Finite Group Databasebeta

← C2×C3×#5246 #11864903

Subgroups — C2×C3×#5246 #11864903

Order 1152.

1813 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×2−1+4C211yesmaximal, characteristic
2C2×C3×#951C212yesmaximal, complemented
3C2×C3×#944C211yesmaximal, characteristic, complemented
4C3×#5246C218yesmaximal, direct product factor
5C2×C3×#837C212yesmaximal, complemented
6C2×C3×#836C211yesmaximal, characteristic, complemented
7C2×C3×#456—31nomaximal
8C2×#5246C311yesmaximal, characteristic, direct product factor
9C32×2−1+4—24no—
10C32×2−1+4—22no—
11C32×2−1+4C2214yes—
12C2×C32×#41—22no—
13C2×C32×#41—22no—
14C2×C32×#41C2212yes—
15C22×C32×Q8—22no—
16C22×C32×Q8C2211yescharacteristic
17C22×C3×Dic6C2211yescharacteristic
18C3×#951—24no—
19C3×#951C2218yescomplemented
20C3×#944—22no—
21C3×#944C2214yescomplemented
22C2×C3×#216—22no—
23C2×C3×#214—22no—
24C2×C3×Dic3⋊C4—22no—
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×#208C2212yes—
27C3×#837C2218yescomplemented
28C3×#836C2214yescomplemented
29C2×C3×2−1+4—21no—
30C2×C3×2−1+4D311yescharacteristic
31C2×C3×2−1+4C2×C311yescharacteristic
32C2×C3×#137—32no—
33C2×C3×C4⋊Q8—31no—
34C2×#951C2×C312yescomplemented
35C2×#944C2×C311yescharacteristic, complemented
36C3×#456—38no—
37C2×C3×#104—32no—
38C2×C3×#101—31no—
39#5246C2×C318yesdirect product factor
40C2×#837C2×C312yescomplemented
41C2×#836C2×C311yescharacteristic, complemented
42C32×#41—28no—
43C32×#41—28no—
44C32×#41—28no—
45C32×#41—28no—
46C32×#41—22no—
47C32×#41—22no—
48C32×#41D414yes—
49C32×#41C2314yes—
50C2×C32×Q8—22no—
51C2×C32×Q8—28no—
52C2×C32×Q8—28no—
53C2×C32×Q8—22no—
54C2×C32×Q8—22no—
55C2×C32×Q8D412yes—
56C2×C32×Q8D412yes—
57C2×C32×Q8C2312yes—
58C2×C32×D4—22no—
59C2×C32×D4—22no—
60C2×C32×D4D412yes—
61C22×C32×C4—24no—
62C22×C32×C4—22no—
63C22×C32×C4D412yes—
64C22×C32×C4C2311yescharacteristic
65C22×C3×Dic3—21no—
66C22×C3×Dic3—21no—
67C2×C3×Dic6—22no—
68C2×C3×Dic6—22no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2312yes—
71C3×#216—28no—
72C3×#214—28no—
73C3×Dic3⋊C4—28no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3C2312yes—
76C3×#208—22no—
77C3×#208C2314yes—
78C2×C3×C3⋊C8—22no—
79C2×#456—31noSylow
80C3×2−1+4—28no—
81C3×2−1+4—24no—
82C3×2−1+4—24no—
83C3×2−1+4—24no—
84C3×2−1+4—22no—
85C3×2−1+4—22no—
86C3×2−1+4—24no—
87C3×2−1+4C2×D314yes—
88C3×2−1+4C22×C314yes—
89C2×C3×#41—24no—
90C2×C3×#41—24no—
91C2×C3×#41—22no—
92C2×C3×#41—22no—
93C2×C3×#41—22no—
94C2×C3×#41—22no—
95C2×C3×#41—22no—
96C2×C3×#41C2×D312yes—
97C2×C3×#41C22×C312yes—
98C22×C3×Q8—24no—
99C22×C3×Q8—22no—
100C22×C3×Q8—22no—
101C22×C3×Q8—21no—
102C22×C3×Q8—31no—
103C22×C3×Q8C2×D311yescharacteristic
104C22×C3×Q8C22×C311yescharacteristic
105C22×Dic6C22×C311yescharacteristic
106C3×#137—38no—
107C3×#137—64no—
108C2×C3×Q16—62no—
109C2×C3×SD16—62no—
110C2×C3×M16—32no—
111C3×C4⋊Q8—34no—
112C3×C4⋊Q8—62no—
113C2×C3×C4⋊C4—62no—
114C2×C3×C42—31no—
115#951—24no—
116#951C22×C318yescomplemented
117#944—22no—
118#944C22×C314yescomplemented
119C2×#216—22no—
120C2×#214—22no—
121C2×Dic3⋊C4—22no—
122C2×C4×Dic3C22×C311yescharacteristic
123C2×#208C22×C312yes—
124C3×#104—38no—
125C3×#101—34no—
126#837C22×C318yescomplemented
127#836C22×C314yescomplemented
128C23×C32—22no—
129C23×C32C2×D411yescharacteristic
130C32×Q8—28no—
131C32×Q8—22no—
132C32×Q8—24no—
133C32×Q8—24no—
134C32×Q8C2×D414yes—
135C32×D4—28no—
136C32×D4—24no—
137C32×D4—24no—
138C32×D4—22no—
139C32×D4C2×D414yes—
140C2×C32×C4—216no—
141C2×C32×C4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—24no—
145C2×C32×C4—24no—
146C2×C32×C4C2×D414yes—
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D412yes—
149C2×C32×C4C2×D412yes—
150C2×C32×C4C2411yescharacteristic
151C2×C3×Dic3—24no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—41no—
155C2×C3×Dic3—41no—
156C3×Dic6—24no—
157C3×Dic6—42no—
158C3×C3⋊C8—24no—
159C2×2−1+4C3×D311yescharacteristic
160C2×#137—32no—
161C2×C4⋊Q8—31no—
162#456—38no—
163C2×#104—32no—
164C2×#101—31no—
165C3×#41—216no—
166C3×#41—216no—
167C3×#41—28no—
168C3×#41—28no—
169C3×#41—28no—
170C3×#41—28no—
171C3×#41—216no—
172C3×#41—216no—
173C3×#41—24no—
174C3×#41—24no—
175C3×#41—28no—
176C3×#41—28no—
177C3×#41—28no—
178C3×#41—22no—
179C3×#41—22no—
180C3×#41—28no—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—24no—
184C3×#41—24no—
185C3×#41#6714yes—
186C3×#41C3×D414yes—
187C3×#41C22×D314yes—
188C3×#41C23×C314yes—
189C2×C3×Q8—24no—
190C2×C3×Q8—22no—
191C2×C3×Q8—22no—
192C2×C3×Q8—24no—
193C2×C3×Q8—216no—
194C2×C3×Q8—28no—
195C2×C3×Q8—28no—
196C2×C3×Q8—216no—
197C2×C3×Q8—24no—
198C2×C3×Q8—28no—
199C2×C3×Q8—28no—
200C2×C3×Q8—22no—
201C2×C3×Q8—22no—
202C2×C3×Q8—22no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—22no—
206C2×C3×Q8—22no—
207C2×C3×Q8—32no—
208C2×C3×Q8—61no—
209C2×C3×Q8—62no—
210C2×C3×Q8—62no—
211C2×C3×Q8#6712yes—
212C2×C3×Q8#6712yes—
213C2×C3×Q8C3×D412yes—
214C2×C3×Q8C3×D412yes—
215C2×C3×Q8C22×D312yes—
216C2×C3×Q8C23×C312yes—
217C2×C3×D4—24no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—24no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—22no—
224C2×C3×D4#6712yes—
225C2×C3×D4C3×D412yes—
226C22×C3×C4—28no—
227C22×C3×C4—24no—
228C22×C3×C4—24no—
229C22×C3×C4—24no—
230C22×C3×C4—22no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4—21no—
234C22×C3×C4—61no—
235C22×C3×C4—61no—
236C22×C3×C4#6712yes—
237C22×C3×C4C3×D412yes—
238C22×C3×C4C22×D311yescharacteristic
239C22×C3×C4C23×C311yescharacteristic
240C22×Dic3—21no—
241C22×Dic3—21no—
242C2×Dic6—22no—
243C2×Dic6—22no—
244C2×Dic6—21no—
245C2×Dic6C23×C312yes—
246C3×Q16—68no—
247C3×SD16—68no—
248C3×M16—34no—
249C3×M16—62no—
250C2×C3×C8—62no—
251C3×C4⋊C4—68no—
252C3×C42—32no—
253C3×C42—61no—
254#216—28no—
255#214—28no—
256Dic3⋊C4—28no—
257C4×Dic3—21no—
258C4×Dic3C23×C312yes—
259#208—22no—
260#208C23×C314yes—
261C2×C3⋊C8—22no—
262C22×C32—24no—
263C22×C32—22no—
264C22×C32—44no—
265C22×C32#12011yescharacteristic
266C22×C32#12011yescharacteristic
267C22×C32C22×D411yescharacteristic
268C32×C4—24no—
269C32×C4—24no—
270C32×C4#12012yes—
271C32×C4C22×D412yes—
272C3×Dic3—24no—
273C3×Dic3—42no—
2742−1+4—24no—
2752−1+4—22no—
2762−1+4C2×C3×D314yes—
277C2×#41—22no—
278C2×#41—22no—
279C2×#41C2×C3×D312yes—
280C22×Q8—22no—
281C22×Q8—31no—
282C22×Q8C2×C3×D311yescharacteristic
283#137—38no—
284#137—64no—
285C2×Q16—62no—
286C2×SD16—62no—
287C2×M16—32no—
288C4⋊Q8—34no—
289C4⋊Q8—62no—
290C2×C4⋊C4—62no—
291C2×C42—31no—
292#104—38no—
293#101—34no—
294C23×C3—24no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—21no—
298C23×C3C2×#6711yescharacteristic
299C23×C3C2×C3×D411yescharacteristic
300C3×Q8—216no—
301C3×Q8—28no—
302C3×Q8—28no—
303C3×Q8—28no—
304C3×Q8—24no—
305C3×Q8—28no—
306C3×Q8—22no—
307C3×Q8—24no—
308C3×Q8—22no—
309C3×Q8—24no—
310C3×Q8—24no—
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—64no—
314C3×Q8—122no—
315C3×Q8C2×#6714yes—
316C3×Q8C2×C3×D414yes—
317C3×D4—216no—
318C3×D4—28no—
319C3×D4—28no—
320C3×D4—28no—
321C3×D4—28no—
322C3×D4—24no—
323C3×D4—24no—
324C3×D4—24no—
325C3×D4—24no—
326C3×D4—24no—
327C3×D4—22no—
328C3×D4—22no—
329C3×D4—24no—
330C3×D4C2×#6714yes—
331C3×D4C2×C3×D414yes—
332C2×C3×C4—24no—
333C2×C3×C4—24no—
334C2×C3×C4—232no—
335C2×C3×C4—216no—
336C2×C3×C4—216no—
337C2×C3×C4—24no—
338C2×C3×C4—28no—
339C2×C3×C4—28no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—64no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—121no—
359C2×C3×C4—121no—
360C2×C3×C4C2×#6714yes—
361C2×C3×C4C2×#6711yescharacteristic
362C2×C3×C4C2×#6712yes—
363C2×C3×C4C2×#6712yes—
364C2×C3×C4C2×C3×D414yes—
365C2×C3×C4C2×C3×D411yescharacteristic
366C2×C3×C4C2×C3×D412yes—
367C2×C3×C4C2×C3×D412yes—
368C2×C3×C4C23×D311yescharacteristic
369C2×C3×C4C24×C311yescharacteristic
370C2×Dic3—24no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375Dic6—24no—
376Dic6—42no—
377C3×C8—64no—
378C3⋊C8—24no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—44no—
382C2×C32#45612yescomplemented
383C2×C32C2×#12011yescharacteristic
384#41—28no—
385#41—28no—
386#41—28no—
387#41—28no—
388#41—22no—
389#41—22no—
390#41C3×#6714yes—
391#41C22×C3×D314yes—
392C2×Q8—22no—
393C2×Q8—28no—
394C2×Q8—28no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—32no—
398C2×Q8—61no—
399C2×Q8—62no—
400C2×Q8—62no—
401C2×Q8C3×#6712yes—
402C2×Q8C3×#6712yes—
403C2×Q8C22×C3×D312yes—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4C3×#6712yes—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4C3×#6712yes—
412C22×C4C22×C3×D311yescharacteristic
413Q16—68no—
414SD16—68no—
415M16—34no—
416M16—62no—
417C2×C8—62no—
418C4⋊C4—68no—
419C42—32no—
420C42—61no—
421C22×C3—22no—
422C22×C3—22no—
423C22×C3—28no—
424C22×C3—24no—
425C22×C3—24no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—42no—
430C22×C3—48no—
431C22×C3—44no—
432C22×C3—44no—
433C22×C3#95311yescharacteristic
434C22×C3#95311yescharacteristic
435C22×C3C3×#12011yescharacteristic
436C22×C3C3×#12011yescharacteristic
437C22×C3C22×#6711yescharacteristic
438C22×C3C22×C3×D411yescharacteristic
439C3×C4—28no—
440C3×C4—28no—
441C3×C4—24no—
442C3×C4—24no—
443C3×C4—24no—
444C3×C4—24no—
445C3×C4—22no—
446C3×C4—22no—
447C3×C4—64no—
448C3×C4—122no—
449C3×C4#95312yes—
450C3×C4C3×#12012yes—
451C3×C4C22×#6712yes—
452C3×C4C22×C3×D412yes—
453Dic3—24no—
454Dic3—42no—
455C32C2×#45611yescharacteristic, complemented, Sylow
456C23—22no—
457C23C2×C3×#6711yescharacteristic
458Q8—28no—
459Q8—22no—
460Q8—24no—
461Q8—24no—
462Q8—64no—
463Q8—122no—
464Q8C2×C3×#6714yes—
465D4—28no—
466D4—24no—
467D4—24no—
468D4—22no—
469D4C2×C3×#6714yes—
470C2×C4—216no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4C2×C3×#6714yes—
482C2×C4C2×C3×#6711yescharacteristic
483C2×C4C2×C3×#6712yes—
484C2×C4C2×C3×#6712yes—
485C2×C4C23×C3×D311yescharacteristic
486C8—64no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—22no—
492C2×C3—21no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—48no—
496C2×C3—44no—
497C2×C3—44no—
498C2×C3#524612yesdirect product factor
499C2×C3C3×#45612yescomplemented
500C2×C3C2×#95311yescharacteristic
501C2×C3C2×C3×#12011yescharacteristic
502C22—22no—
503C22—24no—
504C22—44no—
505C22C3×#95311yescharacteristic
506C22C3×#95311yescharacteristic
507C22C22×C3×#6711yescharacteristic
508C4—24no—
509C4—24no—
510C4—64no—
511C4—122no—
512C4C3×#95312yes—
513C4C22×C3×#6712yes—
514C3—21no—
515C3C2×#524611yescharacteristic, direct product factor
516C3C2×C3×#45611yescharacteristic, complemented
517C2—21no—
518C2—21no—
519C2—44no—
520C2C3×#524612yesdirect product factor
521C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC22×C3#953
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
Fitting, prime radicalC2×C32×2−1+4C2
socleC22×C32#120
hypercenterC2×C3×2−1+4D3
hypocenterC3C2×C3×#456
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×2−1+4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#951C2semidirect
3. C2×C3×#944C2semidirect
4. C3×#5246C2direct (N × H)
5. C2×C3×#837C2semidirect
6. C2×C3×#836C2semidirect
8. C2×#5246C3direct (N × H)
19. C3×#951C22semidirect
21. C3×#944C22semidirect
27. C3×#837C22semidirect
28. C3×#836C22semidirect
34. C2×#951C2×C3semidirect
35. C2×#944C2×C3semidirect
39. #5246C2×C3direct (N × H)
40. C2×#837C2×C3semidirect
41. C2×#836C2×C3semidirect
116. #951C22×C3semidirect
118. #944C22×C3semidirect
126. #837C22×C3semidirect
127. #836C22×C3semidirect
382. C2×C32#456semidirect
455. C32C2×#456semidirect
498. C2×C3#5246direct (N × H)
499. C2×C3C3×#456semidirect
515. C3C2×#5246direct (N × H)
516. C3C2×C3×#456semidirect
520. C2C3×#5246direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×2−1+42
C2×C3×#9512
C2×C3×#9442
C3×#52462
C2×C3×#8372
C2×C3×#8362
C2×C3×#4563
C2×#52463