Finite Group Databasebeta

← C3×#10590479 #13157595

Subgroups — C3×#10590479 #13157595

Order 1728.

686 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#893C211yesmaximal, characteristic
2C3×#63844C211yesmaximal, characteristic, complemented
3C2×C3×#3716C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic, complemented
5C32×#808C211yesmaximal, characteristic, complemented
6C3×#63562C211yesmaximal, characteristic
7C3×Dic3×C3⋊C8C211yesmaximal, characteristic, complemented
8C3×C4×#214—31nomaximal
9C3×#4852—31nomaximal
10#10590479C311yesmaximal, characteristic, direct product factor
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3—21no—
14C33×C4⋊C4C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×(C2×D3)⋊C4—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic32—21no—
22C3×#3716C414yescomplemented
23C32×C4×D4—31no—
24C2×C3×#214—31no—
25C2×C3×#204—31no—
26C3×#893—21no—
27C3×#893D311yescharacteristic
28C3×#893C2×C311yescharacteristic
29C3×#887—31no—
30C3×C4×Dic6—31no—
31#63844C2×C311yescharacteristic, complemented
32C2×#3716C2×C311yescharacteristic
33C32×#106—31no—
34#63637C2×C311yescharacteristic, complemented
35C3×#832—31no—
36C3×C8×Dic3—31no—
37C3×#810—31no—
38C3×#810—31no—
39C3×#810—61no—
40C3×#809—31no—
41C3×#808—21no—
42C3×#808D311yescharacteristic, complemented
43C3×#808C2×C311yescharacteristic, complemented
44C3×C4×C3⋊C8—31no—
45#63562C2×C311yescharacteristic
46Dic3×C3⋊C8C2×C311yescharacteristic, complemented
47C22×C32×D3—21no—
48C2×C33×C4—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×#634—21no—
52C3×#634C2×C412yescomplemented
53C2×C32×Dic3—21no—
54C2×C32×Dic3D411yescharacteristic
55C2×C32×Dic3D411yescharacteristic
56C32×D12—21no—
57C32×D12C2×C412yes—
58C32×C4×D3—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8C2×C412yescomplemented
61C3×C4×SD16—91no—
62C4×#214—31no—
63#4852—31no—
64C2×C32×D4—31no—
65C22×C32×C4—61no—
66C2×#634C22×C311yescharacteristic
67C2×C3×D12—21no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C22×C311yescharacteristic
70C2×C3×C4×D3—22no—
71C2×C3×C4×D3—21no—
72C2×C3×C4×D3—21no—
73C2×C3×Dic6—31no—
74C2×C3×Dic6—31no—
75C2×C3×Dic6—61no—
76C2×C32×C8—31no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4—21no—
79C32×C4⋊C4—21no—
80C32×C4⋊C4—31no—
81C32×C4⋊C4—41no—
82C32×C4⋊C4C2×D311yescharacteristic
83C32×C4⋊C4C2×D311yescharacteristic
84C32×C4⋊C4C22×C311yescharacteristic
85C32×C22⋊C4—61no—
86C32×C42—31no—
87C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
88C3×#214—34no—
89C3×(C2×D3)⋊C4—22no—
90C3×(C2×D3)⋊C4—21no—
91C3×(C2×D3)⋊C4—21no—
92C3×C4⋊Dic3—21no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C22×C311yescharacteristic
95C3×Dic3⋊C4—61no—
96C3×Dic3⋊C4—61no—
97C3×C4×Dic3—21no—
98C3×C4×Dic3—61no—
99C3×C4×Dic3—61no—
100C3×C4×Dic3C2×D311yescharacteristic
101C3×C4×Dic3C22×C311yescharacteristic
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C22×C311yescharacteristic
108C3×#204—34no—
109#3724—21no—
110Dic32—21no—
111#3716C3×C414yescomplemented
112C22×C33C2×D411yescharacteristic
113C2×C32×D3—22no—
114C2×C32×D3—41no—
115C2×C32×D3—41no—
116C33×C4—41no—
117C33×C4C22×C411yescharacteristic
118C33×C4#4111yescharacteristic
119C3×C3⋊Dic3—22no—
120C32×Dic3—21no—
121C32×Dic3—41no—
122C32×Dic3SD1612yescomplemented
123C2×C3×SD16—91no—
124C3×C4×Q8—91no—
125C3×C4×D4—31no—
126C3×C4×D4—31no—
127C3×C4×D4—61no—
128C2×#214—31no—
129C2×#204—31no—
130#893C3×D311yescharacteristic
131#887—31no—
132C4×Dic6—31no—
133C3×#106—31no—
134C3×#106—31no—
135C3×#106—61no—
136C3×#103—91no—
137C3×#102—91no—
138C3×C4×C8—91no—
139#832—31no—
140C8×Dic3—31no—
141#810—31no—
142#810—31no—
143#810—61no—
144#809—31no—
145#808C3×D311yescharacteristic, complemented
146C4×C3⋊C8—31no—
147C23×C32—61no—
148C22×C3×D3—22no—
149C22×C3×D3—21no—
150C22×C3×D3—21no—
151C32×D4—32no—
152C32×D4—61no—
153C2×C32×C4—21no—
154C2×C32×C4—22no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—31no—
161C2×C32×C4—31no—
162C2×C32×C4—41no—
163C2×C32×C4—41no—
164C2×C32×C4—41no—
165C2×C32×C4—42no—
166C2×C32×C4—61no—
167C2×C32×C4—122no—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C23×C311yescharacteristic
171C2×C3⋊Dic3—21no—
172#634—21no—
173#634C2×C3×C412yescomplemented
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—41no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—121no—
182C2×C3×Dic3#6711yescharacteristic
183C2×C3×Dic3#6711yescharacteristic
184C2×C3×Dic3C3×D411yescharacteristic
185C2×C3×Dic3C3×D411yescharacteristic
186C3×D12—22no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—22no—
190C3×D12C4×D312yes—
191C3×D12C2×C3×C412yes—
192C3×C4×D3—44no—
193C3×C4×D3—42no—
194C3×C4×D3—42no—
195C3×Dic6—32no—
196C3×Dic6—32no—
197C3×Dic6—62no—
198C3×Dic6—61no—
199C3×Dic6—61no—
200C3×Dic6—121no—
201C32×C8—32no—
202C3⋊(C3⋊C8)—21no—
203C3×C3⋊C8—22no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—61no—
206C3×C3⋊C8—121no—
207C3×C3⋊C8C4×D312yescomplemented
208C3×C3⋊C8C2×C3×C412yescomplemented
209C4×SD16—91noSylow
210C2×C33C4×D411yescharacteristic
211C2×C33C2×SD1611yescharacteristic
212C2×C33#13511yescharacteristic
213C32×D3—42no—
214C2×C3×Q8—91no—
215C2×C3×D4—31no—
216C2×C3×D4—31no—
217C2×C3×D4—61no—
218C22×C3×C4—61no—
219C22×C3×C4—61no—
220C22×C3×C4—62no—
221C2×D12C2×C3×D311yescharacteristic
222C2×C4×D3—21no—
223C2×Dic6—31no—
224C2×Dic6—31no—
225C2×Dic6—61no—
226C3×SD16—94no—
227C2×C3×C8—31no—
228C2×C3×C8—31no—
229C2×C3×C8—61no—
230C2×C3×C8—91no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—61no—
238C3×C4⋊C4—181no—
239C3×C4⋊C4D3211yescharacteristic
240C3×C4⋊C4C2×C3×D311yescharacteristic
241C3×C4⋊C4C2×C3×D311yescharacteristic
242C3×C22⋊C4—61no—
243C3×C22⋊C4—61no—
244C3×C22⋊C4—62no—
245C3×C42—31no—
246C3×C42—31no—
247C3×C42—61no—
248C3×C42—181no—
249#214—34no—
250(C2×D3)⋊C4—21no—
251C4⋊Dic3C2×C3×D311yescharacteristic
252Dic3⋊C4—61no—
253Dic3⋊C4—61no—
254C4×Dic3—61no—
255C4×Dic3—61no—
256C4×Dic3C2×C3×D311yescharacteristic
257C2×C3⋊C8—31no—
258C2×C3⋊C8—31no—
259C2×C3⋊C8—61no—
260C2×C3⋊C8C2×C3×D311yescharacteristic
261#204—34no—
262C22×C32—21no—
263C22×C32—21no—
264C22×C32—21no—
265C22×C32—41no—
266C22×C32—62no—
267C22×C32—121no—
268C22×C32—121no—
269C22×C32D3×D411yescharacteristic
270C22×C32C2×#6711yescharacteristic
271C22×C32C2×C3×D411yescharacteristic
272C2×C3×D3—24no—
273C2×C3×D3—22no—
274C2×C3×D3—22no—
275C2×C3×D3—42no—
276C2×C3×D3—42no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C2×C3×D3—41no—
280C2×C3×D3—41no—
281C32×C4—21no—
282C32×C4—21no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—32no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4—41no—
291C32×C4—42no—
292C32×C4—41no—
293C32×C4—41no—
294C32×C4—61no—
295C32×C4—81no—
296C32×C4—81no—
297C32×C4—82no—
298C32×C4—121no—
299C32×C4C2×C4×D311yescharacteristic
300C32×C4C2×C4×D311yescharacteristic
301C32×C4#23511yescharacteristic
302C32×C4#23911yescharacteristic
303C32×C4C22×C3×C411yescharacteristic
304C32×C4C3×#4111yescharacteristic
305C3⋊Dic3—22no—
306C3×Dic3—21no—
307C3×Dic3—21no—
308C3×Dic3—22no—
309C3×Dic3—41no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—81no—
315C3×Dic3—122no—
316C3×Dic3#21412yescomplemented
317C3×Dic3C3×SD1612yescomplemented
318C2×SD16—91no—
319C4×Q8—91no—
320C4×D4—31no—
321#106—31no—
322#103—91no—
323#102—91no—
324C4×C8—91no—
325C33C4×SD1611yescharacteristic, complemented, Sylow
326C23×C3—62no—
327C23×C3—61no—
328C23×C3—61no—
329C22×D3—21no—
330C3×Q8—92no—
331C3×Q8—181no—
332C3×D4—32no—
333C3×D4—32no—
334C3×D4—62no—
335C3×D4—62no—
336C3×D4—61no—
337C3×D4—61no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—42no—
352C2×C3×C4—41no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—121no—
358C2×C3×C4—122no—
359C2×C3×C4—122no—
360C2×C3×C4—124no—
361C2×C3×C4—181no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C22×C3×D311yescharacteristic
364C2×C3×C4C22×C3×D311yescharacteristic
365C2×Dic3—21no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—121no—
369C2×Dic3C3×#6711yescharacteristic
370C2×Dic3C3×#6711yescharacteristic
371D12—21no—
372D12C3×C4×D312yes—
373C4×D3—42no—
374Dic6—32no—
375Dic6—32no—
376Dic6—62no—
377Dic6—61no—
378Dic6—61no—
379Dic6—121no—
380C3×C8—32no—
381C3×C8—32no—
382C3×C8—62no—
383C3×C8—181no—
384C3⋊C8—61no—
385C3⋊C8—61no—
386C3⋊C8—121no—
387C3⋊C8C3×C4×D312yescomplemented
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—41no—
398C2×C32—41no—
399C2×C32—41no—
400C2×C32—122no—
401C2×C32#89311yescharacteristic
402C2×C32D3×SD1611yescharacteristic
403C2×C32#91611yescharacteristic
404C2×C32C4×#6711yescharacteristic
405C2×C32C2×#21411yescharacteristic
406C2×C32#95011yescharacteristic
407C2×C32C3×C4×D411yescharacteristic
408C2×C32C2×C3×SD1611yescharacteristic
409C2×C32C3×#13511yescharacteristic
410C3×D3—44no—
411C3×D3—42no—
412C3×D3—42no—
413C2×Q8—91no—
414C2×D4—31no—
415C22×C4—61no—
416SD16—94no—
417C2×C8—31no—
418C2×C8—91no—
419C4⋊C4—31no—
420C4⋊C4—181no—
421C4⋊C4C3×D3211yescharacteristic
422C22⋊C4—61no—
423C42—31no—
424C42—181no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—41no—
429C22×C3—64no—
430C22×C3—62no—
431C22×C3—62no—
432C22×C3—122no—
433C22×C3—122no—
434C22×C3—121no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3—121no—
438C22×C3D3×#6711yescharacteristic
439C22×C3C3×D3×D411yescharacteristic
440C22×C3C2×C3×#6711yescharacteristic
441C2×D3—22no—
442C2×D3—41no—
443C2×D3—41no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—21no—
450C3×C4—32no—
451C3×C4—32no—
452C3×C4—42no—
453C3×C4—41no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—41no—
457C3×C4—41no—
458C3×C4—62no—
459C3×C4—61no—
460C3×C4—61no—
461C3×C4—81no—
462C3×C4—81no—
463C3×C4—82no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4—121no—
467C3×C4—182no—
468C3×C4—241no—
469C3×C4#379911yescharacteristic
470C3×C4C4×D3211yescharacteristic
471C3×C4C2×C3×C4×D311yescharacteristic
472C3×C4C2×C3×C4×D311yescharacteristic
473C3×C4C3×#23511yescharacteristic
474C3×C4C3×#23911yescharacteristic
475Dic3—21no—
476Dic3—41no—
477Dic3—62no—
478Dic3—62no—
479Dic3—122no—
480Dic3C3×#21412yescomplemented
481C32—21no—
482C32—21no—
483C32—21no—
484C32—41no—
485C32#485211yescharacteristic, complemented
486C32C4×#21411yescharacteristic, complemented
487C32C3×C4×SD1611yescharacteristic, complemented
488C23—61no—
489Q8—92no—
490Q8—181no—
491D4—32no—
492D4—61no—
493C2×C4—21no—
494C2×C4—31no—
495C2×C4—31no—
496C2×C4—61no—
497C2×C4—122no—
498C2×C4—181no—
499C2×C4C2×C3×D3211yescharacteristic
500C8—32no—
501C8—181no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—41no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—124no—
515C2×C3—122no—
516C2×C3—122no—
517C2×C3#6390411yescharacteristic
518C2×C3D3×#21411yescharacteristic
519C2×C3#6394711yescharacteristic
520C2×C3C3×#89311yescharacteristic
521C2×C3C3×D3×SD1611yescharacteristic
522C2×C3C3×#91611yescharacteristic
523C2×C3C3×C4×#6711yescharacteristic
524C2×C3C2×C3×#21411yescharacteristic
525C2×C3C3×#95011yescharacteristic
526D3—42no—
527C22—62no—
528C22—121no—
529C22—121no—
530C22C3×D3×#6711yescharacteristic
531C4—32no—
532C4—41no—
533C4—61no—
534C4—121no—
535C4—182no—
536C4C3×#379911yescharacteristic
537C4C3×C4×D3211yescharacteristic
538C3—21no—
539C3—21no—
540C3—21no—
541C3—41no—
542C3#1059047911yescharacteristic, direct product factor
543C3C3×#485211yescharacteristic, complemented
544C3C3×C4×#21411yescharacteristic, complemented
545C2—122no—
546C2C3×#6390411yescharacteristic
547C2C3×D3×#21411yescharacteristic
548C2C3×#6394711yescharacteristic
subgroupG/H
centerC22×C3D3×#67
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C32×C4C23×C3
hypercenterC3×C4⋊C4D32
hypocenterC32C3×C4×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#63844C2semidirect
4. C3×#63637C2semidirect
5. C32×#808C2semidirect
7. C3×Dic3×C3⋊C8C2semidirect
10. #10590479C3direct (N × H)
22. C3×#3716C4semidirect
31. #63844C2×C3semidirect
34. #63637C2×C3semidirect
42. C3×#808D3semidirect
43. C3×#808C2×C3semidirect
46. Dic3×C3⋊C8C2×C3semidirect
52. C3×#634C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
111. #3716C3×C4semidirect
122. C32×Dic3SD16semidirect
145. #808C3×D3semidirect
173. #634C2×C3×C4semidirect
207. C3×C3⋊C8C4×D3semidirect
208. C3×C3⋊C8C2×C3×C4semidirect
316. C3×Dic3#214semidirect
317. C3×Dic3C3×SD16semidirect
325. C33C4×SD16semidirect
387. C3⋊C8C3×C4×D3semidirect
480. Dic3C3×#214semidirect
485. C32#4852semidirect
486. C32C4×#214semidirect
487. C32C3×C4×SD16semidirect
542. C3#10590479direct (N × H)
543. C3C3×#4852semidirect
544. C3C3×C4×#214semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
C3×#638442
C2×C3×#37162
C3×#636372
C32×#8082
C3×#635622
C3×Dic3×C3⋊C82
C3×C4×#2143
C3×#48523
#105904793