Finite Group Databasebeta

← C3×#10591465 #13158512

Subgroups — C3×#10591465 #13158512

Order 1728.

762 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64147C211yesmaximal, characteristic
2C2×C32×#214C211yesmaximal, characteristic, complemented
3C3×#63871C211yesmaximal, characteristic, complemented
4C2×C3×#3710C211yesmaximal, characteristic, complemented
5C32×#832C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8C3×#5171—31nomaximal
9C3×#4778—31nomaximal
10#10591465C311yesmaximal, characteristic, direct product factor
11C2×C33×D4C2211yescharacteristic
12C2×C3×#638—21no—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×#214—22no—
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21C3×Dic3⋊Dic3—21no—
22C3×#3722—21no—
23C3×#3710—22no—
24C2×C32×SD16—31no—
25#64147C2×C311yescharacteristic
26C3×#946—31no—
27C3×#938—31no—
28C3×#938—31no—
29C3×#938—61no—
30C2×C3×#214—21no—
31C2×C3×#214D311yescharacteristic, complemented
32C2×C3×#214C2×C311yescharacteristic, complemented
33C2×C3×D3×C8—31no—
34C2×C3×D3×C8—31no—
35C3×#897—31no—
36#63871C2×C311yescharacteristic, complemented
37C2×#3710C2×C311yescharacteristic, complemented
38C32×#102—31no—
39C3×#835—31no—
40C3×#832—21no—
41C3×#832D311yescharacteristic, complemented
42C3×#832C2×C311yescharacteristic, complemented
43C3×#817—31no—
44C3×#816—31no—
45C3×#808—31no—
46#63573C2×C311yescharacteristic, complemented
47#63566C2×C311yescharacteristic, complemented
48C23×C33—21no—
49C22×C3×C3⋊D3D411yescharacteristic
50C33×D4—21no—
51C33×D4—21no—
52C2×C33×C4C2311yescharacteristic
53C3×#638—42no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4×C3⋊D3—21no—
57C3×C4×C3⋊D3—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8D412yescomplemented
64C3×#465—91no—
65#5171—31no—
66#4778—31no—
67C2×C32×Q8—31no—
68C2×C32×D4—21no—
69C2×C32×D4—21no—
70C2×C32×D4—21no—
71C2×C32×D4—41no—
72C2×C32×D4C2×D311yescharacteristic
73C2×C32×D4C2×D311yescharacteristic
74C2×C32×D4C22×C311yescharacteristic
75C2×#638—21no—
76C2×C4×C3⋊D3C22×C311yescharacteristic
77C2×C3×#67—61no—
78C2×C3×#67—61no—
79C2×C3×#67—62no—
80C2×C3×C4×D3—31no—
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—61no—
83C2×C3×Dic6—21no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C22×C311yescharacteristic
86C32×SD16—62no—
87C2×C32×C8—31no—
88C2×C32×C8—31no—
89C32×C4⋊C4—31no—
90#3757—21no—
91C4⋊(C3⋊Dic3)C22×C311yescharacteristic
92C3×#217—61no—
93C3×#217—61no—
94C3×#217—62no—
95C3×#214—24no—
96C3×#214—22no—
97C3×#214—22no—
98C3×(C2×D3)⋊C4—61no—
99C3×(C2×D3)⋊C4—61no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—61no—
104C3×C4⋊Dic3—61no—
105C3×C4⋊Dic3C2×D311yescharacteristic
106C3×C4⋊Dic3C22×C311yescharacteristic
107C3×Dic3⋊C4—61no—
108C2×C3×C3⋊C8—21no—
109C2×C3×C3⋊C8—21no—
110C2×C3×C3⋊C8C2×D311yescharacteristic
111C2×C3×C3⋊C8C2×D311yescharacteristic
112C2×C3×C3⋊C8C22×C311yescharacteristic
113C2×C3×C3⋊C8C22×C311yescharacteristic
114C3×D3×C8—62no—
115C3×D3×C8—62no—
116Dic3⋊Dic3—21no—
117#3722—21no—
118#3710—22no—
119C22×C33—41no—
120C22×C33—41no—
121C22×C33—41no—
122C22×C33C2×D411yescharacteristic
123C2×C3×C3⋊D3—21no—
124C2×C3×C3⋊D3—21no—
125C2×C3×C3⋊D3SD1612yescomplemented
126C33×C4C2×D411yescharacteristic
127C33×C4#4111yescharacteristic
128C3×C3⋊Dic3—21no—
129C3×C3⋊Dic3—41no—
130C32×Dic3—41no—
131C32×Dic3—41no—
132C2×C3×SD16—31no—
133C2×C3×SD16—31no—
134C2×C3×SD16—61no—
135C22×C3×C8—91no—
136C3×C22⋊Q8—91no—
137C3×#121—91no—
138#946—31no—
139#938—31no—
140#938—31no—
141#938—61no—
142C2×#214C3×D311yescharacteristic, complemented
143C2×D3×C8—31no—
144C2×D3×C8—31no—
145#897—31no—
146C3×#106—91no—
147C3×#103—91no—
148C3×#102—31no—
149C3×#102—31no—
150C3×#102—61no—
151#835—31no—
152#832C3×D311yescharacteristic, complemented
153#817—31no—
154#816—31no—
155#808—31no—
156C23×C32—21no—
157C23×C32—22no—
158C23×C32—21no—
159C23×C32—21no—
160C23×C32—41no—
161C23×C32—41no—
162C23×C32—42no—
163C22×C3⋊D3C3×D411yescharacteristic
164C22×C3×D3—31no—
165C22×C3×D3—31no—
166C22×C3×D3—61no—
167C32×Q8—61no—
168C32×Q8—61no—
169C32×D4—21no—
170C32×D4—21no—
171C32×D4—22no—
172C32×D4—22no—
173C32×D4—21no—
174C32×D4—21no—
175C32×D4—21no—
176C32×D4—21no—
177C32×D4—41no—
178C32×D4—41no—
179C32×D4—41no—
180C32×D4—41no—
181C32×D4—42no—
182C32×D4—42no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—41no—
187C2×C32×C4—61no—
188C2×C32×C4—61no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192#638—42no—
193C2×C3⋊Dic3—21no—
194C2×C3⋊Dic3C3×D411yescharacteristic
195C4×C3⋊D3—21no—
196C4×C3⋊D3—21no—
197C3×#67—122no—
198C3×#67—122no—
199C3×#67—124no—
200C2×C3×Dic3—21no—
201C2×C3×Dic3—21no—
202C2×C3×Dic3—21no—
203C2×C3×Dic3—21no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—41no—
207C2×C3×Dic3—41no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—61no—
211C2×C3×Dic3—61no—
212C3×C4×D3—61no—
213C3×C4×D3—61no—
214C3×C4×D3—61no—
215C3×C4×D3—61no—
216C3×C4×D3—121no—
217C3×C4×D3—121no—
218C3×Dic6—22no—
219C3×Dic6—21no—
220C3×Dic6—21no—
221C3×Dic6—21no—
222C3×Dic6—21no—
223C3×Dic6—41no—
224C32×C8—32no—
225C32×C8—61no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—22no—
229C3×C3⋊C8—41no—
230C3×C3⋊C8#6712yescomplemented
231C3×C3⋊C8C3×D412yescomplemented
232#465—91noSylow
233C2×C33—81no—
234C2×C33#12111yescharacteristic
235C2×C33C2×SD1611yescharacteristic
236C2×C33#13511yescharacteristic
237C3×C3⋊D3—22no—
238C2×C3×Q8—31no—
239C2×C3×Q8—31no—
240C2×C3×Q8—61no—
241C2×C3×D4—21no—
242C2×C3×D4—21no—
243C2×C3×D4—21no—
244C2×C3×D4—41no—
245C2×C3×D4—181no—
246C2×C3×D4D3211yescharacteristic
247C2×C3×D4C2×C3×D311yescharacteristic
248C2×C3×D4C2×C3×D311yescharacteristic
249C22×C3×C4—91no—
250C2×#67—61no—
251C2×#67—61no—
252C2×#67—62no—
253C2×C4×D3—31no—
254C2×C4×D3—31no—
255C2×C4×D3—61no—
256C2×Dic6C2×C3×D311yescharacteristic
257C3×SD16—62no—
258C3×SD16—62no—
259C3×SD16—64no—
260C2×C3×C8—31no—
261C2×C3×C8—31no—
262C2×C3×C8—31no—
263C2×C3×C8—31no—
264C2×C3×C8—61no—
265C2×C3×C8—61no—
266C2×C3×C8—182no—
267C3×C4⋊C4—31no—
268C3×C4⋊C4—31no—
269C3×C4⋊C4—61no—
270C3×C4⋊C4—91no—
271C3×C4⋊C4—181no—
272C3×C22⋊C4—181no—
273C3×C22⋊C4—181no—
274#217—61no—
275#217—61no—
276#217—62no—
277#214—22no—
278(C2×D3)⋊C4—61no—
279(C2×D3)⋊C4—61no—
280C4⋊Dic3—31no—
281C4⋊Dic3—31no—
282C4⋊Dic3—61no—
283C4⋊Dic3—61no—
284C4⋊Dic3C2×C3×D311yescharacteristic
285Dic3⋊C4—61no—
286C2×C3⋊C8C2×C3×D311yescharacteristic
287C2×C3⋊C8C2×C3×D311yescharacteristic
288D3×C8—62no—
289D3×C8—62no—
290C22×C32—21no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—41no—
294C22×C32—42no—
295C22×C32—42no—
296C22×C32—42no—
297C22×C32—44no—
298C22×C32—41no—
299C22×C32—41no—
300C22×C32—41no—
301C22×C32—41no—
302C22×C32—41no—
303C22×C32—42no—
304C22×C32—42no—
305C22×C32—41no—
306C22×C32—41no—
307C22×C32—41no—
308C22×C32—41no—
309C22×C32—81no—
310C22×C32—81no—
311C22×C32—81no—
312C22×C32—81no—
313C22×C32—82no—
314C22×C32—82no—
315C22×C32D3×D411yescharacteristic
316C22×C32D3×D411yescharacteristic
317C22×C32C2×C3×D411yescharacteristic
318C2×C3⋊D3—21no—
319C2×C3⋊D3—21no—
320C2×C3⋊D3C3×SD1612yescomplemented
321C2×C3×D3—32no—
322C2×C3×D3—32no—
323C2×C3×D3—62no—
324C2×C3×D3—61no—
325C2×C3×D3—61no—
326C2×C3×D3—61no—
327C2×C3×D3—61no—
328C2×C3×D3—121no—
329C2×C3×D3—121no—
330C32×C4—21no—
331C32×C4—21no—
332C32×C4—21no—
333C32×C4—21no—
334C32×C4—21no—
335C32×C4—21no—
336C32×C4—41no—
337C32×C4—41no—
338C32×C4—121no—
339C32×C4—121no—
340C32×C4#23511yescharacteristic
341C32×C4D3×D411yescharacteristic
342C32×C4#23711yescharacteristic
343C32×C4C2×#6711yescharacteristic
344C32×C4C2×C3×D411yescharacteristic
345C32×C4C3×#4111yescharacteristic
346C3⋊Dic3—21no—
347C3⋊Dic3—41no—
348C3×Dic3—42no—
349C3×Dic3—41no—
350C3×Dic3—41no—
351C3×Dic3—41no—
352C3×Dic3—41no—
353C3×Dic3—61no—
354C3×Dic3—61no—
355C3×Dic3—81no—
356C3×Dic3—121no—
357C3×Dic3—122no—
358C3×Dic3—121no—
359C3×Dic3—121no—
360C2×SD16—31no—
361C22×C8—91no—
362C22⋊Q8—91no—
363#121—91no—
364#106—91no—
365#103—91no—
366#102—31no—
367C33#46511yescharacteristic, complemented, Sylow
368C23×C3—22no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3—21no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—42no—
375C23×C3—91no—
376C22×D3—31no—
377C22×D3—31no—
378C22×D3—61no—
379C3×Q8—62no—
380C3×Q8—61no—
381C3×Q8—61no—
382C3×Q8—61no—
383C3×Q8—61no—
384C3×Q8—121no—
385C3×D4—22no—
386C3×D4—22no—
387C3×D4—21no—
388C3×D4—21no—
389C3×D4—21no—
390C3×D4—21no—
391C3×D4—21no—
392C3×D4—21no—
393C3×D4—41no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—41no—
397C3×D4—42no—
398C3×D4—42no—
399C3×D4—362no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—41no—
404C2×C3×C4—61no—
405C2×C3×C4—61no—
406C2×C3×C4—61no—
407C2×C3×C4—61no—
408C2×C3×C4—91no—
409C2×C3×C4—121no—
410C2×C3×C4—121no—
411C2×C3×C4—181no—
412C2×C3×C4—181no—
413C2×C3×C4—181no—
414C2×C3×C4C2×D3211yescharacteristic
415C2×C3×C4C22×C3×D311yescharacteristic
416C2×C3×C4C22×C3×D311yescharacteristic
417#67—122no—
418#67—122no—
419#67—124no—
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—31no—
423C2×Dic3—31no—
424C2×Dic3—61no—
425C2×Dic3—62no—
426C2×Dic3—61no—
427C2×Dic3—61no—
428C4×D3—61no—
429C4×D3—61no—
430C4×D3—61no—
431C4×D3—61no—
432C4×D3—121no—
433C4×D3—121no—
434Dic6—21no—
435Dic6—21no—
436C3×C8—32no—
437C3×C8—32no—
438C3×C8—62no—
439C3×C8—61no—
440C3×C8—61no—
441C3×C8—121no—
442C3⋊C8—21no—
443C3⋊C8C3×#6712yescomplemented
444C2×C32—21no—
445C2×C32—21no—
446C2×C32—21no—
447C2×C32—21no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—21no—
451C2×C32—21no—
452C2×C32—21no—
453C2×C32—41no—
454C2×C32—41no—
455C2×C32—41no—
456C2×C32—82no—
457C2×C32—82no—
458C2×C32—82no—
459C2×C32—84no—
460C2×C32—81no—
461C2×C32—81no—
462C2×C32—81no—
463C2×C32#88411yescharacteristic
464C2×C32#91211yescharacteristic
465C2×C32D3×SD1611yescharacteristic
466C2×C32D3×SD1611yescharacteristic
467C2×C32#91611yescharacteristic
468C2×C32#93811yescharacteristic
469C2×C32C3×#12111yescharacteristic
470C2×C32C2×C3×SD1611yescharacteristic
471C2×C32C3×#13511yescharacteristic
472C3⋊D3—22no—
473C3×D3—62no—
474C3×D3—62no—
475C3×D3—122no—
476C2×Q8—31no—
477C2×D4—181no—
478C2×D4C3×D3211yescharacteristic
479C22×C4—91no—
480SD16—62no—
481C2×C8—31no—
482C2×C8—31no—
483C2×C8—182no—
484C4⋊C4—31no—
485C4⋊C4—91no—
486C4⋊C4—181no—
487C22⋊C4—181no—
488C22⋊C4—181no—
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—44no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3—41no—
501C22×C3—41no—
502C22×C3—41no—
503C22×C3—41no—
504C22×C3—41no—
505C22×C3—41no—
506C22×C3—41no—
507C22×C3—41no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—82no—
513C22×C3—82no—
514C22×C3—92no—
515C22×C3—181no—
516C22×C3—181no—
517C22×C3#381111yescharacteristic
518C22×C3C3×D3×D411yescharacteristic
519C22×C3C3×D3×D411yescharacteristic
520C2×D3—32no—
521C2×D3—32no—
522C2×D3—62no—
523C2×D3—61no—
524C2×D3—61no—
525C2×D3—61no—
526C2×D3—61no—
527C2×D3—121no—
528C2×D3—121no—
529C3×C4—21no—
530C3×C4—21no—
531C3×C4—21no—
532C3×C4—21no—
533C3×C4—21no—
534C3×C4—21no—
535C3×C4—41no—
536C3×C4—41no—
537C3×C4—122no—
538C3×C4—121no—
539C3×C4—121no—
540C3×C4—121no—
541C3×C4—121no—
542C3×C4—181no—
543C3×C4—241no—
544C3×C4—361no—
545C3×C4#380411yescharacteristic
546C3×C4D3×#6711yescharacteristic
547C3×C4C3×#23511yescharacteristic
548C3×C4C3×D3×D411yescharacteristic
549C3×C4C3×#23711yescharacteristic
550C3×C4C2×C3×#6711yescharacteristic
551Dic3—41no—
552Dic3—41no—
553Dic3—61no—
554Dic3—61no—
555Dic3—121no—
556Dic3—122no—
557Dic3—121no—
558Dic3—121no—
559C32—21no—
560C32—21no—
561C32—21no—
562C32—41no—
563C32#477811yescharacteristic, complemented
564C32#517111yescharacteristic, complemented
565C32C3×#46511yescharacteristic, complemented
566C23—21no—
567C23—91no—
568Q8—61no—
569Q8—61no—
570D4—21no—
571D4—21no—
572D4—362no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—91no—
576C2×C4—181no—
577C2×C4—181no—
578C2×C4—181no—
579C2×C4C2×C3×D3211yescharacteristic
580C8—32no—
581C8—61no—
582C2×C3—21no—
583C2×C3—21no—
584C2×C3—21no—
585C2×C3—21no—
586C2×C3—21no—
587C2×C3—21no—
588C2×C3—21no—
589C2×C3—21no—
590C2×C3—21no—
591C2×C3—41no—
592C2×C3—41no—
593C2×C3—41no—
594C2×C3—82no—
595C2×C3—82no—
596C2×C3—82no—
597C2×C3—84no—
598C2×C3—81no—
599C2×C3—81no—
600C2×C3—81no—
601C2×C3—182no—
602C2×C3#6393011yescharacteristic
603C2×C3#6393611yescharacteristic
604C2×C3#6395811yescharacteristic
605C2×C3C3×#88411yescharacteristic
606C2×C3C3×#91211yescharacteristic
607C2×C3C3×D3×SD1611yescharacteristic
608C2×C3C3×D3×SD1611yescharacteristic
609C2×C3C3×#91611yescharacteristic
610C2×C3C3×#93811yescharacteristic
611D3—62no—
612D3—62no—
613D3—122no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—92no—
618C22—181no—
619C22—181no—
620C22C3×#381111yescharacteristic
621C4—121no—
622C4—121no—
623C4—181no—
624C4—361no—
625C4C3×#380411yescharacteristic
626C4C3×D3×#6711yescharacteristic
627C3—21no—
628C3—21no—
629C3—21no—
630C3—41no—
631C3#1059146511yescharacteristic, direct product factor
632C3C3×#477811yescharacteristic, complemented
633C3C3×#517111yescharacteristic, complemented
634C2—81no—
635C2—182no—
636C2C3×#6393011yescharacteristic
637C2C3×#6393611yescharacteristic
638C2C3×#6395811yescharacteristic
subgroupG/H
centerC22×C3#3811
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC3×#64147C2
hypercenterC2×C3×D4D32
hypocenterC32C3×#465
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
2. C2×C32×#214C2semidirect
3. C3×#63871C2semidirect
4. C2×C3×#3710C2semidirect
5. C32×#832C2semidirect
6. C3×#63573C2semidirect
7. C3×#63566C2semidirect
10. #10591465C3direct (N × H)
31. C2×C3×#214D3semidirect
32. C2×C3×#214C2×C3semidirect
36. #63871C2×C3semidirect
37. C2×#3710C2×C3semidirect
41. C3×#832D3semidirect
42. C3×#832C2×C3semidirect
46. #63573C2×C3semidirect
47. #63566C2×C3semidirect
63. C32×C3⋊C8D4semidirect
125. C2×C3×C3⋊D3SD16semidirect
142. C2×#214C3×D3semidirect
152. #832C3×D3semidirect
230. C3×C3⋊C8#67semidirect
231. C3×C3⋊C8C3×D4semidirect
320. C2×C3⋊D3C3×SD16semidirect
367. C33#465semidirect
443. C3⋊C8C3×#67semidirect
563. C32#4778semidirect
564. C32#5171semidirect
565. C32C3×#465semidirect
631. C3#10591465direct (N × H)
632. C3C3×#4778semidirect
633. C3C3×#5171semidirect

Maximal subgroups

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

subgroupindex
C3×#641472
C2×C32×#2142
C3×#638712
C2×C3×#37102
C32×#8322
C3×#635732
C3×#635662
C3×#51713
C3×#47783
#105914653