Finite Group Databasebeta

← #13155969

Subgroups — #13155969

Order 1728.

830 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723529C211yesmaximal, characteristic, complemented
2#11723499C211yesmaximal, characteristic, complemented
3C2×C3×#3714C211yesmaximal, characteristic, complemented
4C2×C3×#3713C211yesmaximal, characteristic, complemented
5#11722075C211yesmaximal, characteristic, complemented
6#11722050C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591486—31nomaximal
9#10591433—31nomaximal
10#10590749—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×#6290—21no—
15C2×#6290—21no—
16C2×D3×C3⋊Dic3—21no—
17C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)—21no—
22C3×#3714—22no—
23C3×#3713—22no—
24#64149—31no—
25D4×C3⋊Dic3—31no—
26C2×C3×#213—31no—
27C2×C3×#213—31no—
28C2×C3×#213—31no—
29C2×C3×D24—31no—
30#63914—31no—
31#63914—31no—
32#63914—61no—
33#63898—31no—
34#63898—31no—
35#63898—61no—
36C2×#3714D311yescharacteristic, complemented
37C2×#3713D311yescharacteristic, complemented
38#63641—31no—
39C3×#832—31no—
40C3×#821—31no—
41#63572—31no—
42#63572—31no—
43#63572—61no—
44#63564—31no—
45#63563—31no—
46#63563D311yescharacteristic, complemented
47#63562—31no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C22×C32×D3—21no—
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)D411yescharacteristic
53C2×C3⋊(C3⋊Dic3)D411yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C32×D12—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×D12D412yescomplemented
61#6290—42no—
62#6290—42no—
63D3×C3⋊Dic3—42no—
64C3×C3⋊(C3⋊C8)—21no—
65C32×C3⋊C8—21no—
66#5150—91no—
67#5150—91no—
68#4786—91no—
69C2×C32×D4—31no—
70C2×C32×D4—31no—
71C2×#638—61no—
72C2×#638—61no—
73C22×C3⋊Dic3—61no—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C3×D12—21no—
76C2×C3×D12—21no—
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×D12C2×D311yescharacteristic
81C2×C3×D12C2×D311yescharacteristic
82C2×C3×D12C2×D311yescharacteristic
83C2×C3×D12C2×D311yescharacteristic
84C2×#625—61no—
85C2×#625—61no—
86C2×#625—61no—
87C2×#625—61no—
88C2×#625—62no—
89C2×#625—62no—
90C2×D3×Dic3—61no—
91C2×D3×Dic3—61no—
92C2×D3×Dic3—62no—
93C2×C32×C8—31no—
94#3757—61no—
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic
96C4×C3⋊Dic3—31no—
97C4×C3⋊Dic3—31no—
98C4×C3⋊Dic3—31no—
99C4×C3⋊Dic3—61no—
100C4×C3⋊Dic3—61no—
101C4×C3⋊Dic3—61no—
102C4×C3⋊Dic3—121no—
103C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
104C3×#213—62no—
105C3×#213—62no—
106C3×#213—62no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C2×C3×C3⋊C8—21no—
111C2×C3×C3⋊C8—31no—
112C2×C3×C3⋊C8—31no—
113C2×C3×C3⋊C8—61no—
114C2×C3×C3⋊C8C2×D311yescharacteristic
115C2×C3×C3⋊C8C2×D311yescharacteristic
116C3×D24—62no—
117(C2×D3)⋊Dic3—61no—
118(C2×D3)⋊Dic3—61no—
119(C2×D3)⋊Dic3—62no—
120#3714—22no—
121#3713—22no—
122C22×C33C2×D411yescharacteristic
123C2×C3×C3⋊D3—41no—
124C2×C3×C3⋊D3—41no—
125C2×C3×C3⋊D3—41no—
126C2×C32×D3—22no—
127C2×C32×D3—41no—
128C2×C32×D3—41no—
129C2×C32×D3—41no—
130C2×C32×D3—41no—
131C2×C32×D3—41no—
132C33×C4C2×D411yescharacteristic
133C33×C4#4111yescharacteristic
134C3⋊(C3⋊Dic3)—21no—
135C3⋊(C3⋊Dic3)D812yescomplemented
136C3×C3⋊Dic3—41no—
137C2×C3×D8—91no—
138C2×C3×D8—91no—
139#940—91no—
140#940—91no—
141#940—91no—
142#940—181no—
143D4×Dic3—91no—
144D4×Dic3—91no—
145D4×Dic3—181no—
146C2×#213—31no—
147C2×#213—31no—
148C2×#213—31no—
149C2×D24—31no—
150#893—91no—
151C3×#102—91no—
152#832—31no—
153#832—91no—
154#821—31no—
155#821—91no—
156Dic3⋊C8—91no—
157Dic3⋊C8—91no—
158Dic3⋊C8—91no—
159Dic3⋊C8—181no—
160#809—91no—
161C23×C32—61no—
162C23×C32—61no—
163C22×C3⋊D3—21no—
164C22×C3×D3—22no—
165C22×C3×D3—21no—
166C22×C3×D3—21no—
167C22×C3×D3—22no—
168C22×C3×D3—21no—
169C22×C3×D3—21no—
170C22×C3×D3—62no—
171C22×C3×D3—61no—
172C22×C3×D3—61no—
173C32×D4—32no—
174C32×D4—61no—
175C32×D4—61no—
176C32×D4—61no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—41no—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184#638—122no—
185#638—122no—
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—31no—
192C2×C3⋊Dic3—31no—
193C2×C3⋊Dic3—61no—
194C2×C3⋊Dic3—61no—
195C2×C3⋊Dic3—61no—
196C2×C3⋊Dic3—61no—
197C2×C3⋊Dic3—61no—
198C2×C3⋊Dic3—61no—
199C2×C3⋊Dic3—121no—
200C2×C3⋊Dic3—121no—
201C2×C3⋊Dic3—122no—
202C4⋊(C3⋊D3)—21no—
203C4⋊(C3⋊D3)—21no—
204C2×C3×Dic3—62no—
205C2×C3×Dic3—61no—
206C2×C3×Dic3—61no—
207C3×D12—22no—
208C3×D12—21no—
209C3×D12—21no—
210C3×D12—21no—
211C3×D12—21no—
212C3×D12—21no—
213C3×D12—21no—
214C3×D12—22no—
215C3×D12—41no—
216C3×D12—41no—
217C3×D12—61no—
218C3×D12—61no—
219C3×D12—61no—
220C3×D12—61no—
221C3×D12—121no—
222C3×D12—121no—
223C3×D12#6712yescomplemented
224C3×D12#6712yescomplemented
225#625—122no—
226#625—122no—
227#625—124no—
228#625—124no—
229#625—122no—
230#625—122no—
231D3×Dic3—122no—
232D3×Dic3—122no—
233D3×Dic3—124no—
234C32×C8—61no—
235C3⋊(C3⋊C8)—21no—
236C3×C3⋊C8—21no—
237C3×C3⋊C8—21no—
238C3×C3⋊C8—41no—
239C3×C3⋊C8—61no—
240C3×C3⋊C8—61no—
241C3×C3⋊C8—121no—
242#459—271noSylow
243C2×C33#12111yescharacteristic
244C2×C33C2×D811yescharacteristic
245C2×C33AGL(1,ℤ/8)11yescharacteristic
246C3×C3⋊D3—81no—
247C32×D3—42no—
248C32×D3—81no—
249C2×C3×D4—31no—
250C2×C3×D4—31no—
251C2×C3×D4—31no—
252C2×C3×D4—31no—
253C2×C3×D4—61no—
254C2×C3×D4—61no—
255C2×C3×D4—91no—
256C2×#67—181no—
257C2×#67—181no—
258C2×#67—181no—
259C2×#67—181no—
260C2×#67—181no—
261C2×#67—181no—
262C2×#67—182no—
263C2×#67—182no—
264C22×Dic3—181no—
265C22×Dic3—181no—
266C22×Dic3—182no—
267C2×D12—31no—
268C2×D12—31no—
269C2×D12—61no—
270C2×D12D3211yescharacteristic
271C2×D12D3211yescharacteristic
272C2×C4×D3—181no—
273C3×D8—182no—
274C3×D8—182no—
275C2×C3×C8—31no—
276C2×C3×C8—31no—
277C2×C3×C8—61no—
278C2×C3×C8—91no—
279C3×C4⋊C4—91no—
280#217—181no—
281#217—181no—
282#217—182no—
283#213—62no—
284#213—62no—
285#213—62no—
286(C2×D3)⋊C4—181no—
287C4⋊Dic3—31no—
288C4⋊Dic3—31no—
289C4⋊Dic3—61no—
290C4×Dic3—91no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—181no—
294C4×Dic3—181no—
295C4×Dic3—181no—
296C4×Dic3—361no—
297C2×C3⋊C8—31no—
298C2×C3⋊C8—31no—
299C2×C3⋊C8—61no—
300C2×C3⋊C8D3211yescharacteristic
301D24—62no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—41no—
306C22×C32—62no—
307C22×C32—121no—
308C22×C32—121no—
309C22×C32—121no—
310C22×C32—121no—
311C22×C32—121no—
312C22×C32D3×D411yescharacteristic
313C22×C32D3×D411yescharacteristic
314C22×C32D3×D411yescharacteristic
315C2×C3⋊D3—41no—
316C2×C3⋊D3—41no—
317C2×C3⋊D3—41no—
318C2×C3×D3—24no—
319C2×C3×D3—22no—
320C2×C3×D3—22no—
321C2×C3×D3—42no—
322C2×C3×D3—42no—
323C2×C3×D3—41no—
324C2×C3×D3—41no—
325C2×C3×D3—41no—
326C2×C3×D3—41no—
327C2×C3×D3—42no—
328C2×C3×D3—42no—
329C2×C3×D3—42no—
330C2×C3×D3—41no—
331C2×C3×D3—41no—
332C2×C3×D3—41no—
333C2×C3×D3—41no—
334C2×C3×D3—41no—
335C2×C3×D3—41no—
336C2×C3×D3—122no—
337C2×C3×D3—122no—
338C2×C3×D3—122no—
339C2×C3×D3—121no—
340C2×C3×D3—121no—
341C2×C3×D3—121no—
342C2×C3×D3—121no—
343C2×C3×D3—121no—
344C2×C3×D3—121no—
345C32×C4—21no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—21no—
349C32×C4—21no—
350C32×C4—21no—
351C32×C4—41no—
352C32×C4—41no—
353C32×C4#23511yescharacteristic
354C32×C4D3×D411yescharacteristic
355C32×C4#23711yescharacteristic
356C32×C4#23711yescharacteristic
357C32×C4C2×#6711yescharacteristic
358C32×C4C2×#6711yescharacteristic
359C3⋊Dic3—32no—
360C3⋊Dic3—32no—
361C3⋊Dic3—32no—
362C3⋊Dic3—41no—
363C3⋊Dic3—62no—
364C3⋊Dic3—62no—
365C3⋊Dic3—62no—
366C3⋊Dic3—61no—
367C3⋊Dic3—62no—
368C3⋊Dic3—61no—
369C3⋊Dic3—61no—
370C3⋊Dic3—122no—
371C3⋊Dic3—121no—
372C3⋊Dic3—121no—
373C3⋊Dic3—122no—
374C3×Dic3—122no—
375C3×Dic3—121no—
376C3×Dic3—121no—
377C2×D8—91no—
378C2×D8—91no—
379#127—271no—
380C4×D4—271no—
381C4⋊C8—271no—
382#102—91no—
383#102—271no—
384C33#45911yescharacteristic, complemented, Sylow
385C23×C3—62no—
386C23×C3—62no—
387C23×C3—61no—
388C23×C3—61no—
389C23×C3—61no—
390C23×C3—61no—
391C23×C3—181no—
392C22×D3—21no—
393C22×D3—21no—
394C22×D3—62no—
395C22×D3—61no—
396C22×D3—61no—
397C3×D4—32no—
398C3×D4—32no—
399C3×D4—62no—
400C3×D4—62no—
401C3×D4—61no—
402C3×D4—61no—
403C3×D4—61no—
404C3×D4—61no—
405C3×D4—61no—
406C3×D4—61no—
407C3×D4—121no—
408C3×D4—121no—
409C3×D4—181no—
410C3×D4—181no—
411C2×C3×C4—21no—
412C2×C3×C4—21no—
413C2×C3×C4—21no—
414C2×C3×C4—41no—
415C2×C3×C4—181no—
416C2×C3×C4C2×D3211yescharacteristic
417C2×C3×C4C2×D3211yescharacteristic
418C2×C3×C4C2×D3211yescharacteristic
419#67—362no—
420#67—362no—
421#67—362no—
422#67—362no—
423#67—364no—
424#67—364no—
425#67—362no—
426#67—362no—
427C2×Dic3—62no—
428C2×Dic3—61no—
429C2×Dic3—61no—
430C2×Dic3—91no—
431C2×Dic3—91no—
432C2×Dic3—91no—
433C2×Dic3—91no—
434C2×Dic3—91no—
435C2×Dic3—91no—
436C2×Dic3—181no—
437C2×Dic3—181no—
438C2×Dic3—181no—
439C2×Dic3—181no—
440C2×Dic3—181no—
441C2×Dic3—181no—
442C2×Dic3—361no—
443C2×Dic3—361no—
444C2×Dic3—362no—
445C2×Dic3—362no—
446C2×Dic3—364no—
447D12—21no—
448D12—21no—
449D12—21no—
450D12—61no—
451D12—61no—
452D12—61no—
453D12—61no—
454D12—121no—
455D12—121no—
456D12#62512yescomplemented
457C4×D3—362no—
458C3×C8—61no—
459C3×C8—61no—
460C3×C8—121no—
461C3×C8—181no—
462C3⋊C8—21no—
463C3⋊C8—61no—
464C3⋊C8—61no—
465C3⋊C8—121no—
466C2×C32—21no—
467C2×C32—21no—
468C2×C32—21no—
469C2×C32—21no—
470C2×C32—21no—
471C2×C32—21no—
472C2×C32—21no—
473C2×C32—21no—
474C2×C32—21no—
475C2×C32—41no—
476C2×C32—41no—
477C2×C32—41no—
478C2×C32—122no—
479C2×C32—241no—
480C2×C32#88411yescharacteristic
481C2×C32D3×D811yescharacteristic
482C2×C32D3×D811yescharacteristic
483C2×C32D3×D811yescharacteristic
484C2×C32#91111yescharacteristic
485C2×C32#91111yescharacteristic
486C2×C32#91411yescharacteristic
487C2×C32#93911yescharacteristic
488C2×C32#93911yescharacteristic
489C3⋊D3—81no—
490C3×D3—44no—
491C3×D3—42no—
492C3×D3—42no—
493C3×D3—82no—
494C3×D3—81no—
495C3×D3—81no—
496C3×D3—242no—
497C3×D3—241no—
498C3×D3—241no—
499C2×D4—31no—
500C2×D4—31no—
501C2×D4—91no—
502C2×D4—541no—
503C2×D4—541no—
504C22×C4—541no—
505D8—182no—
506D8—182no—
507C2×C8—31no—
508C2×C8—91no—
509C4⋊C4—91no—
510C22⋊C4—541no—
511C42—271no—
512C22×C3—21no—
513C22×C3—21no—
514C22×C3—21no—
515C22×C3—41no—
516C22×C3—64no—
517C22×C3—62no—
518C22×C3—62no—
519C22×C3—122no—
520C22×C3—122no—
521C22×C3—122no—
522C22×C3—122no—
523C22×C3—122no—
524C22×C3—121no—
525C22×C3—121no—
526C22×C3—121no—
527C22×C3—121no—
528C22×C3—121no—
529C22×C3—121no—
530C22×C3—121no—
531C22×C3—121no—
532C22×C3—121no—
533C22×C3—121no—
534C22×C3—361no—
535C22×C3—361no—
536C22×C3—361no—
537C22×C3#380211yescharacteristic
538C22×C3#381111yescharacteristic
539C22×C3#381111yescharacteristic
540C2×D3—22no—
541C2×D3—41no—
542C2×D3—41no—
543C2×D3—41no—
544C2×D3—41no—
545C2×D3—41no—
546C2×D3—122no—
547C2×D3—122no—
548C2×D3—122no—
549C2×D3—121no—
550C2×D3—121no—
551C2×D3—121no—
552C2×D3—121no—
553C2×D3—121no—
554C2×D3—121no—
555C3×C4—21no—
556C3×C4—21no—
557C3×C4—21no—
558C3×C4—21no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—41no—
562C3×C4—41no—
563C3×C4—361no—
564C3×C4#379511yescharacteristic
565C3×C4#380411yescharacteristic
566C3×C4#380511yescharacteristic
567C3×C4C2×#62511yescharacteristic
568C3×C4D3×#6711yescharacteristic
569C3×C4D3×#6711yescharacteristic
570Dic3—92no—
571Dic3—92no—
572Dic3—92no—
573Dic3—122no—
574Dic3—121no—
575Dic3—121no—
576Dic3—182no—
577Dic3—182no—
578Dic3—182no—
579Dic3—182no—
580Dic3—181no—
581Dic3—181no—
582Dic3—181no—
583Dic3—362no—
584Dic3—361no—
585Dic3—361no—
586Dic3—362no—
587C32—21no—
588C32—21no—
589C32—21no—
590C32—41no—
591C32#478611yescharacteristic, complemented
592C32#515011yescharacteristic, complemented
593C32#515011yescharacteristic, complemented
594C23—61no—
595C23—61no—
596C23—181no—
597D4—32no—
598D4—61no—
599D4—61no—
600D4—61no—
601D4—181no—
602D4—181no—
603D4—1082no—
604D4—1082no—
605C2×C4—181no—
606C2×C4—271no—
607C2×C4—271no—
608C2×C4—1082no—
609C2×C4D3311yescharacteristic
610C8—61no—
611C8—181no—
612C2×C3—21no—
613C2×C3—21no—
614C2×C3—21no—
615C2×C3—21no—
616C2×C3—21no—
617C2×C3—21no—
618C2×C3—21no—
619C2×C3—21no—
620C2×C3—21no—
621C2×C3—41no—
622C2×C3—41no—
623C2×C3—41no—
624C2×C3—124no—
625C2×C3—122no—
626C2×C3—122no—
627C2×C3—242no—
628C2×C3—241no—
629C2×C3—241no—
630C2×C3—721no—
631C2×C3#6379711yescharacteristic
632C2×C3#6379811yescharacteristic
633C2×C3#6388511yescharacteristic
634C2×C3#6392611yescharacteristic
635C2×C3#6392611yescharacteristic
636C2×C3#6392711yescharacteristic
637C2×C3#6393711yescharacteristic
638C2×C3#6397111yescharacteristic
639C2×C3#6398011yescharacteristic
640D3—42no—
641D3—81no—
642D3—242no—
643D3—241no—
644D3—241no—
645C22—62no—
646C22—121no—
647C22—121no—
648C22—121no—
649C22—121no—
650C22—121no—
651C22—361no—
652C22—361no—
653C22—361no—
654C22#8895411yescharacteristic
655C4—272no—
656C4—361no—
657C4—541no—
658C4D3×#62511yescharacteristic
659C4#8896111yescharacteristic
660C3—21no—
661C3—21no—
662C3—21no—
663C3—41no—
664C3#1059074911yescharacteristic, complemented
665C3#1059143311yescharacteristic, complemented
666C3#1059148611yescharacteristic, complemented
667C2—122no—
668C2—241no—
669C2—721no—
670C2#1172283211yescharacteristic
671C2#1172287711yescharacteristic
672C2#1172295411yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
hypocenterC33#459
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. #11723529C2semidirect
2. #11723499C2semidirect
3. C2×C3×#3714C2semidirect
4. C2×C3×#3713C2semidirect
5. #11722075C2semidirect
6. #11722050C2semidirect
7. C3×#63563C2semidirect
36. C2×#3714D3semidirect
37. C2×#3713D3semidirect
46. #63563D3semidirect
60. C32×D12D4semidirect
135. C3⋊(C3⋊Dic3)D8semidirect
223. C3×D12#67semidirect
224. C3×D12#67semidirect
384. C33#459semidirect
456. D12#625semidirect
591. C32#4786semidirect
592. C32#5150semidirect
593. C32#5150semidirect
664. C3#10590749semidirect
665. C3#10591433semidirect
666. C3#10591486semidirect

Maximal subgroups

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

subgroupindex
#117235292
#117234992
C2×C3×#37142
C2×C3×#37132
#117220752
#117220502
C3×#635632
#105914863
#105914333
#105907493