Finite Group Databasebeta

← C2×C3×#4542 #11796455

Subgroups — C2×C3×#4542 #11796455

Order 1152.

685 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#124C211yesmaximal, characteristic
2C2×C3×#836C212yesmaximal, complemented
3C3×#4542C214yesmaximal, direct product factor
4C2×C3×#355—31nomaximal
5C2×#4542C311yesmaximal, characteristic, direct product factor
6C22×C32×Q8C2211yescharacteristic
7C22×C32×D4C411yescharacteristic
8C32×#124—22no—
9C32×#124C412yes—
10C32×#124C2212yes—
11C2×C32×C22⋊C4—22no—
12C2×C32×C42C411yescharacteristic
13C2×C3×#208—22no—
14C3×#836—22no—
15C3×#836C2214yescomplemented
16C2×C3×#124—21no—
17C2×C3×#124D311yescharacteristic
18C2×C3×#124C2×C311yescharacteristic
19C2×C3×#101—32no—
20C2×#836C2×C312yescomplemented
21C3×#355—34no—
22#4542C2×C314yesdirect product factor
23C24×C32—21no—
24C2×C32×Q8—41no—
25C2×C32×Q8—41no—
26C2×C32×Q8C2×C411yescharacteristic
27C2×C32×Q8D412yes—
28C2×C32×Q8C2311yescharacteristic
29C2×C32×D4—21no—
30C2×C32×D4—41no—
31C2×C32×D4—41no—
32C2×C32×D4C2×C412yes—
33C22×C32×C4—21no—
34C22×C32×C4C2×C411yescharacteristic
35C22×C32×C4D412yes—
36C32×C22⋊C4—24no—
37C32×C22⋊C4—24no—
38C32×C42—21no—
39C32×C42C2×C412yes—
40C3×#208—24no—
41C3×#208—42no—
42C2×C3×C3⋊C8—42no—
43C2×#355—31noSylow
44C22×C3×Q8—21no—
45C22×C3×Q8C2×D311yescharacteristic
46C22×C3×Q8C22×C311yescharacteristic
47C22×C3×D4—21no—
48C22×C3×D4Dic311yescharacteristic
49C22×C3×D4C3×C411yescharacteristic
50C2×C3×M16—62no—
51C3×#124—24no—
52C3×#124—22no—
53C3×#124—22no—
54C3×#124—22no—
55C3×#124—22no—
56C3×#124Dic312yes—
57C3×#124C3×C412yes—
58C3×#124C2×D312yes—
59C3×#124C22×C312yes—
60C2×C3×C22⋊C4—24no—
61C2×C3×C22⋊C4—22no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C42—21no—
64C2×C3×C42Dic311yescharacteristic
65C2×C3×C42C3×C411yescharacteristic
66C2×#208—22no—
67C3×#101—34no—
68C3×#101—62no—
69#836—22no—
70#836C22×C314yescomplemented
71C23×C32—22no—
72C23×C32—44no—
73C23×C32—41no—
74C23×C32—41no—
75C23×C32C22⋊C411yescharacteristic
76C32×Q8—41no—
77C32×Q8—42no—
78C32×Q8—41no—
79C32×D4—42no—
80C32×D4—42no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—22no—
84C2×C32×C4—22no—
85C2×C32×C4—22no—
86C2×C32×C4—42no—
87C2×C32×C4C22⋊C411yescharacteristic
88C2×C32×C4C22⋊C412yes—
89C2×C32×C4C22×C411yescharacteristic
90C2×C32×C4C2×D412yes—
91C3×C3⋊C8—44no—
92C2×#124C3×D311yescharacteristic
93C2×#101—32no—
94#355—34no—
95C24×C3—22no—
96C24×C3—21no—
97C24×C3—21no—
98C2×C3×Q8—21no—
99C2×C3×Q8—22no—
100C2×C3×Q8—21no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8—42no—
106C2×C3×Q8—42no—
107C2×C3×Q8C2×Dic311yescharacteristic
108C2×C3×Q8#6712yes—
109C2×C3×Q8C2×C3×C411yescharacteristic
110C2×C3×Q8C3×D412yes—
111C2×C3×Q8C22×D311yescharacteristic
112C2×C3×Q8C23×C311yescharacteristic
113C2×C3×D4—22no—
114C2×C3×D4—21no—
115C2×C3×D4—21no—
116C2×C3×D4—22no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—42no—
122C2×C3×D4—42no—
123C2×C3×D4C2×Dic312yes—
124C2×C3×D4C2×C3×C412yes—
125C22×C3×C4—22no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—22no—
130C22×C3×C4C2×Dic311yescharacteristic
131C22×C3×C4#6712yes—
132C22×C3×C4C2×C3×C411yescharacteristic
133C22×C3×C4C3×D412yes—
134C3×M16—64no—
135C3×M16—122no—
136C2×C3×C8—122no—
137C3×C22⋊C4—28no—
138C3×C22⋊C4—28no—
139C3×C22⋊C4—24no—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—24no—
143C3×C42—22no—
144C3×C42—21no—
145C3×C42—21no—
146C3×C42—22no—
147C3×C42C2×Dic312yes—
148C3×C42C2×C3×C412yes—
149#208—24no—
150#208—42no—
151C2×C3⋊C8—42no—
152C22×C32—22no—
153C22×C32—44no—
154C22×C32—42no—
155C22×C32—82no—
156C22×C32—82no—
157C22×C32#9911yescharacteristic
158C22×C32#9911yescharacteristic
159C22×C32C2×C22⋊C411yescharacteristic
160C32×C4—21no—
161C32×C4—21no—
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—42no—
165C22×Q8C2×C3×D311yescharacteristic
166C22×D4C3×Dic311yescharacteristic
167C2×M16—62no—
168#124—22no—
169#124C3×Dic312yes—
170#124C2×C3×D312yes—
171C2×C22⋊C4—22no—
172C2×C42C3×Dic311yescharacteristic
173#101—34no—
174#101—62no—
175C23×C3—24no—
176C23×C3—22no—
177C23×C3—22no—
178C23×C3—21no—
179C23×C3—42no—
180C23×C3—42no—
181C23×C3—48no—
182C23×C3—44no—
183C23×C3—44no—
184C23×C3—41no—
185C23×C3—41no—
186C23×C3—41no—
187C23×C3—41no—
188C23×C3#21711yescharacteristic
189C23×C3C3×C22⋊C411yescharacteristic
190C3×Q8—41no—
191C3×Q8—42no—
192C3×Q8—41no—
193C3×Q8—42no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—42no—
197C3×Q8—44no—
198C3×Q8—42no—
199C3×D4—42no—
200C3×D4—42no—
201C3×D4—44no—
202C3×D4—44no—
203C3×D4—42no—
204C3×D4—42no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—21no—
209C2×C3×C4—22no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—22no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—44no—
225C2×C3×C4—42no—
226C2×C3×C4—42no—
227C2×C3×C4#21711yescharacteristic
228C2×C3×C4#21712yes—
229C2×C3×C4C3×C22⋊C411yescharacteristic
230C2×C3×C4C3×C22⋊C412yes—
231C2×C3×C4C22×Dic311yescharacteristic
232C2×C3×C4C2×#6712yes—
233C2×C3×C4C22×C3×C411yescharacteristic
234C2×C3×C4C2×C3×D412yes—
235C3×C8—124no—
236C3⋊C8—44no—
237C2×C32—21no—
238C2×C32—21no—
239C2×C32—82no—
240C2×C32#35512yescomplemented
241C2×C32C2×#9911yescharacteristic
242C24—21no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8C2×C3×Dic311yescharacteristic
246C2×Q8C3×#6712yes—
247C2×Q8C22×C3×D311yescharacteristic
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4C2×C3×Dic312yes—
252C22×C4—21no—
253C22×C4C2×C3×Dic311yescharacteristic
254C22×C4C3×#6712yes—
255M16—64no—
256M16—122no—
257C2×C8—122no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C42—21no—
261C42C2×C3×Dic312yes—
262C22×C3—24no—
263C22×C3—22no—
264C22×C3—22no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—44no—
269C22×C3—48no—
270C22×C3—44no—
271C22×C3—44no—
272C22×C3—42no—
273C22×C3—42no—
274C22×C3—84no—
275C22×C3—82no—
276C22×C3—82no—
277C22×C3—84no—
278C22×C3—82no—
279C22×C3—82no—
280C22×C3#83411yescharacteristic
281C22×C3#83411yescharacteristic
282C22×C3C3×#9911yescharacteristic
283C22×C3C3×#9911yescharacteristic
284C22×C3C2×#21711yescharacteristic
285C22×C3C2×C3×C22⋊C411yescharacteristic
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—21no—
290C3×C4—22no—
291C3×C4—21no—
292C3×C4—22no—
293C3×C4—21no—
294C3×C4—22no—
295C3×C4—21no—
296C3×C4—42no—
297C3×C4—42no—
298C3×C4—44no—
299C3×C4—42no—
300C3×C4—42no—
301C32C2×#35511yescharacteristic, complemented, Sylow
302C23—22no—
303C23—44no—
304C23—41no—
305C23—41no—
306C23C3×#21711yescharacteristic
307Q8—41no—
308Q8—42no—
309Q8—41no—
310D4—42no—
311D4—42no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4C3×#21711yescharacteristic
319C2×C4C3×#21712yes—
320C2×C4C22×C3×Dic311yescharacteristic
321C2×C4C2×C3×#6712yes—
322C8—124no—
323C2×C3—22no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—22no—
331C2×C3—84no—
332C2×C3—82no—
333C2×C3—82no—
334C2×C3#454212yesdirect product factor
335C2×C3C3×#35512yescomplemented
336C2×C3C2×#83411yescharacteristic
337C2×C3C2×C3×#9911yescharacteristic
338C22—22no—
339C22—44no—
340C22—42no—
341C22—82no—
342C22—82no—
343C22C3×#83411yescharacteristic
344C22C3×#83411yescharacteristic
345C22C2×C3×#21711yescharacteristic
346C4—22no—
347C4—21no—
348C4—22no—
349C4—21no—
350C4—42no—
351C3—21no—
352C3C2×#454211yescharacteristic, direct product factor
353C3C2×C3×#35511yescharacteristic, complemented
354C2—21no—
355C2—21no—
356C2—82no—
357C2C3×#454212yesdirect product factor
358C2C2×C3×#83411yescharacteristic
subgroupG/H
centerC22×C3#834
commutatorC2×C3×C4C22×C3×C4
FrattiniC2×Q8C22×C3×D3
FittingC2×C32×#124C2
socleC22×C32#99
prime radicalC22×C32×D4C4
cosocleC2×C3×Q8C23×C3
hypercenterC2×C3×#124D3
hypocenterC3C2×C3×#355
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C23×C3 ◁ C22×C3×Q8 ◁ C2×C3×#124

Complements

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

NH ≅ G/Ntype
2. C2×C3×#836C2semidirect
3. C3×#4542C2direct (N × H)
5. C2×#4542C3direct (N × H)
15. C3×#836C22semidirect
20. C2×#836C2×C3semidirect
22. #4542C2×C3direct (N × H)
70. #836C22×C3semidirect
240. C2×C32#355semidirect
301. C32C2×#355semidirect
334. C2×C3#4542direct (N × H)
335. C2×C3C3×#355semidirect
352. C3C2×#4542direct (N × H)
353. C3C2×C3×#355semidirect
357. C2C3×#4542direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#1242
C2×C3×#8362
C3×#45422
C2×C3×#3553
C2×#45423