Finite Group Databasebeta

← C2×C3×#4721 #11869378

Subgroups — C2×C3×#4721 #11869378

Order 1152.

1021 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Dic3C211yesmaximal, characteristic
2C2×C3×#924C211yesmaximal, characteristic
3C2×C32×C22⋊C8C211yesmaximal, characteristic
4C2×C3×#817C212yesmaximal, complemented
5C2×C3×#816C212yesmaximal, complemented
6C3×#4721C218yesmaximal, direct product factor
7C2×C3×#483—31nomaximal
8C2×#4721C311yesmaximal, characteristic, direct product factor
9C23×C32×C4C2211yescharacteristic
10C23×C3×Dic3—21no—
11C22×C3×Dic6C2211yescharacteristic
12C22×C32×C8C2212yes—
13C2×C3×#217—21no—
14C2×C3×C4⋊Dic3—21no—
15C2×C3×C4⋊Dic3—22no—
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C4⋊Dic3C2214yes—
18C2×C3×C4⋊Dic3C2212yes—
19C3×#924—22no—
20C3×#924C2214yes—
21C2×C3×Dic3⋊C4—21no—
22C32×C22⋊C8C2214yes—
23C3×#817C2218yescomplemented
24C3×#816C2218yescomplemented
25C2×C3×C22⋊Q8—31no—
26C22×C3×C4⋊C4—31no—
27C22×C4⋊Dic3C2×C311yescharacteristic
28C2×#924C2×C311yescharacteristic
29C3×#483—38no—
30C2×C3×#106—32no—
31C2×C3×#103—32no—
32C2×C3×C22⋊C8—21no—
33C2×C3×C22⋊C8D311yescharacteristic
34C2×C3×C22⋊C8C2×C311yescharacteristic
35C2×#817C2×C312yescomplemented
36C2×#816C2×C312yescomplemented
37#4721C2×C318yesdirect product factor
38C24×C32D411yescharacteristic
39C22×C32×C4—24no—
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D412yes—
42C22×C32×C4C2312yes—
43C22×C32×C4C2311yescharacteristic
44C22×C3×Dic3—21no—
45C22×C3×Dic3—21no—
46C22×C3×Dic3—24no—
47C22×C3×Dic3—22no—
48C22×C3×Dic3—44no—
49C2×C3×Dic6—24no—
50C2×C3×Dic6—21no—
51C2×C3×Dic6C2312yes—
52C2×C32×C8—24no—
53C2×C32×C8C2314yes—
54C3×#217—24no—
55C3×C4⋊Dic3—21no—
56C3×C4⋊Dic3—24no—
57C3×C4⋊Dic3—22no—
58C3×C4⋊Dic3C2312yes—
59C3×C4⋊Dic3C2314yes—
60C3×Dic3⋊C4—24no—
61C2×#483—31noSylow
62C22×C3×Q8—31no—
63C23×C3×C4—21no—
64C23×C3×C4—61no—
65C23×C3×C4C2×D311yescharacteristic
66C23×C3×C4C22×C311yescharacteristic
67C23×Dic3—21no—
68C22×Dic6C22×C311yescharacteristic
69C22×C3×C8—22no—
70C22×C3×C8C2×D312yes—
71C22×C3×C8C22×C312yes—
72C3×C22⋊Q8—34no—
73C3×C22⋊Q8—62no—
74C2×C3×C4⋊C4—32no—
75C2×C3×C4⋊C4—31no—
76C2×C3×C4⋊C4—34no—
77C2×C3×C4⋊C4—61no—
78C2×C3×C4⋊C4—62no—
79C2×C3×C4⋊C4—61no—
80C2×C3×C22⋊C4—61no—
81C2×#217—21no—
82C2×C4⋊Dic3—21no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3C22×C314yes—
85C2×C4⋊Dic3C22×C311yescharacteristic
86C2×C4⋊Dic3C22×C312yes—
87#924—22no—
88#924C22×C314yes—
89C2×Dic3⋊C4—21no—
90C3×#106—38no—
91C3×#103—38no—
92C3×C22⋊C8—24no—
93C3×C22⋊C8C2×D314yes—
94C3×C22⋊C8C22×C314yes—
95#817C22×C318yescomplemented
96#816C22×C318yescomplemented
97C23×C32—22no—
98C23×C32—22no—
99C23×C32SD1614yes—
100C23×C32C2×D412yes—
101C23×C32C2×D411yescharacteristic
102C2×C32×C4—22no—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4C2×D412yes—
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4#4114yes—
108C2×C32×C4C2411yescharacteristic
109C2×C3×Dic3—22no—
110C2×C3×Dic3—22no—
111C2×C3×Dic3—24no—
112C2×C3×Dic3—42no—
113C2×C3×Dic3—48no—
114C2×C3×Dic3—44no—
115C2×C3×Dic3—42no—
116C3×Dic6—24no—
117C3×Dic6—42no—
118C32×C8—24no—
119C2×C22⋊Q8—31no—
120C22×C4⋊C4—31no—
121#483—38no—
122C2×#106—32no—
123C2×#103—32no—
124C2×C22⋊C8C3×D311yescharacteristic
125C24×C3—21no—
126C24×C3D1211yescharacteristic
127C24×C3C3×D411yescharacteristic
128C2×C3×Q8—32no—
129C2×C3×Q8—61no—
130C2×C3×Q8—64no—
131C22×C3×C4—24no—
132C22×C3×C4—24no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—44no—
138C22×C3×C4—64no—
139C22×C3×C4—61no—
140C22×C3×C4—61no—
141C22×C3×C4—62no—
142C22×C3×C4—124no—
143C22×C3×C4D1212yes—
144C22×C3×C4D1211yescharacteristic
145C22×C3×C4C3×D411yescharacteristic
146C22×C3×C4C3×D412yes—
147C22×C3×C4C22×D312yes—
148C22×C3×C4C22×D311yescharacteristic
149C22×C3×C4C23×C312yes—
150C22×C3×C4C23×C311yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—21no—
153C22×Dic3—24no—
154C22×Dic3—22no—
155C22×Dic3—44no—
156C2×Dic6—24no—
157C2×Dic6—21no—
158C2×Dic6C23×C312yes—
159C2×C3×C8—24no—
160C2×C3×C8—24no—
161C2×C3×C8—24no—
162C2×C3×C8—44no—
163C2×C3×C8C22×D314yes—
164C2×C3×C8C23×C314yes—
165C3×C4⋊C4—34no—
166C3×C4⋊C4—32no—
167C3×C4⋊C4—64no—
168C3×C4⋊C4—64no—
169C3×C4⋊C4—62no—
170C3×C4⋊C4—61no—
171C3×C22⋊C4—64no—
172#217—24no—
173C4⋊Dic3—21no—
174C4⋊Dic3—24no—
175C4⋊Dic3—22no—
176C4⋊Dic3C23×C314yes—
177C4⋊Dic3C23×C312yes—
178Dic3⋊C4—24no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—28no—
182C22×C32#12312yes—
183C22×C32C2×SD1614yes—
184C22×C32C2×SD1612yes—
185C22×C32#13712yes—
186C22×C32C22×D411yescharacteristic
187C32×C4—22no—
188C32×C4#12312yes—
189C32×C4C2×#4112yes—
190C3×Dic3—42no—
191C3×Dic3—44no—
192C3×Dic3—42no—
193C22×Q8—31no—
194C23×C4—61no—
195C23×C4C2×C3×D311yescharacteristic
196C22×C8C2×C3×D312yes—
197C22⋊Q8—34no—
198C22⋊Q8—62no—
199C2×C4⋊C4—32no—
200C2×C4⋊C4—34no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—61no—
205C2×C22⋊C4—61no—
206#106—38no—
207#103—38no—
208C22⋊C8C2×C3×D314yes—
209C23×C3—22no—
210C23×C3—22no—
211C23×C3—22no—
212C23×C3—22no—
213C23×C3—24no—
214C23×C3—21no—
215C23×C3—22no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3#20414yes—
219C23×C3C3×SD1614yes—
220C23×C3C2×D1212yes—
221C23×C3C2×D1211yescharacteristic
222C23×C3C2×C3×D412yes—
223C23×C3C2×C3×D411yescharacteristic
224C3×Q8—64no—
225C3×Q8—122no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—22no—
235C2×C3×C4—24no—
236C2×C3×C4—44no—
237C2×C3×C4—44no—
238C2×C3×C4—42no—
239C2×C3×C4—62no—
240C2×C3×C4—62no—
241C2×C3×C4—64no—
242C2×C3×C4—122no—
243C2×C3×C4—128no—
244C2×C3×C4—124no—
245C2×C3×C4—122no—
246C2×C3×C4C2×D1212yes—
247C2×C3×C4C2×D1211yescharacteristic
248C2×C3×C4#23714yes—
249C2×C3×C4C2×C3×D412yes—
250C2×C3×C4C2×C3×D411yescharacteristic
251C2×C3×C4C3×#4114yes—
252C2×C3×C4C23×D311yescharacteristic
253C2×C3×C4C24×C311yescharacteristic
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—24no—
257C2×Dic3—42no—
258C2×Dic3—48no—
259C2×Dic3—44no—
260C2×Dic3—42no—
261Dic6—24no—
262Dic6—42no—
263C3×C8—24no—
264C3×C8—24no—
265C3×C8—44no—
266C2×C32—24no—
267C2×C32#48314yescomplemented
268C2×C32C2×#12311yescharacteristic
269C2×C32C22×SD1611yescharacteristic
270C2×C32C2×#13711yescharacteristic
271C24C3×D1211yescharacteristic
272C2×Q8—32no—
273C2×Q8—61no—
274C2×Q8—64no—
275C22×C4—24no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—64no—
279C22×C4—62no—
280C22×C4—124no—
281C22×C4C3×D1211yescharacteristic
282C22×C4C3×D1212yes—
283C22×C4C22×C3×D312yes—
284C22×C4C22×C3×D311yescharacteristic
285C2×C8—24no—
286C2×C8C22×C3×D314yes—
287C4⋊C4—34no—
288C4⋊C4—32no—
289C4⋊C4—64no—
290C4⋊C4—62no—
291C4⋊C4—61no—
292C4⋊C4—64no—
293C22⋊C4—64no—
294C22×C3—22no—
295C22×C3—28no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—28no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—24no—
303C22×C3—22no—
304C22×C3—21no—
305C22×C3—42no—
306C22×C3—42no—
307C22×C3—48no—
308C22×C3#88612yes—
309C22×C3C2×#20414yes—
310C22×C3C2×#20412yes—
311C22×C3#90912yes—
312C22×C3C3×#12312yes—
313C22×C3C2×C3×SD1614yes—
314C22×C3C2×C3×SD1612yes—
315C22×C3C3×#13712yes—
316C22×C3C22×D1211yescharacteristic
317C22×C3C22×C3×D411yescharacteristic
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—22no—
321C3×C4—22no—
322C3×C4—42no—
323C3×C4—124no—
324C3×C4—122no—
325C3×C4—122no—
326C3×C4#88612yes—
327C3×C4C3×#12312yes—
328C3×C4C2×#23712yes—
329C3×C4C2×C3×#4112yes—
330Dic3—44no—
331Dic3—42no—
332Dic3—42no—
333C32C2×#48311yescharacteristic, complemented, Sylow
334C23—22no—
335C23—22no—
336C23C3×#20414yes—
337C23C2×C3×D1212yes—
338C23C2×C3×D1211yescharacteristic
339Q8—64no—
340Q8—122no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—62no—
345C2×C4—62no—
346C2×C4—64no—
347C2×C4—122no—
348C2×C4—128no—
349C2×C4—124no—
350C2×C4—122no—
351C2×C4C2×C3×D1212yes—
352C2×C4C2×C3×D1211yescharacteristic
353C2×C4C3×#23714yes—
354C2×C4C23×C3×D311yescharacteristic
355C8—24no—
356C2×C3—24no—
357C2×C3—24no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—24no—
361C2×C3—21no—
362C2×C3—44no—
363C2×C3#472114yesdirect product factor
364C2×C3C3×#48314yescomplemented
365C2×C3C2×#88611yescharacteristic
366C2×C3C22×#20411yescharacteristic
367C2×C3C2×#90911yescharacteristic
368C2×C3C2×C3×#12311yescharacteristic
369C2×C3C22×C3×SD1611yescharacteristic
370C2×C3C2×C3×#13711yescharacteristic
371C22—22no—
372C22—22no—
373C22—28no—
374C22C3×#88612yes—
375C22C2×C3×#20414yes—
376C22C2×C3×#20412yes—
377C22C3×#90912yes—
378C22C22×C3×D1211yescharacteristic
379C4—22no—
380C4—122no—
381C4—122no—
382C4—124no—
383C4C3×#88612yes—
384C4C2×C3×#23712yes—
385C3—21no—
386C3C2×#472111yescharacteristic, direct product factor
387C3C2×C3×#48311yescharacteristic, complemented
388C2—24no—
389C2C3×#472114yesdirect product factor
390C2C2×C3×#88611yescharacteristic
391C2C22×C3×#20411yescharacteristic
392C2C2×C3×#90911yescharacteristic
subgroupG/H
centerC23×C3C2×D12
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×C22⋊C8C2
socleC23×C32C2×D4
prime radicalC24×C32D4
hypercenterC2×C3×C22⋊C8D3
hypocenterC3C2×C3×#483
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C23×C3×C4 ◁ C2×C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
4. C2×C3×#817C2semidirect
5. C2×C3×#816C2semidirect
6. C3×#4721C2direct (N × H)
8. C2×#4721C3direct (N × H)
23. C3×#817C22semidirect
24. C3×#816C22semidirect
35. C2×#817C2×C3semidirect
36. C2×#816C2×C3semidirect
37. #4721C2×C3direct (N × H)
95. #817C22×C3semidirect
96. #816C22×C3semidirect
267. C2×C32#483semidirect
333. C32C2×#483semidirect
363. C2×C3#4721direct (N × H)
364. C2×C3C3×#483semidirect
386. C3C2×#4721direct (N × H)
387. C3C2×C3×#483semidirect
389. C2C3×#4721direct (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⋊Dic32
C2×C3×#9242
C2×C32×C22⋊C82
C2×C3×#8172
C2×C3×#8162
C3×#47212
C2×C3×#4833
C2×#47213