes—
263C22×C3C2×#23514yes—
264C22×C3C23×C3×C411yescharacteristic
265C22×C3C23×C3×C412yes—
266C22×C3C2×C3×#4114yes—
267C2×D3—24no—
268C2×D3—28no—
269C2×D3C2×C3×C4214yes—
270C3×C4—216no—
271C3×C4—28no—
272C3×C4—28no—
273C3×C4—24no—
274C3×C4—28no—
275C3×C4—34no—
276C3×C4—64no—
277C3×C4—68no—
278C3×C4C42×D314yes—
279C3×C4#87218yesdirect product factor
280C3×C4C2×C3×C4214yes—
281C3×C4C3×#11718yescomplemented
282Dic3—24no—
283Dic3—28no—
284Dic3C2×C3×C4214yes—
285C32C4×#11711yescharacteristic, complemented, Sylow
286C23—34no—
287C23—32no—
288C23—64no—
289C23C23×C3×D311yescharacteristic
290C2×C4—28no—
291C2×C4—32no—
292C2×C4—34no—
293C2×C4—316no—
294C2×C4—34no—
295C2×C4—38no—
296C2×C4—68no—
297C2×C4—64no—
298C2×C4—616no—
299C2×C4C2×C3×C4×D318yes—
300C2×C4C2×C3×C4×D312yes—
301C2×C4C2×C3×C4×D314yes—
302C2×C4C2×C3×C4×D314yes—
303C2×C4C3×#23518yes—
304C2×C3—21no—
305C2×C3—22no—
306C2×C3—24no—
307C2×C3—64no—
308C2×C3C2×C42×D311yescharacteristic
309C2×C3C2×#87212yes—
310C2×C3C4×#23514yes—
311C2×C3C22×C3×C4211yescharacteristic
312C2×C3C2×C3×#11712yes—
313C2×C3C3×C4×#4114yes—
314D3—24no—
315C22—34no—
316C22—64no—
317C22—68no—
318C22C22×C3×C4×D311yescharacteristic
319C22C22×C3×C4×D312yes—
320C22C2×C3×#23514yes—
321C4—28no—
322C4—34no—
323C4—64no—
324C4—68no—
325C4C3×C42×D314yes—
326C4C3×#87218yesdirect product factor
327C3—21no—
328C3C4×#87211yescharacteristic, direct product factor
329C3C3×C4×#11711yescharacteristic, complemented
330C2—64no—
331C2C2×C3×C42×D311yescharacteristic
332C2C2×C3×#87212yes—
333C2C3×C4×#23514yes—
subgroupG/H
centerC2×C3×C42C2×D3
commutatorC2×C3C22×C3×C42
FrattiniC23C23×C3×D3
FittingC32×C43C2
socleC23×C32C24
prime radicalC23×C3×D3C23
cosocleC23×C3C24×C3
hypercenterC3×C43D3
hypocenterC3C3×C4×#117
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3×C42 ◁ C3×C43

Complements

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

NH ≅ G/Ntype
3. C32×C43C2semidirect
5. C3×#4932C2semidirect
6. C3×C4×Dic3⋊C4C2semidirect
7. C3×C42×Dic3C2semidirect
9. C4×#872C3direct (N × H)
21. C3×#872C4direct (N × H)
22. C3×C42×D3C4semidirect
33. C3×C43D3semidirect
34. C3×C43C2×C3semidirect
36. #4932C2×C3semidirect
37. C4×Dic3⋊C4C2×C3semidirect
38. C42×Dic3C2×C3semidirect
52. C32×C42C2×C4semidirect
54. C3×Dic3⋊C4C2×C4semidirect
58. C3×C4×Dic3C2×C4semidirect
87. #872C3×C4direct (N × H)
88. C42×D3C3×C4semidirect
108. C3×C4×D3C42semidirect
115. C43C3×D3semidirect
153. C3×C42C4×D3semidirect
155. C3×C42C2×C3×C4semidirect
157. Dic3⋊C4C2×C3×C4semidirect
161. C4×Dic3C2×C3×C4semidirect
170. C32×C4#117semidirect
230. C4×D3C3×C42semidirect
254. C42C3×C4×D3semidirect
279. C3×C4#872direct (N × H)
281. C3×C4C3×#117semidirect
285. C32C4×#117semidirect
326. C4C3×#872direct (N × H)
328. C3C4×#872direct (N × H)
329. C3C3×C4×#117semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8722
C2×C3×C42×D32
C32×C432
C3×C4×(C2×D3)⋊C42
C3×#49322
C3×C4×Dic3⋊C42
C3×C42×Dic32
C3×C4×#1173
C4×#8723