Finite Group Databasebeta

← C3×C4×#3717 #13157377

Subgroups — C3×C4×#3717 #13157377

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
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic, complemented
3C2×C3×#3717C211yesmaximal, characteristic
4C32×C4×C3⋊C8C211yesmaximal, characteristic, complemented
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8C3×C4×#215—31nomaximal
9C3×C4×#204—31nomaximal
10C4×#3717C311yesmaximal, characteristic, direct product factor
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3—21no—
13C2×C32×Dic6C2211yescharacteristic
14C33×C42C2211yescharacteristic
15C3×#3752—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×Dic3⋊C4—21no—
19C32×C4×Dic3—21no—
20C2×C32×C3⋊C8C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C3×#3717C414yesdirect product factor
23C32×C4×Q8—31no—
24C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
25C2×C3×#215—31no—
26C2×C3×#204—31no—
27C3×C4×D12—31no—
28C3×C4×D12—31no—
29C3×C4×D12—61no—
30C3×C4×Dic6—21no—
31C3×C4×Dic6D311yescharacteristic, complemented
32C3×C4×Dic6C2×C311yescharacteristic, complemented
33C2×#3717C2×C311yescharacteristic
34C32×C4×C8—31no—
35C3×#835—31no—
36C3×#821—31no—
37C3×#817—31no—
38C3×#816—31no—
39C3×#809—31no—
40C3×#808—31no—
41C3×C4×C3⋊C8—21no—
42C3×C4×C3⋊C8D311yescharacteristic, complemented
43C3×C4×C3⋊C8C2×C311yescharacteristic, complemented
44#63573C2×C311yescharacteristic, complemented
45#63566C2×C311yescharacteristic, complemented
46#63564C2×C311yescharacteristic
47C22×C3×C3⋊D3—21no—
48C2×C33×C4D411yescharacteristic
49C2×C33×C4D411yescharacteristic
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)C2×C412yes—
54C3×C4×C3⋊D3—42no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57C32×Dic6—21no—
58C32×Dic6C2×C412yescomplemented
59C32×C3⋊C8—21no—
60C32×C3⋊C8C2×C412yescomplemented
61C3×C4×SD16—91no—
62C4×#215—31no—
63C4×#204—31no—
64C2×C32×Q8—31no—
65C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
66C2×C4×C3⋊D3—21no—
67C2×C3×D12—31no—
68C2×C3×D12—31no—
69C2×C3×D12—61no—
70C2×C3×C4×D3—62no—
71C2×C3×C4×D3—61no—
72C2×C3×C4×D3—61no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6C2×D311yescharacteristic
75C2×C3×Dic6C22×C311yescharacteristic
76C2×C32×C8—31no—
77C2×C32×C8—31no—
78C32×C4⋊C4—31no—
79C32×C4⋊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
88#3752—21no—
89C4⋊(C3⋊Dic3)C22×C311yescharacteristic
90C3×#215—34no—
91C3×(C2×D3)⋊C4—62no—
92C3×(C2×D3)⋊C4—61no—
93C3×(C2×D3)⋊C4—61no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—31no—
97C3×C4⋊Dic3—61no—
98C3×C4⋊Dic3C2×D311yescharacteristic
99C3×C4⋊Dic3C22×C311yescharacteristic
100C3×Dic3⋊C4—22no—
101C3×Dic3⋊C4—21no—
102C3×Dic3⋊C4—21no—
103C3×C4×Dic3—22no—
104C3×C4×Dic3—21no—
105C3×C4×Dic3—21no—
106C2×C3×C3⋊C8—21no—
107C2×C3×C3⋊C8—21no—
108C2×C3×C3⋊C8C2×D311yescharacteristic
109C2×C3×C3⋊C8C2×D311yescharacteristic
110C2×C3×C3⋊C8C22×C311yescharacteristic
111C2×C3×C3⋊C8C22×C311yescharacteristic
112C3×#204—34no—
113#3717C3×C414yesdirect product factor
114C22×C33C2×D411yescharacteristic
115C2×C3×C3⋊D3—22no—
116C2×C3×C3⋊D3—41no—
117C2×C3×C3⋊D3—41no—
118C33×C4—21no—
119C33×C4SD1612yescomplemented
120C33×C4C22×C411yescharacteristic
121C33×C4#4111yescharacteristic
122C3×C3⋊Dic3—41no—
123C32×Dic3—22no—
124C32×Dic3—41no—
125C2×C3×SD16—91no—
126C3×C4×Q8—31no—
127C3×C4×Q8—31no—
128C3×C4×Q8—61no—
129C3×C4×D4—91no—
130C2×#215—31no—
131C2×#204—31no—
132C4×D12—31no—
133C4×D12—31no—
134C4×D12—61no—
135C4×Dic6C3×D311yescharacteristic, complemented
136C3×#106—91no—
137C3×#103—91no—
138C3×#102—91no—
139C3×C4×C8—31no—
140C3×C4×C8—31no—
141C3×C4×C8—61no—
142#835—31no—
143#821—31no—
144#817—31no—
145#816—31no—
146#809—31no—
147#808—31no—
148C4×C3⋊C8C3×D311yescharacteristic, complemented
149C22×C3⋊D3—21no—
150C22×C3×D3—62no—
151C22×C3×D3—61no—
152C22×C3×D3—61no—
153C32×Q8—32no—
154C32×Q8—61no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
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—41no—
165C2×C32×C4—41no—
166C2×C32×C4—41no—
167C2×C32×C4—61no—
168C2×C32×C4—61no—
169C2×C32×C4D1211yescharacteristic
170C2×C32×C4D1211yescharacteristic
171C2×C32×C4#6711yescharacteristic
172C2×C32×C4#6711yescharacteristic
173C2×C32×C4C3×D411yescharacteristic
174C2×C32×C4C3×D411yescharacteristic
175C2×C32×C4C22×D311yescharacteristic
176C2×C32×C4C22×D311yescharacteristic
177C2×C32×C4C23×C311yescharacteristic
178C2×C3⋊Dic3—21no—
179C4⋊(C3⋊D3)—21no—
180C4⋊(C3⋊D3)C2×C3×C412yes—
181C4×C3⋊D3—42no—
182C2×C3×Dic3—22no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C3×D12—32no—
192C3×D12—32no—
193C3×D12—62no—
194C3×D12—62no—
195C3×D12—61no—
196C3×D12—61no—
197C3×C4×D3—124no—
198C3×C4×D3—122no—
199C3×C4×D3—122no—
200C3×Dic6—22no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—22no—
204C3×Dic6C4×D312yescomplemented
205C3×Dic6C2×C3×C412yescomplemented
206C32×C8—32no—
207C32×C8—61no—
208C3×C3⋊C8—22no—
209C3×C3⋊C8—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—22no—
212C3×C3⋊C8C4×D312yescomplemented
213C3×C3⋊C8C2×C3×C412yescomplemented
214C4×SD16—91noSylow
215C2×C33C4×D411yescharacteristic
216C2×C33C2×SD1611yescharacteristic
217C2×C33#13511yescharacteristic
218C3×C3⋊D3—42no—
219C2×C3×Q8—31no—
220C2×C3×Q8—31no—
221C2×C3×Q8—61no—
222C2×C3×D4—91no—
223C22×C3×C4—181no—
224C2×D12—31no—
225C2×D12—31no—
226C2×D12—61no—
227C2×C4×D3—62no—
228C2×C4×D3—61no—
229C2×C4×D3—61no—
230C2×Dic6C2×C3×D311yescharacteristic
231C3×SD16—94no—
232C2×C3×C8—31no—
233C2×C3×C8—31no—
234C2×C3×C8—31no—
235C2×C3×C8—31no—
236C2×C3×C8—61no—
237C2×C3×C8—61no—
238C3×C4⋊C4—31no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—61no—
241C3×C4⋊C4—62no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—61no—
244C3×C4⋊C4—91no—
245C3×C22⋊C4—181no—
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(C2×D3)⋊C4—62no—
258(C2×D3)⋊C4—61no—
259(C2×D3)⋊C4—61no—
260C4⋊Dic3—31no—
261C4⋊Dic3—31no—
262C4⋊Dic3—61no—
263C4⋊Dic3C2×C3×D311yescharacteristic
264Dic3⋊C4—21no—
265C4×Dic3—21no—
266C2×C3⋊C8C2×C3×D311yescharacteristic
267C2×C3⋊C8C2×C3×D311yescharacteristic
268#204—34no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—41no—
273C22×C32C2×D1211yescharacteristic
274C22×C32C2×#6711yescharacteristic
275C22×C32C2×C3×D411yescharacteristic
276C2×C3⋊D3—22no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—64no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—122no—
283C2×C3×D3—122no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
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#20412yescomplemented
310C32×C4#21512yescomplemented
311C32×C4C3×SD1612yescomplemented
312C32×C4C2×C4×D311yescharacteristic
313C32×C4C2×C4×D311yescharacteristic
314C32×C4#23511yescharacteristic
315C32×C4#23511yescharacteristic
316C32×C4C22×C3×C411yescharacteristic
317C32×C4C3×#4111yescharacteristic
318C3⋊Dic3—41no—
319C3×Dic3—24no—
320C3×Dic3—22no—
321C3×Dic3—22no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—122no—
326C3×Dic3—121no—
327C3×Dic3—121no—
328C2×SD16—91no—
329C4×Q8—31no—
330C4×D4—91no—
331#106—91no—
332#103—91no—
333#102—91no—
334C4×C8—31no—
335C33C4×SD1611yescharacteristic, complemented, Sylow
336C23×C3—181no—
337C22×D3—62no—
338C22×D3—61no—
339C22×D3—61no—
340C3×Q8—32no—
341C3×Q8—32no—
342C3×Q8—62no—
343C3×Q8—62no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×D4—92no—
347C3×D4—181no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—62no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—62no—
364C2×C3×C4—61no—
365C2×C3×C4—61no—
366C2×C3×C4—181no—
367C2×C3×C4—362no—
368C2×C3×C4#62611yescharacteristic
369C2×C3×C4#62611yescharacteristic
370C2×C3×C4C3×D1211yescharacteristic
371C2×C3×C4C3×D1211yescharacteristic
372C2×C3×C4C3×#6711yescharacteristic
373C2×C3×C4C3×#6711yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C22×C3×D311yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—62no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382D12—32no—
383D12—32no—
384D12—62no—
385D12—62no—
386D12—61no—
387D12—61no—
388C4×D3—124no—
389C4×D3—122no—
390C4×D3—122no—
391Dic6—21no—
392Dic6C3×C4×D312yescomplemented
393C3×C8—32no—
394C3×C8—32no—
395C3×C8—62no—
396C3×C8—62no—
397C3×C8—61no—
398C3×C8—61no—
399C3⋊C8—21no—
400C3⋊C8C3×C4×D312yescomplemented
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×C32C4×D1211yescharacteristic
414C2×C32C2×#20411yescharacteristic
415C2×C32#90411yescharacteristic
416C2×C32C4×#6711yescharacteristic
417C2×C32C2×#21511yescharacteristic
418C2×C32#95011yescharacteristic
419C2×C32C3×C4×D411yescharacteristic
420C2×C32C2×C3×SD1611yescharacteristic
421C2×C32C3×#13511yescharacteristic
422C3⋊D3—42no—
423C3×D3—124no—
424C3×D3—122no—
425C3×D3—122no—
426C2×Q8—31no—
427C2×D4—91no—
428C22×C4—181no—
429SD16—94no—
430C2×C8—31no—
431C2×C8—31no—
432C4⋊C4—31no—
433C4⋊C4—61no—
434C4⋊C4—91no—
435C22⋊C4—181no—
436C42—61no—
437C42C3×D3211yescharacteristic
438C22×C3—21no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—41no—
442C22×C3—182no—
443C22×C3—361no—
444C22×C3—361no—
445C22×C3C2×#62611yescharacteristic
446C22×C3C2×C3×D1211yescharacteristic
447C22×C3C2×C3×#6711yescharacteristic
448C2×D3—64no—
449C2×D3—62no—
450C2×D3—62no—
451C2×D3—122no—
452C2×D3—122no—
453C2×D3—121no—
454C2×D3—121no—
455C2×D3—121no—
456C2×D3—121no—
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—121no—
481C3×C4—121no—
482C3×C4—361no—
483C3×C4#371712yesdirect product factor
484C3×C4C3×#20412yescomplemented
485C3×C4C3×#21512yescomplemented
486C3×C4#379811yescharacteristic
487C3×C4C4×D3211yescharacteristic
488C3×C4C2×C3×C4×D311yescharacteristic
489C3×C4C2×C3×C4×D311yescharacteristic
490C3×C4C3×#23511yescharacteristic
491C3×C4C3×#23511yescharacteristic
492Dic3—22no—
493Dic3—41no—
494Dic3—122no—
495Dic3—121no—
496Dic3—121no—
497C32—21no—
498C32—21no—
499C32—21no—
500C32—41no—
501C32C4×#20411yescharacteristic, complemented
502C32C4×#21511yescharacteristic, complemented
503C32C3×C4×SD1611yescharacteristic, complemented
504C23—181no—
505Q8—32no—
506Q8—61no—
507D4—92no—
508D4—181no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—181no—
512C2×C4—362no—
513C2×C4C3×#62611yescharacteristic
514C2×C4C3×#62611yescharacteristic
515C2×C4C2×C3×D3211yescharacteristic
516C8—32no—
517C8—61no—
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—362no—
531C2×C3#6382911yescharacteristic
532C2×C3C2×#371711yescharacteristic
533C2×C3C4×#62611yescharacteristic
534C2×C3C3×C4×D1211yescharacteristic
535C2×C3C2×C3×#20411yescharacteristic
536C2×C3C3×#90411yescharacteristic
537C2×C3C3×C4×#6711yescharacteristic
538C2×C3C2×C3×#21511yescharacteristic
539C2×C3C3×#95011yescharacteristic
540D3—124no—
541D3—122no—
542D3—122no—
543C22—182no—
544C22—361no—
545C22—361no—
546C22C2×C3×#62611yescharacteristic
547C4—21no—
548C4—62no—
549C4—121no—
550C4—361no—
551C4C3×#371712yesdirect product factor
552C4C3×#379811yescharacteristic
553C4C3×C4×D3211yescharacteristic
554C3—21no—
555C3—21no—
556C3—21no—
557C3—41no—
558C3C4×#371711yescharacteristic, direct product factor
559C3C3×C4×#20411yescharacteristic, complemented
560C3C3×C4×#21511yescharacteristic, complemented
561C2—362no—
562C2C3×#6382911yescharacteristic
563C2C2×C3×#371711yescharacteristic
564C2C3×C4×#62611yescharacteristic
subgroupG/H
centerC2×C3×C4#626
commutatorC32×C4C22×C3×C4
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
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. C32×C4×C3⋊C8C2semidirect
5. C3×#63573C2semidirect
6. C3×#63566C2semidirect
10. C4×#3717C3direct (N × H)
22. C3×#3717C4direct (N × H)
31. C3×C4×Dic6D3semidirect
32. C3×C4×Dic6C2×C3semidirect
42. C3×C4×C3⋊C8D3semidirect
43. C3×C4×C3⋊C8C2×C3semidirect
44. #63573C2×C3semidirect
45. #63566C2×C3semidirect
58. C32×Dic6C2×C4semidirect
60. C32×C3⋊C8C2×C4semidirect
113. #3717C3×C4direct (N × H)
119. C33×C4SD16semidirect
135. C4×Dic6C3×D3semidirect
148. C4×C3⋊C8C3×D3semidirect
204. C3×Dic6C4×D3semidirect
205. C3×Dic6C2×C3×C4semidirect
212. C3×C3⋊C8C4×D3semidirect
213. C3×C3⋊C8C2×C3×C4semidirect
309. C32×C4#204semidirect
310. C32×C4#215semidirect
311. C32×C4C3×SD16semidirect
335. C33C4×SD16semidirect
392. Dic6C3×C4×D3semidirect
400. C3⋊C8C3×C4×D3semidirect
483. C3×C4#3717direct (N × H)
484. C3×C4C3×#204semidirect
485. C3×C4C3×#215semidirect
501. C32C4×#204semidirect
502. C32C4×#215semidirect
503. C32C3×C4×SD16semidirect
551. C4C3×#3717direct (N × H)
558. C3C4×#3717direct (N × H)
559. C3C3×C4×#204semidirect
560. C3C3×C4×#215semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C32×C4×Dic62
C2×C3×#37172
C32×C4×C3⋊C82
C3×#635732
C3×#635662
C3×#635642
C3×C4×#2153
C3×C4×#2043
C4×#37173