Finite Group Databasebeta

← C3×#10591686 #13158702

Subgroups — C3×#10591686 #13158702

Order 1728.

882 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64367C211yesmaximal, characteristic, complemented
2C32×#949C211yesmaximal, characteristic, complemented
3C3×#63860C211yesmaximal, characteristic, complemented
4C3×#63830C211yesmaximal, characteristic, complemented
5C32×#837C211yesmaximal, characteristic, complemented
6C3×#63570C211yesmaximal, characteristic, complemented
7C3×#63561C211yesmaximal, characteristic, complemented
8C3×#5201—31nomaximal
9C3×#4827—31nomaximal
10#10591686C311yesmaximal, characteristic, direct product factor
11C33×#41C2211yescharacteristic
12C3×Q8×C3⋊D3—21no—
13C3×#3830—21no—
14C3×#3830—22no—
15C3×#3828—21no—
16C3×#3828C2211yescharacteristic
17C2×C3×#634—21no—
18C2×C3×#634C2211yescharacteristic
19C2×C32×D12C2211yescharacteristic
20C32×#215—21no—
21C32×#213—21no—
22C32×C4×Dic3C2211yescharacteristic
23C32×#208C2211yescharacteristic
24C32×#208C2211yescharacteristic
25C3×(C2×D3)⋊Dic3—22no—
26C3×#3716—21no—
27C3×#3714—21no—
28#64367C2×C311yescharacteristic, complemented
29C3×#1010—31no—
30C3×#1010—31no—
31C3×#1010—61no—
32C32×AGL(1,ℤ/8)—31no—
33C3×#949—21no—
34C3×#949D311yescharacteristic, complemented
35C3×#949C2×C311yescharacteristic, complemented
36C3×#936—31no—
37C3×#932—31no—
38C3×#908—31no—
39C3×#875—31no—
40#63860C2×C311yescharacteristic, complemented
41#63830C2×C311yescharacteristic, complemented
42C32×#104—31no—
43C3×#837—21no—
44C3×#837D311yescharacteristic, complemented
45C3×#837C2×C311yescharacteristic, complemented
46C3×#825—31no—
47C3×#824—31no—
48C3×#824—31no—
49C3×#805—31no—
50#63570C2×C311yescharacteristic, complemented
51#63561C2×C311yescharacteristic, complemented
52C22×C32×D3—21no—
53C33×Q8D411yescharacteristic
54C33×D4—21no—
55C33×D4D411yescharacteristic
56C2×C33×C4—21no—
57C2×C33×C4C2311yescharacteristic
58C3×#638—22no—
59C3×#638—21no—
60C2×C3×C3⋊Dic3—22no—
61C2×C3×C3⋊Dic3D412yes—
62C3×C4⋊(C3⋊D3)D411yescharacteristic
63C3×C4×C3⋊D3—22no—
64C3×C4×C3⋊D3—21no—
65C3×#634—22no—
66C3×#634—21no—
67C3×#634—21no—
68C3×#634D411yescharacteristic
69C2×C32×Dic3—21no—
70C32×D12—21no—
71C32×D12—21no—
72C32×C3⋊C8—21no—
73C32×C3⋊C8—21no—
74C3×#454—91no—
75#5201—31no—
76#4827—31no—
77C32×#41—21no—
78C32×#41—21no—
79C32×#41—21no—
80C32×#41—41no—
81C32×#41C2×D311yescharacteristic
82C32×#41C2×D311yescharacteristic
83C32×#41C22×C311yescharacteristic
84C2×C32×D4—31no—
85Q8×C3⋊D3—21no—
86#3830—21no—
87#3830—22no—
88#3828—21no—
89#3828C22×C311yescharacteristic
90C2×#634—21no—
91C2×#634C22×C311yescharacteristic
92C3×D3×Q8—62no—
93C3×D3×Q8—61no—
94C3×D3×Q8—61no—
95C3×#237—62no—
96C3×#237—62no—
97C3×#237—64no—
98C3×#237—62no—
99C3×#237—61no—
100C3×#237—61no—
101C3×#235—31no—
102C3×#235—31no—
103C3×#235—61no—
104C3×#235—61no—
105C3×#235—61no—
106C3×#235—62no—
107C2×C3×D12—21no—
108C2×C3×D12C2×D311yescharacteristic
109C2×C3×D12C22×C311yescharacteristic
110C2×C3×Dic6—31no—
111C2×C3×Dic6—31no—
112C2×C3×Dic6—61no—
113C2×C3×Dic6—61no—
114C2×C3×Dic6—61no—
115C2×C3×Dic6—62no—
116C32×SD16—61no—
117C32×D8—61no—
118C32×M16—31no—
119C32×M16—31no—
120C32×C42—31no—
121C3×#217—62no—
122C3×#215—22no—
123C3×#215—21no—
124C3×#215—21no—
125C3×#214—61no—
126C3×#213—22no—
127C3×#213—21no—
128C3×#213—21no—
129C3×#213—61no—
130C3×(C2×D3)⋊C4—62no—
131C3×C4×Dic3—21no—
132C3×C4×Dic3C2×D311yescharacteristic
133C3×C4×Dic3C22×C311yescharacteristic
134C3×#208—21no—
135C3×#208—21no—
136C3×#208C2×D311yescharacteristic
137C3×#208C2×D311yescharacteristic
138C3×#208C22×C311yescharacteristic
139C3×#208C22×C311yescharacteristic
140C3×D24—61no—
141C3×#204—61no—
142(C2×D3)⋊Dic3—22no—
143#3716—21no—
144#3714—21no—
145C22×C33—21no—
146C22×C33C2×D411yescharacteristic
147C2×C3×C3⋊D3—21no—
148C2×C32×D3—42no—
149C2×C32×D3—41no—
150C33×C4—21no—
151C33×C4C2×D411yescharacteristic
152C33×C4C2×D411yescharacteristic
153C3×C3⋊Dic3—22no—
154C3×C3⋊Dic3—21no—
155C32×Dic3—41no—
156C3×2−1+4—91no—
157#1010—31no—
158#1010—31no—
159#1010—61no—
160C3×AGL(1,ℤ/8)—31no—
161C3×AGL(1,ℤ/8)—31no—
162C3×AGL(1,ℤ/8)—61no—
163C3×AGL(1,ℤ/8)—91no—
164C3×#124—91no—
165#949C3×D311yescharacteristic, complemented
166#936—31no—
167#932—31no—
168#908—31no—
169#875—31no—
170C3×#104—31no—
171C3×#104—31no—
172C3×#104—61no—
173C3×#104—91no—
174C3×#101—91no—
175#837C3×D311yescharacteristic, complemented
176#825—31no—
177#824—31no—
178#824—31no—
179#805—31no—
180C23×C32—61no—
181C22×C3×D3—21no—
182C22×C3×D3—21no—
183C22×C3×D3—41no—
184C32×Q8—21no—
185C32×Q8—21no—
186C32×Q8—21no—
187C32×Q8—41no—
188C32×Q8D1211yescharacteristic
189C32×Q8#6711yescharacteristic
190C32×Q8C3×D411yescharacteristic
191C32×D4—21no—
192C32×D4—22no—
193C32×D4—21no—
194C32×D4—21no—
195C32×D4—21no—
196C32×D4—21no—
197C32×D4—21no—
198C32×D4—41no—
199C32×D4—41no—
200C32×D4—41no—
201C32×D4—42no—
202C32×D4—61no—
203C32×D4—61no—
204C32×D4D1211yescharacteristic
205C32×D4#6711yescharacteristic
206C32×D4C3×D411yescharacteristic
207C2×C32×C4—21no—
208C2×C32×C4—22no—
209C2×C32×C4—21no—
210C2×C32×C4—21no—
211C2×C32×C4—21no—
212C2×C32×C4—21no—
213C2×C32×C4—21no—
214C2×C32×C4—41no—
215C2×C32×C4—41no—
216C2×C32×C4—41no—
217C2×C32×C4—42no—
218C2×C32×C4—61no—
219C2×C32×C4C22×D311yescharacteristic
220C2×C32×C4C22×D311yescharacteristic
221C2×C32×C4C23×C311yescharacteristic
222#638—22no—
223#638—21no—
224C2×C3⋊Dic3—22no—
225C2×C3⋊Dic3C3×D412yes—
226C4⋊(C3⋊D3)C3×D411yescharacteristic
227C4×C3⋊D3—22no—
228C4×C3⋊D3—21no—
229#634—22no—
230#634—21no—
231#634—21no—
232#634C3×D411yescharacteristic
233C3×#67—64no—
234C3×#67—62no—
235C3×#67—62no—
236C3×#67—62no—
237C3×#67—61no—
238C3×#67—61no—
239C2×C3×Dic3—21no—
240C2×C3×Dic3—21no—
241C2×C3×Dic3—32no—
242C2×C3×Dic3—32no—
243C2×C3×Dic3—41no—
244C2×C3×Dic3—62no—
245C2×C3×Dic3—64no—
246C2×C3×Dic3—62no—
247C2×C3×Dic3—62no—
248C3×D12—22no—
249C3×D12—21no—
250C3×D12—21no—
251C3×D12—21no—
252C3×D12—21no—
253C3×D12—31no—
254C3×D12—31no—
255C3×D12—41no—
256C3×D12—61no—
257C3×C4×D3—64no—
258C3×C4×D3—62no—
259C3×C4×D3—62no—
260C3×C4×D3—62no—
261C3×C4×D3—61no—
262C3×C4×D3—61no—
263C3×Dic6—31no—
264C3×Dic6—31no—
265C3×Dic6—61no—
266C3×Dic6—62no—
267C3×Dic6—64no—
268C3×Dic6—62no—
269C3×Dic6—62no—
270C3×Dic6—62no—
271C3×Dic6—61no—
272C3×Dic6—61no—
273C3×Dic6—61no—
274C3×Dic6—61no—
275C32×C8—61no—
276C32×C8—61no—
277C3×C3⋊C8—22no—
278C3×C3⋊C8—21no—
279C3×C3⋊C8—21no—
280C3×C3⋊C8—21no—
281C3×C3⋊C8—21no—
282C3×C3⋊C8—41no—
283#454—91noSylow
284C2×C33—21no—
285C2×C33—41no—
286C2×C33#12011yescharacteristic
287C3×C3⋊D3—41no—
288C32×D3—81no—
289C3×#41—21no—
290C3×#41—21no—
291C3×#41—21no—
292C3×#41—41no—
293C3×#41—91no—
294C3×#41—181no—
295C3×#41—182no—
296C3×#41—181no—
297C3×#41D3211yescharacteristic
298C3×#41C2×C3×D311yescharacteristic
299C3×#41C2×C3×D311yescharacteristic
300C2×C3×Q8—91no—
301C2×C3×Q8—181no—
302C2×C3×Q8—181no—
303C2×C3×D4—31no—
304C2×C3×D4—31no—
305C2×C3×D4—61no—
306D3×Q8—62no—
307D3×Q8—61no—
308D3×Q8—61no—
309#237—62no—
310#237—62no—
311#237—64no—
312#237—62no—
313#237—61no—
314#237—61no—
315#235—31no—
316#235—31no—
317#235—61no—
318#235—61no—
319#235—61no—
320#235—62no—
321C2×D12C2×C3×D311yescharacteristic
322C2×Dic6—31no—
323C2×Dic6—31no—
324C2×Dic6—61no—
325C2×Dic6—61no—
326C2×Dic6—61no—
327C2×Dic6—62no—
328C3×SD16—62no—
329C3×SD16—61no—
330C3×SD16—61no—
331C3×SD16—181no—
332C3×D8—62no—
333C3×D8—61no—
334C3×D8—61no—
335C3×D8—181no—
336C3×M16—31no—
337C3×M16—31no—
338C3×M16—31no—
339C3×M16—31no—
340C3×M16—61no—
341C3×M16—61no—
342C3×C22⋊C4—182no—
343C3×C42—31no—
344C3×C42—31no—
345C3×C42—61no—
346#217—62no—
347#215—21no—
348#214—61no—
349#213—21no—
350#213—61no—
351(C2×D3)⋊C4—62no—
352C4×Dic3C2×C3×D311yescharacteristic
353#208C2×C3×D311yescharacteristic
354#208C2×C3×D311yescharacteristic
355D24—61no—
356#204—61no—
357C22×C32—21no—
358C22×C32—22no—
359C22×C32—21no—
360C22×C32—21no—
361C22×C32—21no—
362C22×C32—21no—
363C22×C32—21no—
364C22×C32—41no—
365C22×C32—41no—
366C22×C32—41no—
367C22×C32—42no—
368C22×C32—121no—
369C22×C32—122no—
370C22×C32D3×D411yescharacteristic
371C22×C32D3×D411yescharacteristic
372C22×C32C2×C3×D411yescharacteristic
373C2×C3⋊D3—21no—
374C2×C3×D3—42no—
375C2×C3×D3—42no—
376C2×C3×D3—42no—
377C2×C3×D3—41no—
378C2×C3×D3—41no—
379C2×C3×D3—62no—
380C2×C3×D3—61no—
381C2×C3×D3—61no—
382C2×C3×D3—82no—
383C32×C4—21no—
384C32×C4—22no—
385C32×C4—21no—
386C32×C4—21no—
387C32×C4—21no—
388C32×C4—21no—
389C32×C4—21no—
390C32×C4—21no—
391C32×C4—21no—
392C32×C4—21no—
393C32×C4—41no—
394C32×C4—41no—
395C32×C4—41no—
396C32×C4—41no—
397C32×C4—42no—
398C32×C4—121no—
399C32×C4C2×D1211yescharacteristic
400C32×C4D3×D411yescharacteristic
401C32×C4D3×D411yescharacteristic
402C32×C4C2×#6711yescharacteristic
403C32×C4C2×C3×D411yescharacteristic
404C32×C4C2×C3×D411yescharacteristic
405C3⋊Dic3—22no—
406C3⋊Dic3—21no—
407C3×Dic3—41no—
408C3×Dic3—41no—
409C3×Dic3—42no—
410C3×Dic3—64no—
411C3×Dic3—62no—
412C3×Dic3—61no—
413C3×Dic3—62no—
414C3×Dic3—61no—
415C3×Dic3—62no—
4162−1+4—91no—
417AGL(1,ℤ/8)—31no—
418AGL(1,ℤ/8)—91no—
419#124—91no—
420#104—31no—
421#104—91no—
422#101—91no—
423C33#45411yescharacteristic, complemented, Sylow
424C23×C3—61no—
425C23×C3—61no—
426C23×C3—121no—
427C22×D3—21no—
428C3×Q8—21no—
429C3×Q8—21no—
430C3×Q8—21no—
431C3×Q8—41no—
432C3×Q8—91no—
433C3×Q8—182no—
434C3×Q8—181no—
435C3×Q8—181no—
436C3×Q8#62611yescharacteristic
437C3×Q8C3×D1211yescharacteristic
438C3×Q8C3×#6711yescharacteristic
439C3×D4—22no—
440C3×D4—21no—
441C3×D4—21no—
442C3×D4—21no—
443C3×D4—21no—
444C3×D4—21no—
445C3×D4—21no—
446C3×D4—41no—
447C3×D4—41no—
448C3×D4—42no—
449C3×D4—41no—
450C3×D4—62no—
451C3×D4—61no—
452C3×D4—61no—
453C3×D4—61no—
454C3×D4—61no—
455C3×D4—91no—
456C3×D4—121no—
457C3×D4—182no—
458C3×D4—181no—
459C3×D4#62611yescharacteristic
460C3×D4C3×D1211yescharacteristic
461C3×D4C3×#6711yescharacteristic
462C2×C3×C4—22no—
463C2×C3×C4—21no—
464C2×C3×C4—21no—
465C2×C3×C4—21no—
466C2×C3×C4—21no—
467C2×C3×C4—21no—
468C2×C3×C4—21no—
469C2×C3×C4—41no—
470C2×C3×C4—41no—
471C2×C3×C4—42no—
472C2×C3×C4—41no—
473C2×C3×C4—61no—
474C2×C3×C4—61no—
475C2×C3×C4—92no—
476C2×C3×C4—121no—
477C2×C3×C4—182no—
478C2×C3×C4—182no—
479C2×C3×C4—181no—
480C2×C3×C4C2×D3211yescharacteristic
481C2×C3×C4C22×C3×D311yescharacteristic
482C2×C3×C4C22×C3×D311yescharacteristic
483#67—64no—
484#67—62no—
485#67—62no—
486#67—62no—
487#67—61no—
488#67—61no—
489C2×Dic3—21no—
490C2×Dic3—32no—
491C2×Dic3—32no—
492C2×Dic3—62no—
493C2×Dic3—64no—
494C2×Dic3—62no—
495C2×Dic3—62no—
496D12—21no—
497D12—21no—
498D12—31no—
499D12—31no—
500D12—61no—
501C4×D3—64no—
502C4×D3—62no—
503C4×D3—62no—
504C4×D3—62no—
505C4×D3—61no—
506C4×D3—61no—
507Dic6—31no—
508Dic6—31no—
509Dic6—61no—
510Dic6—62no—
511Dic6—64no—
512Dic6—62no—
513Dic6—62no—
514Dic6—62no—
515Dic6—61no—
516Dic6—61no—
517Dic6—61no—
518Dic6—61no—
519C3×C8—62no—
520C3×C8—61no—
521C3×C8—61no—
522C3×C8—61no—
523C3×C8—61no—
524C3×C8—121no—
525C3⋊C8—21no—
526C3⋊C8—21no—
527C2×C32—21no—
528C2×C32—21no—
529C2×C32—21no—
530C2×C32—21no—
531C2×C32—21no—
532C2×C32—21no—
533C2×C32—41no—
534C2×C32—41no—
535C2×C32—41no—
536C2×C32—41no—
537C2×C32—41no—
538C2×C32—42no—
539C2×C32—41no—
540C2×C32—41no—
541C2×C32—81no—
542C2×C32—81no—
543C2×C32—81no—
544C2×C32—82no—
545C2×C32—241no—
546C2×C32#88211yescharacteristic
547C2×C32#93711yescharacteristic
548C2×C32C3×#12011yescharacteristic
549C3⋊D3—41no—
550C3×D3—82no—
551C3×D3—81no—
552C3×D3—81no—
553C3×D3—122no—
554C3×D3—121no—
555C3×D3—121no—
556#41—91no—
557#41—181no—
558#41—182no—
559#41—181no—
560#41C3×D3211yescharacteristic
561C2×Q8—91no—
562C2×Q8—181no—
563C2×Q8—181no—
564C2×D4—31no—
565SD16—61no—
566SD16—181no—
567D8—61no—
568D8—181no—
569M16—31no—
570M16—31no—
571C22⋊C4—182no—
572C42—31no—
573C22×C3—22no—
574C22×C3—21no—
575C22×C3—21no—
576C22×C3—21no—
577C22×C3—21no—
578C22×C3—21no—
579C22×C3—21no—
580C22×C3—41no—
581C22×C3—41no—
582C22×C3—42no—
583C22×C3—41no—
584C22×C3—122no—
585C22×C3—122no—
586C22×C3—122no—
587C22×C3—121no—
588C22×C3—121no—
589C22×C3—181no—
590C22×C3—242no—
591C22×C3#381111yescharacteristic
592C22×C3C3×D3×D411yescharacteristic
593C22×C3C3×D3×D411yescharacteristic
594C2×D3—42no—
595C2×D3—41no—
596C2×D3—62no—
597C2×D3—61no—
598C2×D3—61no—
599C3×C4—22no—
600C3×C4—21no—
601C3×C4—21no—
602C3×C4—21no—
603C3×C4—21no—
604C3×C4—21no—
605C3×C4—21no—
606C3×C4—21no—
607C3×C4—21no—
608C3×C4—21no—
609C3×C4—41no—
610C3×C4—41no—
611C3×C4—42no—
612C3×C4—41no—
613C3×C4—41no—
614C3×C4—121no—
615C3×C4—121no—
616C3×C4—122no—
617C3×C4—182no—
618C3×C4—181no—
619C3×C4C2×#62611yescharacteristic
620C3×C4#381111yescharacteristic
621C3×C4C2×C3×D1211yescharacteristic
622C3×C4C3×D3×D411yescharacteristic
623C3×C4C3×D3×D411yescharacteristic
624C3×C4C2×C3×#6711yescharacteristic
625Dic3—41no—
626Dic3—64no—
627Dic3—62no—
628Dic3—61no—
629Dic3—62no—
630Dic3—61no—
631Dic3—62no—
632C32—21no—
633C32—21no—
634C32—21no—
635C32—41no—
636C32#482711yescharacteristic, complemented
637C32#520111yescharacteristic, complemented
638C32C3×#45411yescharacteristic, complemented
639C23—61no—
640Q8—91no—
641Q8—182no—
642Q8—181no—
643Q8—181no—
644Q8C3×#62611yescharacteristic
645D4—21no—
646D4—61no—
647D4—61no—
648D4—91no—
649D4—182no—
650D4—181no—
651D4C3×#62611yescharacteristic
652C2×C4—21no—
653C2×C4—61no—
654C2×C4—92no—
655C2×C4—182no—
656C2×C4—182no—
657C2×C4—181no—
658C2×C4C2×C3×D3211yescharacteristic
659C8—61no—
660C8—61no—
661C2×C3—21no—
662C2×C3—21no—
663C2×C3—21no—
664C2×C3—21no—
665C2×C3—21no—
666C2×C3—21no—
667C2×C3—41no—
668C2×C3—41no—
669C2×C3—41no—
670C2×C3—42no—
671C2×C3—41no—
672C2×C3—41no—
673C2×C3—41no—
674C2×C3—41no—
675C2×C3—81no—
676C2×C3—81no—
677C2×C3—81no—
678C2×C3—82no—
679C2×C3—242no—
680C2×C3—241no—
681C2×C3—241no—
682C2×C3—361no—
683C2×C3#6398111yescharacteristic
684C2×C3C3×#88211yescharacteristic
685C2×C3C3×#93711yescharacteristic
686D3—81no—
687D3—122no—
688D3—121no—
689D3—121no—
690C22—21no—
691C22—122no—
692C22—121no—
693C22—181no—
694C22C3×#381111yescharacteristic
695C4—21no—
696C4—121no—
697C4—182no—
698C4—181no—
699C4C2×C3×#62611yescharacteristic
700C4C3×#381111yescharacteristic
701C3—21no—
702C3—21no—
703C3—21no—
704C3—41no—
705C3#1059168611yescharacteristic, direct product factor
706C3C3×#482711yescharacteristic, complemented
707C3C3×#520111yescharacteristic, complemented
708C2—21no—
709C2—41no—
710C2—241no—
711C2—361no—
712C2C3×#6398111yescharacteristic
subgroupG/H
centerC2×C3#63981
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×#41C22
socleC2×C33#120
hypercenterC3×#41D32
hypocenterC32C3×#454
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
1. C3×#64367C2semidirect
2. C32×#949C2semidirect
3. C3×#63860C2semidirect
4. C3×#63830C2semidirect
5. C32×#837C2semidirect
6. C3×#63570C2semidirect
7. C3×#63561C2semidirect
10. #10591686C3direct (N × H)
28. #64367C2×C3semidirect
34. C3×#949D3semidirect
35. C3×#949C2×C3semidirect
40. #63860C2×C3semidirect
41. #63830C2×C3semidirect
44. C3×#837D3semidirect
45. C3×#837C2×C3semidirect
50. #63570C2×C3semidirect
51. #63561C2×C3semidirect
165. #949C3×D3semidirect
175. #837C3×D3semidirect
423. C33#454semidirect
636. C32#4827semidirect
637. C32#5201semidirect
638. C32C3×#454semidirect
705. C3#10591686direct (N × H)
706. C3C3×#4827semidirect
707. C3C3×#5201semidirect

Maximal subgroups

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

subgroupindex
C3×#643672
C32×#9492
C3×#638602
C3×#638302
C32×#8372
C3×#635702
C3×#635612
C3×#52013
C3×#48273
#105916863