Finite Group Databasebeta

← C2×C3×#5245 #11864857

Subgroups — C2×C3×#5245 #11864857

Order 1152.

1853 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×2−1+4C211yesmaximal, characteristic, complemented
2C2×C3×#949C212yesmaximal, complemented
3C2×C3×#947C211yesmaximal, characteristic, complemented
4C3×#5245C218yesmaximal, direct product factor
5C2×C3×#837C212yesmaximal, complemented
6C2×C3×#836C211yesmaximal, characteristic, complemented
7C2×C3×#454—31nomaximal
8C2×#5245C311yesmaximal, characteristic, direct product factor
9C32×2−1+4—24no—
10C32×2−1+4—22no—
11C32×2−1+4C2214yescomplemented
12C2×C32×#41—22no—
13C2×C32×#41—22no—
14C2×C32×#41C2212yes—
15C22×C32×Q8—22no—
16C22×C32×Q8C2211yescharacteristic
17C22×C3×D12C2211yescharacteristic
18C3×#949—24no—
19C3×#949C2218yescomplemented
20C3×#947—22no—
21C3×#947C2214yescomplemented
22C2×C3×#215—22no—
23C2×C3×#213—22no—
24C2×C3×(C2×D3)⋊C4—22no—
25C2×C3×C4×Dic3C2211yescharacteristic
26C2×C3×#208C2212yes—
27C3×#837C2218yescomplemented
28C3×#836C2214yescomplemented
29C2×C3×2−1+4—21no—
30C2×C3×2−1+4D311yescharacteristic, complemented
31C2×C3×2−1+4C2×C311yescharacteristic, complemented
32C2×C3×AGL(1,ℤ/8)—32no—
33C2×C3×#124—31no—
34C2×#949C2×C312yescomplemented
35C2×#947C2×C311yescharacteristic, complemented
36C3×#454—38no—
37C2×C3×#104—32no—
38C2×C3×#101—31no—
39#5245C2×C318yesdirect product factor
40C2×#837C2×C312yescomplemented
41C2×#836C2×C311yescharacteristic, complemented
42C23×C3×D3—21no—
43C32×#41—28no—
44C32×#41—28no—
45C32×#41—28no—
46C32×#41—28no—
47C32×#41—22no—
48C32×#41—22no—
49C32×#41D414yes—
50C32×#41C2314yes—
51C2×C32×Q8—22no—
52C2×C32×Q8—28no—
53C2×C32×Q8—28no—
54C2×C32×Q8—22no—
55C2×C32×Q8—22no—
56C2×C32×Q8D412yes—
57C2×C32×Q8D412yes—
58C2×C32×Q8C2312yes—
59C2×C32×D4—22no—
60C2×C32×D4—22no—
61C2×C32×D4D412yes—
62C22×C32×C4—24no—
63C22×C32×C4—22no—
64C22×C32×C4D412yes—
65C22×C32×C4C2311yescharacteristic
66C22×C3×Dic3—21no—
67C2×C3×D12—22no—
68C2×C3×D12—22no—
69C2×C3×D12—21no—
70C2×C3×D12C2312yes—
71C3×#215—28no—
72C3×#213—28no—
73C3×(C2×D3)⋊C4—28no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3C2312yes—
76C3×#208—22no—
77C3×#208C2314yes—
78C2×C3×C3⋊C8—22no—
79C2×#454—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×D314yescomplemented
88C3×2−1+4C22×C314yescomplemented
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×Q8C2×D311yescharacteristic
103C22×C3×Q8C22×C311yescharacteristic
104C22×C3×D4—31no—
105C22×D12C22×C311yescharacteristic
106C3×AGL(1,ℤ/8)—38no—
107C3×AGL(1,ℤ/8)—64no—
108C2×C3×SD16—62no—
109C2×C3×D8—62no—
110C2×C3×M16—32no—
111C3×#124—34no—
112C3×#124—62no—
113C2×C3×C22⋊C4—62no—
114C2×C3×C42—31no—
115#949—24no—
116#949C22×C318yescomplemented
117#947—22no—
118#947C22×C314yescomplemented
119C2×#215—22no—
120C2×#213—22no—
121C2×(C2×D3)⋊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
130C22×C3×D3—22no—
131C22×C3×D3—44no—
132C22×C3×D3—41no—
133C22×C3×D3—41no—
134C32×Q8—28no—
135C32×Q8—22no—
136C32×Q8—24no—
137C32×Q8—24no—
138C32×Q8C2×D414yes—
139C32×D4—28no—
140C32×D4—24no—
141C32×D4—24no—
142C32×D4—22no—
143C32×D4C2×D414yes—
144C2×C32×C4—216no—
145C2×C32×C4—22no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—24no—
149C2×C32×C4—24no—
150C2×C32×C4C2×D414yes—
151C2×C32×C4C2×D411yescharacteristic
152C2×C32×C4C2×D412yes—
153C2×C32×C4C2×D412yes—
154C2×C32×C4C2411yescharacteristic
155C2×C3×Dic3—22no—
156C2×C3×Dic3—42no—
157C3×D12—24no—
158C3×D12—42no—
159C3×C3⋊C8—24no—
160C2×2−1+4C3×D311yescharacteristic, complemented
161C2×AGL(1,ℤ/8)—32no—
162C2×#124—31no—
163#454—38no—
164C2×#104—32no—
165C2×#101—31no—
166C24×C3—61no—
167C23×D3—21no—
168C3×#41—216no—
169C3×#41—216no—
170C3×#41—28no—
171C3×#41—28no—
172C3×#41—28no—
173C3×#41—28no—
174C3×#41—216no—
175C3×#41—216no—
176C3×#41—24no—
177C3×#41—24no—
178C3×#41—28no—
179C3×#41—28no—
180C3×#41—28no—
181C3×#41—22no—
182C3×#41—22no—
183C3×#41—28no—
184C3×#41—22no—
185C3×#41—22no—
186C3×#41—24no—
187C3×#41—24no—
188C3×#41#6714yes—
189C3×#41C3×D414yes—
190C3×#41C22×D314yes—
191C3×#41C23×C314yes—
192C2×C3×Q8—24no—
193C2×C3×Q8—22no—
194C2×C3×Q8—22no—
195C2×C3×Q8—24no—
196C2×C3×Q8—216no—
197C2×C3×Q8—28no—
198C2×C3×Q8—28no—
199C2×C3×Q8—216no—
200C2×C3×Q8—24no—
201C2×C3×Q8—28no—
202C2×C3×Q8—28no—
203C2×C3×Q8—22no—
204C2×C3×Q8—22no—
205C2×C3×Q8—22no—
206C2×C3×Q8—22no—
207C2×C3×Q8—22no—
208C2×C3×Q8—22no—
209C2×C3×Q8—22no—
210C2×C3×Q8#6712yes—
211C2×C3×Q8#6712yes—
212C2×C3×Q8C3×D412yes—
213C2×C3×Q8C3×D412yes—
214C2×C3×Q8C22×D312yes—
215C2×C3×Q8C23×C312yes—
216C2×C3×D4—24no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—24no—
220C2×C3×D4—22no—
221C2×C3×D4—22no—
222C2×C3×D4—22no—
223C2×C3×D4—32no—
224C2×C3×D4—62no—
225C2×C3×D4—62no—
226C2×C3×D4—61no—
227C2×C3×D4#6712yes—
228C2×C3×D4C3×D412yes—
229C22×C3×C4—28no—
230C22×C3×C4—24no—
231C22×C3×C4—24no—
232C22×C3×C4—24no—
233C22×C3×C4—22no—
234C22×C3×C4—22no—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C22×C3×C4—61no—
238C22×C3×C4#6712yes—
239C22×C3×C4C3×D412yes—
240C22×C3×C4C22×D311yescharacteristic
241C22×C3×C4C23×C311yescharacteristic
242C22×Dic3—21no—
243C2×D12—22no—
244C2×D12—22no—
245C2×D12—21no—
246C2×D12C23×C312yes—
247C3×SD16—68no—
248C3×D8—68no—
249C3×M16—34no—
250C3×M16—62no—
251C2×C3×C8—62no—
252C3×C22⋊C4—68no—
253C3×C42—32no—
254C3×C42—61no—
255#215—28no—
256#213—28no—
257(C2×D3)⋊C4—28no—
258C4×Dic3—21no—
259C4×Dic3C23×C312yes—
260#208—22no—
261#208C23×C314yes—
262C2×C3⋊C8—22no—
263C22×C32—24no—
264C22×C32—22no—
265C22×C32—44no—
266C22×C32#12011yescharacteristic
267C22×C32#12011yescharacteristic
268C22×C32C22×D411yescharacteristic
269C2×C3×D3—44no—
270C2×C3×D3—42no—
271C2×C3×D3—82no—
272C2×C3×D3—82no—
273C32×C4—24no—
274C32×C4—24no—
275C32×C4#12012yes—
276C32×C4C22×D412yes—
277C3×Dic3—42no—
2782−1+4—24no—
2792−1+4—22no—
2802−1+4C2×C3×D314yescomplemented
281C2×#41—22no—
282C2×#41—22no—
283C2×#41C2×C3×D312yes—
284C22×Q8—22no—
285C22×Q8C2×C3×D311yescharacteristic
286C22×D4—31no—
287AGL(1,ℤ/8)—38no—
288AGL(1,ℤ/8)—64no—
289C2×SD16—62no—
290C2×D8—62no—
291C2×M16—32no—
292#124—34no—
293#124—62no—
294C2×C22⋊C4—62no—
295C2×C42—31no—
296#104—38no—
297#101—34no—
298C23×C3—24no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—21no—
302C23×C3—62no—
303C23×C3—124no—
304C23×C3—121no—
305C23×C3—121no—
306C23×C3C2×#6711yescharacteristic
307C23×C3C2×C3×D411yescharacteristic
308C22×D3—22no—
309C22×D3—44no—
310C22×D3—41no—
311C22×D3—41no—
312C3×Q8—216no—
313C3×Q8—28no—
314C3×Q8—28no—
315C3×Q8—28no—
316C3×Q8—24no—
317C3×Q8—28no—
318C3×Q8—22no—
319C3×Q8—24no—
320C3×Q8—22no—
321C3×Q8—24no—
322C3×Q8—24no—
323C3×Q8—24no—
324C3×Q8—24no—
325C3×Q8C2×#6714yes—
326C3×Q8C2×C3×D414yes—
327C3×D4—216no—
328C3×D4—28no—
329C3×D4—28no—
330C3×D4—28no—
331C3×D4—28no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—24no—
335C3×D4—24no—
336C3×D4—24no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—24no—
340C3×D4—64no—
341C3×D4—122no—
342C3×D4C2×#6714yes—
343C3×D4C2×C3×D414yes—
344C2×C3×C4—24no—
345C2×C3×C4—24no—
346C2×C3×C4—232no—
347C2×C3×C4—216no—
348C2×C3×C4—216no—
349C2×C3×C4—24no—
350C2×C3×C4—28no—
351C2×C3×C4—28no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—21no—
366C2×C3×C4—22no—
367C2×C3×C4—62no—
368C2×C3×C4—122no—
369C2×C3×C4C2×#6714yes—
370C2×C3×C4C2×#6711yescharacteristic
371C2×C3×C4C2×#6712yes—
372C2×C3×C4C2×#6712yes—
373C2×C3×C4C2×C3×D414yes—
374C2×C3×C4C2×C3×D411yescharacteristic
375C2×C3×C4C2×C3×D412yes—
376C2×C3×C4C2×C3×D412yes—
377C2×C3×C4C23×D311yescharacteristic
378C2×C3×C4C24×C311yescharacteristic
379C2×Dic3—22no—
380C2×Dic3—42no—
381D12—24no—
382D12—42no—
383C3×C8—64no—
384C3⋊C8—24no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—44no—
388C2×C32#45412yescomplemented
389C2×C32C2×#12011yescharacteristic
390C3×D3—82no—
391C24—61no—
392#41—28no—
393#41—28no—
394#41—28no—
395#41—28no—
396#41—22no—
397#41—22no—
398#41C3×#6714yes—
399#41C22×C3×D314yes—
400C2×Q8—22no—
401C2×Q8—28no—
402C2×Q8—28no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8C3×#6712yes—
406C2×Q8C3×#6712yes—
407C2×Q8C22×C3×D312yes—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—32no—
411C2×D4—62no—
412C2×D4—62no—
413C2×D4—61no—
414C2×D4C3×#6712yes—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—61no—
418C22×C4C3×#6712yes—
419C22×C4C22×C3×D311yescharacteristic
420SD16—68no—
421D8—68no—
422M16—34no—
423M16—62no—
424C2×C8—62no—
425C22⋊C4—68no—
426C42—32no—
427C42—61no—
428C22×C3—22no—
429C22×C3—22no—
430C22×C3—28no—
431C22×C3—24no—
432C22×C3—24no—
433C22×C3—21no—
434C22×C3—21no—
435C22×C3—21no—
436C22×C3—42no—
437C22×C3—48no—
438C22×C3—44no—
439C22×C3—44no—
440C22×C3—124no—
441C22×C3—122no—
442C22×C3—242no—
443C22×C3—242no—
444C22×C3#95311yescharacteristic
445C22×C3#95311yescharacteristic
446C22×C3C3×#12011yescharacteristic
447C22×C3C3×#12011yescharacteristic
448C22×C3C22×#6711yescharacteristic
449C22×C3C22×C3×D411yescharacteristic
450C2×D3—44no—
451C2×D3—42no—
452C2×D3—82no—
453C2×D3—82no—
454C3×C4—28no—
455C3×C4—28no—
456C3×C4—24no—
457C3×C4—24no—
458C3×C4—24no—
459C3×C4—24no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—122no—
463C3×C4#95312yes—
464C3×C4C3×#12012yes—
465C3×C4C22×#6712yes—
466C3×C4C22×C3×D412yes—
467Dic3—42no—
468C32C2×#45411yescharacteristic, complemented, Sylow
469C23—22no—
470C23—62no—
471C23—124no—
472C23—121no—
473C23—121no—
474C23C2×C3×#6711yescharacteristic
475Q8—28no—
476Q8—22no—
477Q8—24no—
478Q8—24no—
479Q8C2×C3×#6714yes—
480D4—28no—
481D4—24no—
482D4—24no—
483D4—22no—
484D4—64no—
485D4—122no—
486D4C2×C3×#6714yes—
487C2×C4—216no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—62no—
494C2×C4—122no—
495C2×C4C2×C3×#6714yes—
496C2×C4C2×C3×#6711yescharacteristic
497C2×C4C2×C3×#6712yes—
498C2×C4C2×C3×#6712yes—
499C2×C4C23×C3×D311yescharacteristic
500C8—64no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—22no—
506C2×C3—21no—
507C2×C3—41no—
508C2×C3—41no—
509C2×C3—48no—
510C2×C3—44no—
511C2×C3—44no—
512C2×C3—242no—
513C2×C3#524512yesdirect product factor
514C2×C3C3×#45412yescomplemented
515C2×C3C2×#95311yescharacteristic
516C2×C3C2×C3×#12011yescharacteristic
517D3—82no—
518C22—22no—
519C22—24no—
520C22—44no—
521C22—124no—
522C22—122no—
523C22—242no—
524C22—242no—
525C22C3×#95311yescharacteristic
526C22C3×#95311yescharacteristic
527C22C22×C3×#6711yescharacteristic
528C4—24no—
529C4—24no—
530C4—122no—
531C4C3×#95312yes—
532C4C22×C3×#6712yes—
533C3—21no—
534C3C2×#524511yescharacteristic, direct product factor
535C3C2×C3×#45411yescharacteristic, complemented
536C2—21no—
537C2—21no—
538C2—44no—
539C2—242no—
540C2C3×#524512yesdirect product factor
541C2C2×C3×#95311yescharacteristic
subgroupG/H
centerC22×C3#953
commutator, cosocleC2×C3×C4C24×C3
FrattiniC2×C4C23×C3×D3
FittingC2×C32×2−1+4C2
socleC22×C32#120
hypercenterC2×C3×2−1+4D3
hypocenterC3C2×C3×#454
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
1. C2×C32×2−1+4C2semidirect
2. C2×C3×#949C2semidirect
3. C2×C3×#947C2semidirect
4. C3×#5245C2direct (N × H)
5. C2×C3×#837C2semidirect
6. C2×C3×#836C2semidirect
8. C2×#5245C3direct (N × H)
11. C32×2−1+4C22semidirect
19. C3×#949C22semidirect
21. C3×#947C22semidirect
27. C3×#837C22semidirect
28. C3×#836C22semidirect
30. C2×C3×2−1+4D3semidirect
31. C2×C3×2−1+4C2×C3semidirect
34. C2×#949C2×C3semidirect
35. C2×#947C2×C3semidirect
39. #5245C2×C3direct (N × H)
40. C2×#837C2×C3semidirect
41. C2×#836C2×C3semidirect
87. C3×2−1+4C2×D3semidirect
88. C3×2−1+4C22×C3semidirect
116. #949C22×C3semidirect
118. #947C22×C3semidirect
126. #837C22×C3semidirect
127. #836C22×C3semidirect
160. C2×2−1+4C3×D3semidirect
280. 2−1+4C2×C3×D3semidirect
388. C2×C32#454semidirect
468. C32C2×#454semidirect
513. C2×C3#5245direct (N × H)
514. C2×C3C3×#454semidirect
534. C3C2×#5245direct (N × H)
535. C3C2×C3×#454semidirect
540. C2C3×#5245direct (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×#9492
C2×C3×#9472
C3×#52452
C2×C3×#8372
C2×C3×#8362
C2×C3×#4543
C2×#52453