Finite Group Databasebeta

← C3×C4×#3715 #13157373

Subgroups — C3×C4×#3715 #13157373

Order 1728.

726 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×C4×D12C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#3715C211yesmaximal, characteristic
4C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic
8C3×C4×#215—31nomaximal
9C3×C4×#214—31nomaximal
10C4×#3715C311yesmaximal, characteristic, direct product factor
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C2×C32×Dic6C2211yescharacteristic
14C33×C42C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×(C2×D3)⋊C4—21no—
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C32×Dic3⋊C4—21no—
21C32×C4×Dic3—21no—
22C3×#3715C414yesdirect product factor
23C32×C4×Q8—31no—
24C32×C4×D4—31no—
25C2×C3×#215—31no—
26C2×C3×#214—31no—
27C3×C4×D12—21no—
28C3×C4×D12D311yescharacteristic
29C3×C4×D12C2×C311yescharacteristic
30C3×C4×Dic6—21no—
31C3×C4×Dic6D311yescharacteristic, complemented
32C3×C4×Dic6C2×C311yescharacteristic, complemented
33C2×#3715C2×C311yescharacteristic
34C4×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
35C3×#835—31no—
36C3×#832—31no—
37C3×#810—31no—
38C3×#809—31no—
39C3×#808—31no—
40C3×#808—31no—
41C3×C4×C3⋊C8—31no—
42C3×C4×C3⋊C8—31no—
43C3×C4×C3⋊C8—61no—
44#63574C2×C311yescharacteristic, complemented
45#63565C2×C311yescharacteristic, complemented
46#63562C2×C311yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4D411yescharacteristic
49C2×C33×C4D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3—21no—
54C32×D12—21no—
55C32×D12C2×C412yes—
56C32×C4×D3—42no—
57C32×Dic6—21no—
58C32×Dic6C2×C412yescomplemented
59C3×C3⋊(C3⋊C8)—21no—
60C3×C3⋊(C3⋊C8)C2×C412yescomplemented
61C3×C4×SD16—91no—
62C4×#215—31no—
63C4×#214—31no—
64C2×C32×Q8—31no—
65C2×C32×D4—31no—
66C22×C32×C4—61no—
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—21no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C22×C311yescharacteristic
76C32×C4⋊C4—31no—
77C32×C4⋊C4—31no—
78C32×C4⋊C4—61no—
79C32×C22⋊C4—61no—
80C32×C42—21no—
81C32×C42—21no—
82C32×C42—21no—
83C32×C42—41no—
84C32×C42—61no—
85C32×C42C2×D311yescharacteristic
86C32×C42C2×D311yescharacteristic
87C32×C42C22×C311yescharacteristic
88C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
89C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
90C3×#215—34no—
91C3×#214—34no—
92C3×(C2×D3)⋊C4—22no—
93C3×(C2×D3)⋊C4—21no—
94C3×(C2×D3)⋊C4—21no—
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—21no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C22×C311yescharacteristic
100C3×C4⋊Dic3C22×C311yescharacteristic
101C3×Dic3⋊C4—22no—
102C3×Dic3⋊C4—21no—
103C3×Dic3⋊C4—21no—
104C3×C4×Dic3—22no—
105C3×C4×Dic3—21no—
106C3×C4×Dic3—21no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—31no—
109C2×C3×C3⋊C8—31no—
110C2×C3×C3⋊C8—31no—
111C2×C3×C3⋊C8—61no—
112C2×C3×C3⋊C8—61no—
113#3715C3×C414yesdirect product factor
114C22×C33C2×D411yescharacteristic
115C2×C32×D3—22no—
116C2×C32×D3—41no—
117C2×C32×D3—41no—
118C33×C4—21no—
119C33×C4SD1612yescomplemented
120C33×C4C22×C411yescharacteristic
121C33×C4#4111yescharacteristic
122C32×Dic3—22no—
123C32×Dic3—41no—
124C32×Dic3—41no—
125C2×C3×SD16—91no—
126C3×C4×Q8—31no—
127C3×C4×Q8—31no—
128C3×C4×Q8—61no—
129C3×C4×D4—31no—
130C3×C4×D4—31no—
131C3×C4×D4—61no—
132C2×#215—31no—
133C2×#214—31no—
134C4×D12C3×D311yescharacteristic
135C4×Dic6C3×D311yescharacteristic, complemented
136C3×#106—91no—
137C3×#103—91no—
138C3×#102—91no—
139C3×C4×C8—91no—
140#835—31no—
141#832—31no—
142#810—31no—
143#809—31no—
144#808—31no—
145#808—31no—
146C4×C3⋊C8—31no—
147C4×C3⋊C8—31no—
148C4×C3⋊C8—61no—
149C23×C32—61no—
150C22×C3×D3—22no—
151C22×C3×D3—21no—
152C22×C3×D3—21no—
153C32×Q8—32no—
154C32×Q8—61no—
155C32×D4—32no—
156C32×D4—61no—
157C2×C32×C4—21no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—41no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4—61no—
171C2×C32×C4—61no—
172C2×C32×C4—122no—
173C2×C32×C4#6711yescharacteristic
174C2×C32×C4#6711yescharacteristic
175C2×C32×C4#6711yescharacteristic
176C2×C32×C4#6711yescharacteristic
177C2×C32×C4C3×D411yescharacteristic
178C2×C32×C4C3×D411yescharacteristic
179C2×C32×C4C22×D311yescharacteristic
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C23×C311yescharacteristic
182C2×C3×Dic3—22no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C3×D12—22no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—22no—
195C3×D12C4×D312yes—
196C3×D12C2×C3×C412yes—
197C3×C4×D3—44no—
198C3×C4×D3—42no—
199C3×C4×D3—42no—
200C3×Dic6—22no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—22no—
204C3×Dic6C4×D312yescomplemented
205C3×Dic6C2×C3×C412yescomplemented
206C3⋊(C3⋊C8)—21no—
207C3⋊(C3⋊C8)C2×C3×C412yescomplemented
208C3×C3⋊C8—32no—
209C3×C3⋊C8—32no—
210C3×C3⋊C8—62no—
211C3×C3⋊C8—62no—
212C3×C3⋊C8—61no—
213C3×C3⋊C8—61no—
214C4×SD16—91noSylow
215C2×C33C4×D411yescharacteristic
216C2×C33C2×SD1611yescharacteristic
217C2×C33#13511yescharacteristic
218C32×D3—42no—
219C2×C3×Q8—31no—
220C2×C3×Q8—31no—
221C2×C3×Q8—61no—
222C2×C3×D4—31no—
223C2×C3×D4—31no—
224C2×C3×D4—61no—
225C22×C3×C4—62no—
226C22×C3×C4—61no—
227C22×C3×C4—61no—
228C2×D12C2×C3×D311yescharacteristic
229C2×C4×D3—21no—
230C2×Dic6C2×C3×D311yescharacteristic
231C3×SD16—94no—
232C2×C3×C8—91no—
233C2×C3×C8—91no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—31no—
236C3×C4⋊C4—31no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—61no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4—62no—
241C3×C4⋊C4—61no—
242C3×C4⋊C4—61no—
243C3×C22⋊C4—62no—
244C3×C22⋊C4—61no—
245C3×C22⋊C4—61no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—41no—
250C3×C42—62no—
251C3×C42—61no—
252C3×C42—61no—
253C3×C42D3211yescharacteristic
254C3×C42C2×C3×D311yescharacteristic
255C3×C42C2×C3×D311yescharacteristic
256#215—34no—
257#214—34no—
258(C2×D3)⋊C4—21no—
259C4⋊Dic3C2×C3×D311yescharacteristic
260C4⋊Dic3C2×C3×D311yescharacteristic
261Dic3⋊C4—21no—
262C4×Dic3—21no—
263C2×C3⋊C8—31no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—61no—
268C2×C3⋊C8—61no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32—62no—
274C22×C32—121no—
275C22×C32—121no—
276C22×C32C2×#6711yescharacteristic
277C22×C32C2×#6711yescharacteristic
278C22×C32C2×C3×D411yescharacteristic
279C2×C3×D3—24no—
280C2×C3×D3—22no—
281C2×C3×D3—22no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C2×C3×D3—41no—
285C2×C3×D3—41no—
286C2×C3×D3—41no—
287C2×C3×D3—41no—
288C32×C4—21no—
289C32×C4—22no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—22no—
293C32×C4—22no—
294C32×C4—22no—
295C32×C4—21no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—42no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—41no—
305C32×C4—41no—
306C32×C4—42no—
307C32×C4—62no—
308C32×C4—121no—
309C32×C4—121no—
310C32×C4#21412yescomplemented
311C32×C4#21512yescomplemented
312C32×C4C3×SD1612yescomplemented
313C32×C4C2×C4×D311yescharacteristic
314C32×C4C2×C4×D311yescharacteristic
315C32×C4#23511yescharacteristic
316C32×C4#23511yescharacteristic
317C32×C4C22×C3×C411yescharacteristic
318C32×C4C3×#4111yescharacteristic
319C3×Dic3—24no—
320C3×Dic3—22no—
321C3×Dic3—22no—
322C3×Dic3—42no—
323C3×Dic3—42no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—41no—
328C2×SD16—91no—
329C4×Q8—31no—
330C4×D4—31no—
331#106—91no—
332#103—91no—
333#102—91no—
334C4×C8—91no—
335C33C4×SD1611yescharacteristic, complemented, Sylow
336C23×C3—62no—
337C23×C3—61no—
338C23×C3—61no—
339C22×D3—21no—
340C3×Q8—32no—
341C3×Q8—32no—
342C3×Q8—62no—
343C3×Q8—62no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×D4—32no—
347C3×D4—32no—
348C3×D4—62no—
349C3×D4—62no—
350C3×D4—61no—
351C3×D4—61no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—62no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—62no—
368C2×C3×C4—62no—
369C2×C3×C4—61no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—124no—
374C2×C3×C4—122no—
375C2×C3×C4—122no—
376C2×C3×C4#62511yescharacteristic
377C2×C3×C4#62511yescharacteristic
378C2×C3×C4C3×#6711yescharacteristic
379C2×C3×C4C3×#6711yescharacteristic
380C2×C3×C4C3×#6711yescharacteristic
381C2×C3×C4C3×#6711yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C22×C3×D311yescharacteristic
384C2×C3×C4C22×C3×D311yescharacteristic
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388D12—21no—
389D12C3×C4×D312yes—
390C4×D3—42no—
391Dic6—21no—
392Dic6C3×C4×D312yescomplemented
393C3×C8—92no—
394C3×C8—181no—
395C3⋊C8—32no—
396C3⋊C8—32no—
397C3⋊C8—62no—
398C3⋊C8—62no—
399C3⋊C8—61no—
400C3⋊C8—61no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32—21no—
409C2×C32—21no—
410C2×C32—41no—
411C2×C32—41no—
412C2×C32—41no—
413C2×C32—122no—
414C2×C32C4×#6711yescharacteristic
415C2×C32C4×#6711yescharacteristic
416C2×C32C2×#21411yescharacteristic
417C2×C32C2×#21511yescharacteristic
418C2×C32#95011yescharacteristic
419C2×C32#95011yescharacteristic
420C2×C32C3×C4×D411yescharacteristic
421C2×C32C2×C3×SD1611yescharacteristic
422C2×C32C3×#13511yescharacteristic
423C3×D3—44no—
424C3×D3—42no—
425C3×D3—42no—
426C2×Q8—31no—
427C2×D4—31no—
428C22×C4—61no—
429SD16—94no—
430C2×C8—91no—
431C2×C8—91no—
432C4⋊C4—31no—
433C4⋊C4—31no—
434C4⋊C4—61no—
435C22⋊C4—61no—
436C42—61no—
437C42C3×D3211yescharacteristic
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—41no—
442C22×C3—64no—
443C22×C3—62no—
444C22×C3—62no—
445C22×C3—122no—
446C22×C3—122no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—121no—
450C22×C3—121no—
451C22×C3C2×#62511yescharacteristic
452C22×C3C2×C3×#6711yescharacteristic
453C22×C3C2×C3×#6711yescharacteristic
454C2×D3—22no—
455C2×D3—41no—
456C2×D3—41no—
457C3×C4—22no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—22no—
468C3×C4—22no—
469C3×C4—22no—
470C3×C4—41no—
471C3×C4—41no—
472C3×C4—42no—
473C3×C4—41no—
474C3×C4—41no—
475C3×C4—42no—
476C3×C4—64no—
477C3×C4—62no—
478C3×C4—62no—
479C3×C4—122no—
480C3×C4—122no—
481C3×C4—121no—
482C3×C4—121no—
483C3×C4—121no—
484C3×C4—121no—
485C3×C4#371512yesdirect product factor
486C3×C4C3×#21412yescomplemented
487C3×C4C3×#21512yescomplemented
488C3×C4#379811yescharacteristic
489C3×C4C4×D3211yescharacteristic
490C3×C4C2×C3×C4×D311yescharacteristic
491C3×C4C2×C3×C4×D311yescharacteristic
492C3×C4C3×#23511yescharacteristic
493C3×C4C3×#23511yescharacteristic
494Dic3—22no—
495Dic3—41no—
496Dic3—41no—
497C32—21no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32C4×#21411yescharacteristic, complemented
502C32C4×#21511yescharacteristic, complemented
503C32C3×C4×SD1611yescharacteristic, complemented
504C23—61no—
505Q8—32no—
506Q8—61no—
507D4—32no—
508D4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—122no—
513C2×C4C3×#62511yescharacteristic
514C2×C4C3×#62511yescharacteristic
515C2×C4C2×C3×D3211yescharacteristic
516C8—92no—
517C8—181no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—21no—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3—41no—
528C2×C3—41no—
529C2×C3—41no—
530C2×C3—124no—
531C2×C3—122no—
532C2×C3—122no—
533C2×C3#6382211yescharacteristic
534C2×C3C2×#371511yescharacteristic
535C2×C3C4×#62511yescharacteristic
536C2×C3C3×C4×#6711yescharacteristic
537C2×C3C3×C4×#6711yescharacteristic
538C2×C3C2×C3×#21411yescharacteristic
539C2×C3C2×C3×#21511yescharacteristic
540C2×C3C3×#95011yescharacteristic
541C2×C3C3×#95011yescharacteristic
542D3—42no—
543C22—62no—
544C22—121no—
545C22—121no—
546C22C2×C3×#62511yescharacteristic
547C4—21no—
548C4—62no—
549C4—121no—
550C4—121no—
551C4C3×#371512yesdirect product factor
552C4C3×#379811yescharacteristic
553C4C3×C4×D3211yescharacteristic
554C3—21no—
555C3—21no—
556C3—21no—
557C3—41no—
558C3C4×#371511yescharacteristic, direct product factor
559C3C3×C4×#21411yescharacteristic, complemented
560C3C3×C4×#21511yescharacteristic, complemented
561C2—122no—
562C2C3×#6382211yescharacteristic
563C2C2×C3×#371511yescharacteristic
564C2C3×C4×#62511yescharacteristic
subgroupG/H
centerC2×C3×C4#625
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
cosocleC2×C32×C4C23×C3
hypercenterC3×C42D32
hypocenterC32C3×C4×SD16
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
2. C32×C4×Dic6C2semidirect
4. C3×C4×C3⋊(C3⋊C8)C2semidirect
5. C3×#63574C2semidirect
6. C3×#63565C2semidirect
10. C4×#3715C3direct (N × H)
22. C3×#3715C4direct (N × H)
31. C3×C4×Dic6D3semidirect
32. C3×C4×Dic6C2×C3semidirect
34. C4×C3⋊(C3⋊C8)C2×C3semidirect
44. #63574C2×C3semidirect
45. #63565C2×C3semidirect
58. C32×Dic6C2×C4semidirect
60. C3×C3⋊(C3⋊C8)C2×C4semidirect
113. #3715C3×C4direct (N × H)
119. C33×C4SD16semidirect
135. C4×Dic6C3×D3semidirect
204. C3×Dic6C4×D3semidirect
205. C3×Dic6C2×C3×C4semidirect
207. C3⋊(C3⋊C8)C2×C3×C4semidirect
310. C32×C4#214semidirect
311. C32×C4#215semidirect
312. C32×C4C3×SD16semidirect
335. C33C4×SD16semidirect
392. Dic6C3×C4×D3semidirect
485. C3×C4#3715direct (N × H)
486. C3×C4C3×#214semidirect
487. C3×C4C3×#215semidirect
501. C32C4×#214semidirect
502. C32C4×#215semidirect
503. C32C3×C4×SD16semidirect
551. C4C3×#3715direct (N × H)
558. C3C4×#3715direct (N × H)
559. C3C3×C4×#214semidirect
560. C3C3×C4×#215semidirect

Maximal subgroups

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

subgroupindex
C32×C4×D122
C32×C4×Dic62
C2×C3×#37152
C3×C4×C3⋊(C3⋊C8)2
C3×#635742
C3×#635652
C3×#635622
C3×C4×#2153
C3×C4×#2143
C4×#37153