Finite Group Databasebeta

← C3×C4×#3750 #13162967

Subgroups — C3×C4×#3750 #13162967

Order 1728.

1596 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×C42C211yesmaximal, characteristic
2C2×C3×#3750C212yesmaximal
3C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
4C2×C3×C4×C3⋊Dic3C211yesmaximal, characteristic
5C3×#63658C212yesmaximal
6C4×#3750C311yesmaximal, characteristic, direct product factor
7C3×C4×Dic3⋊C4—34nomaximal
8C22×C33×C4C2212yes—
9C22×C33×C4C2211yescharacteristic
10C22×C3×C3⋊Dic3C2212yes—
11C22×C3×C3⋊Dic3C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C33×C42—22no—
14C3×#3750C418yesdirect product factor
15C3×C4×C3⋊Dic3—22no—
16C3×C4×C3⋊Dic3C414yescomplemented
17C2×C32×C42—24no—
18C2×C32×C42D314yes—
19C2×C32×C42C2×C311yescharacteristic
20C2×#3750C2×C312yes—
21C2×C4×C3⋊Dic3C2×C311yescharacteristic
22C2×C4×C3⋊Dic3C2×C311yescharacteristic
23C2×C3×Dic3⋊C4—38no—
24C2×C3×C4×Dic3—34no—
25C2×C3×C4×Dic3—34no—
26#63658C2×C312yes—
27C3×C3⋊#95—38no—
28C23×C33C2311yescharacteristic
29C2×C33×C4—22no—
30C2×C33×C4—22no—
31C2×C33×C4C2×C414yes—
32C2×C33×C4C2×C412yes—
33C2×C33×C4D412yes—
34C2×C33×C4Q812yes—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3—22no—
38C2×C3×C3⋊Dic3C2×C412yes—
39C2×C3×C3⋊Dic3C2×C414yes—
40C2×C3×C3⋊Dic3C2×C412yes—
41C2×C3×C3⋊Dic3C2×C414yes—
42C3×C4×C4⋊C4—91no—
43C4×Dic3⋊C4—34no—
44C22×C32×C4—28no—
45C22×C32×C4—24no—
46C22×C32×C4C2×D318yes—
47C22×C32×C4C2×D314yes—
48C22×C32×C4C22×C312yes—
49C22×C32×C4C22×C311yescharacteristic
50C22×C3⋊Dic3C22×C312yes—
51C22×C3⋊Dic3C22×C311yescharacteristic
52C22×C3⋊Dic3C22×C311yescharacteristic
53C22×C3×Dic3—34no—
54C22×C3×Dic3—38no—
55C22×C3×Dic3—34no—
56C32×C42—216no—
57C32×C42—28no—
58C32×C42—22no—
59#3750C3×C418yesdirect product factor
60C4×C3⋊Dic3—22no—
61C4×C3⋊Dic3C3×C414yescomplemented
62C3×Dic3⋊C4—332no—
63C3×C4×Dic3—316no—
64C3×C4×Dic3—68no—
65C22×C33C22×C411yescharacteristic
66C22×C33C22×C412yes—
67C22×C33C2×D411yescharacteristic
68C22×C33C2×Q811yescharacteristic
69C22×C33#4112yes—
70C33×C4—24no—
71C33×C4C4⋊C414yescomplemented
72C3×C3⋊Dic3—22no—
73C3×C3⋊Dic3—24no—
74C3×C3⋊Dic3C4214yescomplemented
75C2×C3×C4⋊C4—92no—
76C2×C3×C42—24no—
77C2×C3×C42—91no—
78C2×C3×C42—91no—
79C2×C3×C42C3×D314yes—
80C2×C3×C42C3⋊D311yescharacteristic
81C2×Dic3⋊C4—38no—
82C2×C4×Dic3—34no—
83C2×C4×Dic3—34no—
84C3×#95—92no—
85C3⋊#95—38no—
86C23×C32—24no—
87C23×C32C22×D314yes—
88C23×C32C23×C311yescharacteristic
89C2×C32×C4—216no—
90C2×C32×C4—216no—
91C2×C32×C4—28no—
92C2×C32×C4—28no—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—28no—
96C2×C32×C4—216no—
97C2×C32×C4—28no—
98C2×C32×C4—28no—
99C2×C32×C4Dic618yes—
100C2×C32×C4C4×D3116yes—
101C2×C32×C4C4×D318yes—
102C2×C32×C4#6718yes—
103C2×C32×C4C2×C3×C414yes—
104C2×C32×C4C2×C3×C412yes—
105C2×C32×C4C3×D412yes—
106C2×C32×C4C3×Q812yes—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3—22no—
109C2×C3⋊Dic3—22no—
110C2×C3⋊Dic3C2×C3×C412yes—
111C2×C3⋊Dic3C2×C3×C414yes—
112C2×C3⋊Dic3C2×C3×C412yes—
113C2×C3⋊Dic3C2×C3×C414yes—
114C2×C3×Dic3—38no—
115C2×C3×Dic3—316no—
116C2×C3×Dic3—38no—
117C2×C3×Dic3—316no—
118C2×C3×Dic3—68no—
119C2×C3×Dic3—68no—
120C2×C3×Dic3—68no—
121C4×C4⋊C4—91noSylow
122C2×C33C2×C4211yescharacteristic
123C2×C33C2×C4⋊C411yescharacteristic
124C2×C33#11711yescharacteristic
125C2×C33C4×D412yes—
126C2×C33C4×Q812yes—
127C22×C3×C4—28no—
128C22×C3×C4—24no—
129C22×C3×C4—91no—
130C22×C3×C4—91no—
131C22×C3×C4—92no—
132C22×C3×C4C2×C3×D318yes—
133C22×C3×C4C2×C3×D314yes—
134C22×C3×C4C2×C3⋊D312yes—
135C22×C3×C4C2×C3⋊D311yescharacteristic
136C22×Dic3—34no—
137C22×Dic3—34no—
138C22×Dic3—38no—
139C3×C4⋊C4—98no—
140C3×C42—216no—
141C3×C42—28no—
142C3×C42—22no—
143C3×C42—94no—
144C3×C42—182no—
145Dic3⋊C4—332no—
146C4×Dic3—316no—
147C4×Dic3—68no—
148C22×C32—24no—
149C22×C32—24no—
150C22×C32—28no—
151C22×C32—28no—
152C22×C32—24no—
153C22×C32C2×Dic614yes—
154C22×C32C2×C4×D318yes—
155C22×C32C2×C4×D314yes—
156C22×C32#23518yes—
157C22×C32C2×#6714yes—
158C22×C32C22×C3×C411yescharacteristic
159C22×C32C22×C3×C412yes—
160C22×C32C2×C3×D411yescharacteristic
161C22×C32C2×C3×Q811yescharacteristic
162C22×C32C3×#4112yes—
163C32×C4—232no—
164C32×C4—216no—
165C32×C4—24no—
166C32×C4—216no—
167C32×C4Dic3⋊C4116yescomplemented
168C32×C4C3×C4⋊C414yescomplemented
169C3⋊Dic3—22no—
170C3⋊Dic3—24no—
171C3⋊Dic3C3×C4214yescomplemented
172C3×Dic3—316no—
173C3×Dic3—68no—
174C3×Dic3—616no—
175C2×C4⋊C4—92no—
176C2×C42—91no—
177C2×C42—91no—
178C2×C42C3×C3⋊D311yescharacteristic
179#95—92no—
180C33C4×C4⋊C411yescharacteristic, complemented, Sylow
181C23×C3—24no—
182C23×C3C22×C3×D314yes—
183C23×C3C22×C3⋊D311yescharacteristic
184C2×C3×C4—216no—
185C2×C3×C4—216no—
186C2×C3×C4—28no—
187C2×C3×C4—28no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—28no—
191C2×C3×C4—216no—
192C2×C3×C4—28no—
193C2×C3×C4—28no—
194C2×C3×C4—92no—
195C2×C3×C4—94no—
196C2×C3×C4—92no—
197C2×C3×C4—94no—
198C2×C3×C4—182no—
199C2×C3×C4—182no—
200C2×C3×C4—182no—
201C2×C3×C4C3×Dic618yes—
202C2×C3×C4C3×C4×D3116yes—
203C2×C3×C4C3×C4×D318yes—
204C2×C3×C4C3×#6718yes—
205C2×C3×C4#63412yes—
206C2×C3×C4C4×C3⋊D314yes—
207C2×C3×C4C4×C3⋊D312yes—
208C2×C3×C4#63812yes—
209C2×Dic3—38no—
210C2×Dic3—316no—
211C2×Dic3—38no—
212C2×Dic3—316no—
213C2×Dic3—68no—
214C2×Dic3—68no—
215C2×Dic3—68no—
216C2×C32—24no—
217C2×C32—28no—
218C2×C32—28no—
219C2×C32—24no—
220C2×C32—24no—
221C2×C32C4×Dic618yes—
222C2×C32C42×D314yes—
223C2×C32#87214yes—
224C2×C32C2×Dic3⋊C414yes—
225C2×C32C4×#6718yes—
226C2×C32C2×C3×C4211yescharacteristic
227C2×C32C2×C3×C4⋊C411yescharacteristic
228C2×C32C3×#11711yescharacteristic
229C2×C32C3×C4×D412yes—
230C2×C32C3×C4×Q812yes—
231C22×C4—91no—
232C22×C4—91no—
233C22×C4—92no—
234C22×C4C2×C3×C3⋊D312yes—
235C22×C4C2×C3×C3⋊D311yescharacteristic
236C4⋊C4—98no—
237C42—22no—
238C42—94no—
239C42—182no—
240C22×C3—24no—
241C22×C3—28no—
242C22×C3—24no—
243C22×C3—28no—
244C22×C3—24no—
245C22×C3C2×C3×Dic614yes—
246C22×C3C2×C3×C4×D314yes—
247C22×C3C2×C3×C4×D318yes—
248C22×C3C3×#23518yes—
249C22×C3C2×C3×#6714yes—
250C22×C3C2×#63411yescharacteristic
251C22×C3C2×C4×C3⋊D312yes—
252C22×C3C2×C4×C3⋊D311yescharacteristic
253C22×C3#382812yes—
254C22×C3C2×#63811yescharacteristic
255C3×C4—232no—
256C3×C4—216no—
257C3×C4—24no—
258C3×C4—216no—
259C3×C4—94no—
260C3×C4—182no—
261C3×C4—184no—
262C3×C4C3×Dic3⋊C4116yescomplemented
263C3×C4#375014yesdirect product factor
264Dic3—316no—
265Dic3—616no—
266Dic3—68no—
267C32—24no—
268C32C4×Dic3⋊C414yescomplemented
269C32C3×C4×C4⋊C411yescharacteristic, complemented
270C23C22×C3×C3⋊D311yescharacteristic
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—92no—
274C2×C4—94no—
275C2×C4—92no—
276C2×C4—94no—
277C2×C4—182no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4C3×#63412yes—
281C2×C4C3×C4×C3⋊D314yes—
282C2×C4C3×C4×C3⋊D312yes—
283C2×C4C3×#63812yes—
284C2×C3—28no—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3—28no—
289C2×C3C3×C4×Dic618yes—
290C2×C3C3×C42×D314yes—
291C2×C3C3×#87214yes—
292C2×C3C2×C3×Dic3⋊C414yes—
293C2×C3C3×C4×#6718yes—
294C2×C3C4×#63412yes—
295C2×C3C42×C3⋊D311yescharacteristic
296C2×C3#6408111yescharacteristic
297C2×C3C2×#375011yescharacteristic
298C2×C3C4×#63812yes—
299C22C2×C3×#63411yescharacteristic
300C22C2×C3×C4×C3⋊D312yes—
301C22C2×C3×C4×C3⋊D311yescharacteristic
302C22C3×#382812yes—
303C22C2×C3×#63811yescharacteristic
304C4—24no—
305C4—94no—
306C4—182no—
307C4—184no—
308C4C3×#375014yesdirect product factor
309C3—24no—
310C3C3×C4×Dic3⋊C414yescomplemented
311C3C4×#375011yescharacteristic, direct product factor
312C2C3×C4×#63412yes—
313C2C3×C42×C3⋊D311yescharacteristic
314C2C3×#6408111yescharacteristic
315C2C2×C3×#375011yescharacteristic
316C2C3×C4×#63812yes—
subgroupG/H
centerC22×C3×C4C2×C3⋊D3
commutatorC2×C32C2×C3×C42
FrattiniC23C22×C3×C3⋊D3
FittingC2×C33×C42C2
socle, prime radicalC23×C33C23
cosocleC23×C32C23×C3
hypercenterC2×C3×C42C3⋊D3
hypocenterC32C3×C4×C4⋊C4
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4 ◁ C2×C3×C42

Complements

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

NH ≅ G/Ntype
6. C4×#3750C3direct (N × H)
14. C3×#3750C4direct (N × H)
16. C3×C4×C3⋊Dic3C4semidirect
59. #3750C3×C4direct (N × H)
61. C4×C3⋊Dic3C3×C4semidirect
71. C33×C4C4⋊C4semidirect
74. C3×C3⋊Dic3C42semidirect
167. C32×C4Dic3⋊C4semidirect
168. C32×C4C3×C4⋊C4semidirect
171. C3⋊Dic3C3×C42semidirect
180. C33C4×C4⋊C4semidirect
262. C3×C4C3×Dic3⋊C4semidirect
263. C3×C4#3750direct (N × H)
268. C32C4×Dic3⋊C4semidirect
269. C32C3×C4×C4⋊C4semidirect
308. C4C3×#3750direct (N × H)
310. C3C3×C4×Dic3⋊C4semidirect
311. C3C4×#3750direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C33×C422
C2×C3×#37502
C2×C3×C4×C3⋊Dic32
C3×#636582
C4×#37503
C3×C4×Dic3⋊C43