Finite Group Databasebeta

← C2×C3×#4816 #11867000

Subgroups — C2×C3×#4816 #11867000

Order 1152.

1065 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#946C211yesmaximal, characteristic
2C22×C3×#216C211yesmaximal, characteristic, complemented
3C22×C3×ΓL(1,𝔽25)C211yesmaximal, characteristic
4C2×C3×#897C211yesmaximal, characteristic
5C2×C32×#103C211yesmaximal, characteristic, complemented
6C2×C3×#816C211yesmaximal, characteristic, complemented
7C2×C3×#808C211yesmaximal, characteristic, complemented
8C3×#4816C218yesmaximal, direct product factor
9C2×C3×#470—31nomaximal
10C2×#4816C311yesmaximal, characteristic, direct product factor
11C22×C32×Q8C2211yescharacteristic
12C22×C3×C4×D3C2211yescharacteristic
13C22×C3×Dic6C2211yescharacteristic
14C22×C32×C8C2211yescharacteristic
15C2×C32×C4⋊C4C2211yescharacteristic
16C3×#946—22no—
17C3×#946C2214yes—
18C2×C3×#216—24no—
19C2×C3×#216C2214yescomplemented
20C2×C3×(C2×D3)⋊C4—21no—
21C2×C3×(C2×D3)⋊C4—21no—
22C2×C3×C4⋊Dic3—21no—
23C2×C3×C4⋊Dic3C2211yescharacteristic
24C2×C3×Dic3⋊C4—21no—
25C22×C3×C3⋊C8C2211yescharacteristic
26C2×C3×ΓL(1,𝔽25)—24no—
27C2×C3×ΓL(1,𝔽25)C2214yes—
28C3×#897—22no—
29C3×#897C2214yes—
30C32×#103C2214yescomplemented
31C3×#816C2214yescomplemented
32C3×#808C2214yescomplemented
33C22×C3×Q16—31no—
34C22×C3×M16—31no—
35C2×C3×C22⋊Q8—31no—
36C2×C3×C22⋊Q8—31no—
37C2×#946C2×C311yescharacteristic
38C22×#216C2×C311yescharacteristic, complemented
39C22×ΓL(1,𝔽25)C2×C311yescharacteristic
40C2×#897C2×C311yescharacteristic
41C3×#470—38no—
42C2×C3×#106—31no—
43C2×C3×#103—21no—
44C2×C3×#103—31no—
45C2×C3×#103D311yescharacteristic, complemented
46C2×C3×#103C2×C311yescharacteristic, complemented
47C2×#816C2×C311yescharacteristic, complemented
48C2×#808C2×C311yescharacteristic, complemented
49#4816C2×C318yesdirect product factor
50C23×C3×D3D411yescharacteristic
51C2×C32×Q8—21no—
52C2×C32×Q8—22no—
53C2×C32×Q8—22no—
54C2×C32×Q8C2312yes—
55C22×C32×C4—21no—
56C22×C32×C4—21no—
57C22×C32×C4C2311yescharacteristic
58C22×C3×Dic3—21no—
59C22×C3×Dic3—21no—
60C22×C3×Dic3D411yescharacteristic
61C2×C3×C4×D3—22no—
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3D412yes—
64C2×C3×C4×D3C2312yes—
65C2×C3×Dic6—22no—
66C2×C3×Dic6—22no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2312yes—
69C2×C32×C8—22no—
70C2×C32×C8C2312yes—
71C32×C4⋊C4—21no—
72C32×C4⋊C4C2312yes—
73C3×#216—28no—
74C3×(C2×D3)⋊C4—24no—
75C3×(C2×D3)⋊C4—24no—
76C3×C4⋊Dic3—21no—
77C3×C4⋊Dic3—24no—
78C3×C4⋊Dic3C2312yes—
79C3×Dic3⋊C4—24no—
80C2×C3×C3⋊C8D414yes—
81C2×C3×C3⋊C8C2312yes—
82C3×ΓL(1,𝔽25)—28no—
83C2×#470—31noSylow
84C22×C3×Q8—21no—
85C22×C3×Q8—31no—
86C22×C3×Q8C2×D311yescharacteristic
87C22×C3×Q8C22×C311yescharacteristic
88C23×C3×C4—31no—
89C22×C4×D3C22×C311yescharacteristic
90C22×Dic6C22×C311yescharacteristic
91C2×C3×Q16—34no—
92C2×C3×Q16—64no—
93C2×C3×M16—34no—
94C2×C3×M16—64no—
95C22×C3×C8—21no—
96C22×C3×C8—31no—
97C22×C3×C8C2×D311yescharacteristic
98C22×C3×C8C22×C311yescharacteristic
99C3×C22⋊Q8—34no—
100C3×C22⋊Q8—34no—
101C3×C22⋊Q8—62no—
102C3×C22⋊Q8—62no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—61no—
106C2×C3×C4⋊C4—61no—
107C2×C3×C4⋊C4C2×D311yescharacteristic
108C2×C3×C4⋊C4C22×C311yescharacteristic
109C2×C3×C22⋊C4—61no—
110C2×C3×C22⋊C4—61no—
111#946—22no—
112#946C22×C314yes—
113C2×#216—24no—
114C2×#216C22×C314yescomplemented
115C2×(C2×D3)⋊C4—21no—
116C2×(C2×D3)⋊C4—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3C22×C311yescharacteristic
119C2×Dic3⋊C4—21no—
120C22×C3⋊C8C22×C311yescharacteristic
121C2×ΓL(1,𝔽25)—24no—
122C2×ΓL(1,𝔽25)C22×C314yes—
123#897—22no—
124#897C22×C314yes—
125C3×#106—34no—
126C3×#103—24no—
127C3×#103—34no—
128C3×#103C2×D314yescomplemented
129C3×#103C22×C314yescomplemented
130#816C22×C314yescomplemented
131#808C22×C314yescomplemented
132C23×C32C2×D411yescharacteristic
133C22×C3×D3—22no—
134C22×C3×D3—22no—
135C22×C3×D3—22no—
136C22×C3×D3C2×D412yes—
137C32×Q8—22no—
138C32×Q8—22no—
139C32×Q8—42no—
140C2×C32×C4—22no—
141C2×C32×C4—22no—
142C2×C32×C4—42no—
143C2×C32×C4—42no—
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D412yes—
146C2×C32×C4#4112yes—
147C2×C32×C4C2411yescharacteristic
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—42no—
152C2×C3×Dic3—42no—
153C2×C3×Dic3C2×D412yes—
154C3×C4×D3—24no—
155C3×C4×D3—22no—
156C3×C4×D3—22no—
157C3×Dic6—22no—
158C3×Dic6—22no—
159C3×Dic6—42no—
160C32×C8—22no—
161C3×C3⋊C8C2×D414yes—
162C22×Q16—31no—
163C22×M16—31no—
164C2×C22⋊Q8—31no—
165C2×C22⋊Q8—31no—
166#470—38no—
167C2×#106—31no—
168C2×#103—31no—
169C2×#103C3×D311yescharacteristic, complemented
170C24×C3—31no—
171C23×D3C3×D411yescharacteristic
172C2×C3×Q8—21no—
173C2×C3×Q8—21no—
174C2×C3×Q8—24no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—22no—
180C2×C3×Q8—32no—
181C2×C3×Q8—41no—
182C2×C3×Q8—42no—
183C2×C3×Q8—61no—
184C2×C3×Q8—62no—
185C2×C3×Q8—62no—
186C2×C3×Q8C22×D312yes—
187C2×C3×Q8C23×C312yes—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—31no—
194C22×C3×C4—32no—
195C22×C3×C4—32no—
196C22×C3×C4—41no—
197C22×C3×C4—41no—
198C22×C3×C4—62no—
199C22×C3×C4—62no—
200C22×C3×C4—61no—
201C22×C3×C4—61no—
202C22×C3×C4C22×D311yescharacteristic
203C22×C3×C4C23×C311yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3C3×D411yescharacteristic
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3C3×D412yes—
210C2×C4×D3C23×C312yes—
211C2×Dic6—22no—
212C2×Dic6—22no—
213C2×Dic6—21no—
214C2×Dic6C23×C312yes—
215C3×Q16—68no—
216C3×M16—68no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—34no—
221C2×C3×C8—32no—
222C2×C3×C8—42no—
223C2×C3×C8C22×D312yes—
224C2×C3×C8C23×C312yes—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4—64no—
231C3×C4⋊C4—64no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4C22×D312yes—
234C3×C4⋊C4C23×C312yes—
235C3×C22⋊C4—64no—
236C3×C22⋊C4—64no—
237#216—28no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240C4⋊Dic3—21no—
241C4⋊Dic3—24no—
242C4⋊Dic3C23×C312yes—
243Dic3⋊C4—24no—
244C2×C3⋊C8C3×D414yes—
245C2×C3⋊C8C23×C312yes—
246ΓL(1,𝔽25)—28no—
247C22×C32#12112yes—
248C22×C32#13712yes—
249C22×C32#13712yes—
250C22×C32C22×D411yescharacteristic
251C2×C3×D3—22no—
252C2×C3×D3—22no—
253C2×C3×D3—22no—
254C2×C3×D3—44no—
255C32×C4—42no—
256C32×C4—42no—
257C32×C4#12112yes—
258C32×C4C22×D411yescharacteristic
259C32×C4C2×#4111yescharacteristic
260C3×Dic3—22no—
261C3×Dic3—42no—
262C3×Dic3—42no—
263C22×Q8—31no—
264C22×Q8C2×C3×D311yescharacteristic
265C23×C4—31no—
266C2×Q16—34no—
267C2×Q16—64no—
268C2×M16—34no—
269C2×M16—64no—
270C22×C8—31no—
271C22×C8C2×C3×D311yescharacteristic
272C22⋊Q8—34no—
273C22⋊Q8—34no—
274C22⋊Q8—62no—
275C22⋊Q8—62no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C2×C3×D311yescharacteristic
280C2×C22⋊C4—61no—
281C2×C22⋊C4—61no—
282#106—34no—
283#103—34no—
284#103C2×C3×D314yescomplemented
285C23×C3—21no—
286C23×C3—32no—
287C23×C3—62no—
288C23×C3—62no—
289C23×C3—62no—
290C23×C3D3×D411yescharacteristic
291C23×C3C2×C3×D411yescharacteristic
292C22×D3—22no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3C2×C3×D412yes—
296C3×Q8—24no—
297C3×Q8—22no—
298C3×Q8—22no—
299C3×Q8—22no—
300C3×Q8—22no—
301C3×Q8—42no—
302C3×Q8—44no—
303C3×Q8—42no—
304C3×Q8—42no—
305C3×Q8—62no—
306C3×Q8—62no—
307C3×Q8—122no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—21no—
313C2×C3×C4—22no—
314C2×C3×C4—21no—
315C2×C3×C4—22no—
316C2×C3×C4—32no—
317C2×C3×C4—42no—
318C2×C3×C4—42no—
319C2×C3×C4—44no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—62no—
325C2×C3×C4—64no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—82no—
331C2×C3×C4—122no—
332C2×C3×C4—122no—
333C2×C3×C4#23512yes—
334C2×C3×C4D3×D411yescharacteristic
335C2×C3×C4D3×D412yes—
336C2×C3×C4C2×C3×D411yescharacteristic
337C2×C3×C4C2×C3×D412yes—
338C2×C3×C4C3×#4112yes—
339C2×C3×C4C23×D311yescharacteristic
340C2×C3×C4C24×C311yescharacteristic
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3C2×C3×D412yes—
347C4×D3—24no—
348C4×D3—22no—
349C4×D3—22no—
350Dic6—22no—
351Dic6—22no—
352Dic6—42no—
353C3×C8—22no—
354C3×C8—22no—
355C3×C8—34no—
356C3×C8—42no—
357C3⋊C8C2×C3×D414yes—
358C2×C32#47014yescomplemented
359C2×C32C2×#12111yescharacteristic
360C2×C32C2×#13711yescharacteristic
361C2×C32C2×#13711yescharacteristic
362C3×D3—42no—
363C24—31no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—32no—
368C2×Q8—61no—
369C2×Q8—62no—
370C2×Q8—62no—
371C2×Q8C22×C3×D312yes—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—31no—
375C22×C4—32no—
376C22×C4—32no—
377C22×C4—62no—
378C22×C4—62no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4C22×C3×D311yescharacteristic
382Q16—68no—
383M16—68no—
384C2×C8—22no—
385C2×C8—32no—
386C2×C8—34no—
387C2×C8C22×C3×D312yes—
388C4⋊C4—21no—
389C4⋊C4—32no—
390C4⋊C4—64no—
391C4⋊C4—64no—
392C4⋊C4—61no—
393C4⋊C4C22×C3×D312yes—
394C22⋊C4—64no—
395C22⋊C4—64no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—21no—
400C22×C3—62no—
401C22×C3—62no—
402C22×C3—62no—
403C22×C3—124no—
404C22×C3#88412yes—
405C22×C3#91512yes—
406C22×C3#91812yes—
407C22×C3C3×#12112yes—
408C22×C3C3×#13712yes—
409C22×C3C3×#13712yes—
410C22×C3C2×D3×D411yescharacteristic
411C22×C3C22×C3×D411yescharacteristic
412C2×D3—22no—
413C2×D3—22no—
414C2×D3—22no—
415C2×D3—44no—
416C3×C4—22no—
417C3×C4—21no—
418C3×C4—21no—
419C3×C4—44no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—42no—
423C3×C4—42no—
424C3×C4—62no—
425C3×C4—82no—
426C3×C4—122no—
427C3×C4—122no—
428C3×C4#88412yes—
429C3×C4C3×#12112yes—
430C3×C4C2×#23511yescharacteristic
431C3×C4C2×D3×D411yescharacteristic
432C3×C4C22×C3×D411yescharacteristic
433C3×C4C2×C3×#4111yescharacteristic
434Dic3—22no—
435Dic3—42no—
436Dic3—42no—
437C32C2×#47011yescharacteristic, complemented, Sylow
438C23—32no—
439C23—62no—
440C23—62no—
441C23—62no—
442C23C3×D3×D411yescharacteristic
443Q8—22no—
444Q8—22no—
445Q8—42no—
446Q8—62no—
447Q8—62no—
448Q8—122no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—32no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—62no—
455C2×C4—64no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4C3×#23512yes—
463C2×C4C3×D3×D411yescharacteristic
464C2×C4C3×D3×D412yes—
465C2×C4C23×C3×D311yescharacteristic
466C8—22no—
467C8—34no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—24no—
472C2×C3—122no—
473C2×C3#481614yesdirect product factor
474C2×C3C3×#47014yescomplemented
475C2×C3C2×#88411yescharacteristic
476C2×C3C2×#91511yescharacteristic
477C2×C3C2×#91811yescharacteristic
478C2×C3C2×C3×#12111yescharacteristic
479C2×C3C2×C3×#13711yescharacteristic
480C2×C3C2×C3×#13711yescharacteristic
481D3—42no—
482C22—62no—
483C22—62no—
484C22—62no—
485C22—124no—
486C22C3×#88412yes—
487C22C3×#91512yes—
488C22C3×#91812yes—
489C22C2×C3×D3×D411yescharacteristic
490C4—42no—
491C4—42no—
492C4—62no—
493C4—122no—
494C4—122no—
495C4C3×#88412yes—
496C4C2×C3×#23511yescharacteristic
497C4C2×C3×D3×D411yescharacteristic
498C3—21no—
499C3C2×#481611yescharacteristic, direct product factor
500C3C2×C3×#47011yescharacteristic, complemented
501C2—122no—
502C2C3×#481614yesdirect product factor
503C2C2×C3×#88411yescharacteristic
504C2C2×C3×#91511yescharacteristic
505C2C2×C3×#91811yescharacteristic
subgroupG/H
centerC23×C3D3×D4
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×#103C2
socleC23×C32C2×D4
prime radicalC23×C3×D3D4
hypercenterC2×C3×#103D3
hypocenterC3C2×C3×#470
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C22×C3×C4 ◁ C2×C3×#103

Complements

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

NH ≅ G/Ntype
2. C22×C3×#216C2semidirect
5. C2×C32×#103C2semidirect
6. C2×C3×#816C2semidirect
7. C2×C3×#808C2semidirect
8. C3×#4816C2direct (N × H)
10. C2×#4816C3direct (N × H)
19. C2×C3×#216C22semidirect
30. C32×#103C22semidirect
31. C3×#816C22semidirect
32. C3×#808C22semidirect
38. C22×#216C2×C3semidirect
45. C2×C3×#103D3semidirect
46. C2×C3×#103C2×C3semidirect
47. C2×#816C2×C3semidirect
48. C2×#808C2×C3semidirect
49. #4816C2×C3direct (N × H)
114. C2×#216C22×C3semidirect
128. C3×#103C2×D3semidirect
129. C3×#103C22×C3semidirect
130. #816C22×C3semidirect
131. #808C22×C3semidirect
169. C2×#103C3×D3semidirect
284. #103C2×C3×D3semidirect
358. C2×C32#470semidirect
437. C32C2×#470semidirect
473. C2×C3#4816direct (N × H)
474. C2×C3C3×#470semidirect
499. C3C2×#4816direct (N × H)
500. C3C2×C3×#470semidirect
502. C2C3×#4816direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#9462
C22×C3×#2162
C22×C3×ΓL(1,𝔽25)2
C2×C3×#8972
C2×C32×#1032
C2×C3×#8162
C2×C3×#8082
C3×#48162
C2×C3×#4703
C2×#48163