Finite Group Databasebeta

← #10770243

Subgroups — #10770243

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3536C211yesmaximal, characteristic, complemented
2#82929C211yesmaximal, characteristic, complemented
3#82824C211yesmaximal, characteristic, complemented
4#82627C211yesmaximal, characteristic, complemented
5#71888C212yesmaximal, complemented
6#71887C212yesmaximal, complemented
7#71878C212yesmaximal, complemented
8#71875C212yesmaximal, complemented
9#71708C211yesmaximal, characteristic, complemented
10#71691C211yesmaximal, characteristic, complemented
11C2×#4537C211yesmaximal, characteristic, complemented
12#18424—31nomaximal, Sylow
13C3×#562—22no—
14C3×#554—21no—
15C3×#548—21no—
16C3×#547—21no—
17C3×#546—21no—
18C3×#546—22no—
19C3×#543—21no—
20C3×#542—21no—
21C3×#539C2212yes—
22C3×#538—21no—
23C3×#537—21no—
24C3×#537C2212yes—
25C3×#525C2211yescharacteristic
26C2×C3×C22⋊Q8C2211yescharacteristic
27C2×C3×#121C2211yescharacteristic
28C2×#930C2211yescharacteristic
29#5238—21no—
30#5231—21no—
31#5217—21no—
32C2×C3⋊#95C2211yescharacteristic
33#5207C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5048—21no—
36#5046C2211yescharacteristic, complemented
37#5044C2211yescharacteristic, complemented
38#5039—22no—
39#5038—21no—
40#5036C2212yes—
41#5036C2211yescharacteristic, complemented
42#5034C2212yes—
43#5034C2211yescharacteristic, complemented
44#5003—21no—
45#4992—21no—
46#4991C2212yescomplemented
47#4986C2212yescomplemented
48#4968—21no—
49#4958—22no—
50#4538C2212yescomplemented
51#4537C2212yes—
52#4537C2212yes—
53#4537C2212yescomplemented
54#4528C2212yescomplemented
55#4528C2212yescomplemented
56#3536D311yescharacteristic, complemented
57#3192—31no—
58#3188—31no—
59#2639—31no—
60#1616—32no—
61#1615—32no—
62#1598—32no—
63#1597—32no—
64#1504—31no—
65C2×#327—31no—
66#1472—31no—
67C2×C3×#41—21no—
68C2×C3×#41—22no—
69C2×C3×#41D411yescharacteristic
70C22×C3×Q8D411yescharacteristic
71C22×C3×D4—21no—
72C22×C3×D4D411yescharacteristic
73C23×C3×C4C2311yescharacteristic
74C22×#67—21no—
75C23×Dic3—21no—
76C22×D12C2311yescharacteristic
77C3×C4⋊Q8—21no—
78C3×#126—22no—
79C3×#125—22no—
80C3×#124—22no—
81C3×#124—22no—
82C3×#124—21no—
83C3×#123—22no—
84C3×#123—22no—
85C3×#123—22no—
86C3×#123—21no—
87C3×#123—22no—
88C3×C22⋊Q8—22no—
89C3×C22⋊Q8—22no—
90C3×C22⋊Q8—21no—
91C3×C22⋊Q8—22no—
92C3×C22⋊Q8—22no—
93C3×C22⋊Q8—21no—
94C3×C22⋊Q8—21no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8D411yescharacteristic
100C3×C22⋊Q8C2311yescharacteristic
101C3×#121—22no—
102C3×#121—22no—
103C3×#121—21no—
104C3×#121—22no—
105C3×#121—22no—
106C3×#121—21no—
107C3×#121—21no—
108C3×#121—22no—
109C3×#121D412yes—
110C3×#121D411yescharacteristic
111C3×#121C2312yes—
112C3×#121C2311yescharacteristic
113C3×#120—22no—
114C3×#120—21no—
115C3×C4×Q8—21no—
116C3×C4×D4—22no—
117C3×C4×D4—21no—
118C3×C4×D4—21no—
119C3×C4×D4—21no—
120C3×C4×D4—22no—
121C3×#117D411yescharacteristic
122C2×C3×C4⋊C4—21no—
123C2×C3×C4⋊C4D411yescharacteristic
124C2×C3×C4⋊C4D411yescharacteristic
125C2×C3×C4⋊C4C2312yes—
126C2×C3×C22⋊C4—22no—
127C2×C3×C22⋊C4—21no—
128C2×C3×C22⋊C4D412yes—
129C2×#215—21no—
130C2×#213—22no—
131C2×#213—21no—
132#930—21no—
133#930—21no—
134#930—21no—
135#930—21no—
136C2×(C2×D3)⋊C4—21no—
137C2×(C2×D3)⋊C4—22no—
138C2×(C2×D3)⋊C4—22no—
139C2×C4⋊Dic3C2311yescharacteristic
140C2×#208—21no—
141C22×C3⋊C8—21no—
142#835—21no—
143#832—22no—
144#832—21no—
145C3⋊#95—21no—
146C3⋊#95—22no—
147C3⋊#95—22no—
148C3⋊#95C2312yes—
149#830—21no—
150#830—21no—
151#830C2311yescharacteristic
152#830C2311yescharacteristic
153#830C2312yes—
154#809—22no—
155#809—21no—
156#809—21no—
157#808—22no—
158#808—21no—
159#807—21no—
160#562—22no—
161#554—21no—
162#548—21no—
163#547—21no—
164#546—21no—
165#546—22no—
166#543—21no—
167#542—21no—
168#539C2×D312yes—
169#538—21no—
170#537—21no—
171#537C2×D312yes—
172#525C2×D311yescharacteristic
173C2×C22⋊Q8C2×D311yescharacteristic
174C2×#121—31no—
175C2×#121C2×D311yescharacteristic
176#484—31no—
177#482—61no—
178#481—32no—
179#481—31no—
180#481—61no—
181#469—62no—
182#466—61no—
183#465—61no—
184#450—31no—
185#449—61no—
186#447—32no—
187#447—31no—
188#447—61no—
189#407—31no—
190C2×C22⋊C8—31no—
191#400—32no—
192#397—61no—
193#394—32no—
194#393—61no—
195#386—62no—
196C2×#95—31no—
197#328—32no—
198#327—32no—
199#327—32no—
200#327—32no—
201#324—32no—
202#324—32no—
203C24×C3—21no—
204C24×C3C2×D411yescharacteristic
205C23×D3—21no—
206C3×#41—82no—
207C3×#41—82no—
208C3×#41—81no—
209C3×#41—82no—
210C2×C3×Q8—22no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—81no—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—22no—
218C2×C3×D4—22no—
219C2×C3×D4—22no—
220C2×C3×D4—21no—
221C2×C3×D4—22no—
222C2×C3×D4—21no—
223C2×C3×D4—81no—
224C2×C3×D4—81no—
225C2×C3×D4—81no—
226C2×C3×D4C2×D411yescharacteristic
227C2×C3×D4C2×D412yes—
228C22×C3×C4—22no—
229C22×C3×C4—22no—
230C22×C3×C4—22no—
231C22×C3×C4—21no—
232C22×C3×C4—21no—
233C22×C3×C4—81no—
234C22×C3×C4C2×D412yes—
235C22×C3×C4C2×D412yes—
236C22×C3×C4C2×D411yescharacteristic
237C22×C3×C4C2×D411yescharacteristic
238C22×C3×C4C2×D411yescharacteristic
239C22×C3×C4C2×D411yescharacteristic
240C22×C3×C4C2×D411yescharacteristic
241C22×C3×C4C2411yescharacteristic
242C2×#67—44no—
243C2×#67—41no—
244C2×#67—41no—
245C22×Dic3—21no—
246C22×Dic3—22no—
247C22×Dic3—42no—
248C22×Dic3—81no—
249C22×Dic3—81no—
250C2×D12—21no—
251C2×D12—21no—
252C2×D12—81no—
253C3×C4⋊C4—22no—
254C3×C4⋊C4—22no—
255C3×C4⋊C4—22no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—22no—
262C3×C4⋊C4—22no—
263C3×C4⋊C4—21no—
264C3×C4⋊C4C2×D411yescharacteristic
265C3×C4⋊C4C2×D412yes—
266C3×C4⋊C4C2×D411yescharacteristic
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—22no—
270C3×C22⋊C4—22no—
271C3×C22⋊C4—22no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—22no—
274C3×C22⋊C4—22no—
275C3×C22⋊C4—21no—
276C3×C22⋊C4—21no—
277C3×C22⋊C4—22no—
278C3×C22⋊C4—21no—
279C3×C22⋊C4—22no—
280C3×C22⋊C4—21no—
281C3×C42—22no—
282C3×C42—21no—
283#215—81no—
284#213—81no—
285#213—82no—
286(C2×D3)⋊C4—41no—
287(C2×D3)⋊C4—41no—
288(C2×D3)⋊C4—82no—
289(C2×D3)⋊C4—82no—
290C4⋊Dic3—21no—
291C4⋊Dic3—21no—
292#208—81no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—22no—
296C2×C3⋊C8—81no—
297C2×#41—21no—
298C2×#41—22no—
299C2×#41#6711yescharacteristic
300C22×Q8#6711yescharacteristic
301C22×D4—21no—
302C22×D4—31no—
303C22×D4—61no—
304C22×D4#6711yescharacteristic
305C23×C4—61no—
306C23×C4C22×D311yescharacteristic
307C2×SD16—61no—
308C2×D8—62no—
309C2×D8—61no—
310C2×M16—61no—
311C22×C8—61no—
312C4⋊Q8—21no—
313#126—22no—
314#125—22no—
315#124—22no—
316#124—21no—
317#124—22no—
318#123—22no—
319#123—22no—
320#123—22no—
321#123—21no—
322#123—22no—
323C22⋊Q8—21no—
324C22⋊Q8—22no—
325C22⋊Q8—22no—
326C22⋊Q8—22no—
327C22⋊Q8—22no—
328C22⋊Q8—21no—
329C22⋊Q8—21no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8#6711yescharacteristic
335C22⋊Q8C22×D311yescharacteristic
336#121—21no—
337#121—22no—
338#121—22no—
339#121—22no—
340#121—22no—
341#121—21no—
342#121—21no—
343#121—22no—
344#121—61no—
345#121—61no—
346#121—61no—
347#121—61no—
348#121#6712yes—
349#121#6711yescharacteristic
350#121C22×D312yes—
351#121C22×D311yescharacteristic
352#120—22no—
353#120—21no—
354C4×Q8—21no—
355C4×D4—22no—
356C4×D4—21no—
357C4×D4—21no—
358C4×D4—21no—
359C4×D4—22no—
360#117#6711yescharacteristic
361C2×C4⋊C4—21no—
362C2×C4⋊C4—31no—
363C2×C4⋊C4#6711yescharacteristic
364C2×C4⋊C4#6711yescharacteristic
365C2×C4⋊C4C22×D312yes—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—61no—
369C2×C22⋊C4—62no—
370C2×C22⋊C4—62no—
371C2×C22⋊C4#6712yes—
372#107—61no—
373#106—61no—
374#106—62no—
375#103—61no—
376#102—62no—
377#102—61no—
378#102—61no—
379#102—62no—
380#102—61no—
381C22⋊C8—31no—
382C22⋊C8—31no—
383C22⋊C8—32no—
384C22⋊C8—61no—
385C22⋊C8—61no—
386#95—32no—
387#95—61no—
388#95—62no—
389#95—62no—
390C23×C3—21no—
391C23×C3—21no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—21no—
395C23×C3—41no—
396C23×C3—42no—
397C23×C3—81no—
398C23×C3—81no—
399C23×C3#12011yescharacteristic
400C23×C3#12011yescharacteristic
401C23×C3C22×D411yescharacteristic
402C22×D3—42no—
403C22×D3—42no—
404C22×D3—41no—
405C22×D3—41no—
406C22×D3—41no—
407C3×Q8—41no—
408C3×Q8—42no—
409C3×Q8—81no—
410C3×Q8—82no—
411C3×D4—42no—
412C3×D4—41no—
413C3×D4—42no—
414C3×D4—42no—
415C3×D4—41no—
416C3×D4—42no—
417C3×D4—41no—
418C3×D4—42no—
419C3×D4—82no—
420C3×D4—81no—
421C3×D4—82no—
422C3×D4—82no—
423C2×C3×C4—21no—
424C2×C3×C4—22no—
425C2×C3×C4—21no—
426C2×C3×C4—22no—
427C2×C3×C4—21no—
428C2×C3×C4—22no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—41no—
432C2×C3×C4—41no—
433C2×C3×C4—41no—
434C2×C3×C4—41no—
435C2×C3×C4—41no—
436C2×C3×C4—41no—
437C2×C3×C4—42no—
438C2×C3×C4—42no—
439C2×C3×C4—81no—
440C2×C3×C4—82no—
441C2×C3×C4—82no—
442C2×C3×C4—82no—
443C2×C3×C4#12011yescharacteristic
444C2×C3×C4#12011yescharacteristic
445C2×C3×C4C22×D411yescharacteristic
446C2×C3×C4C22×D411yescharacteristic
447#67—82no—
448#67—82no—
449C2×Dic3—42no—
450C2×Dic3—42no—
451C2×Dic3—42no—
452C2×Dic3—41no—
453C2×Dic3—42no—
454C2×Dic3—84no—
455D12—42no—
456D12—41no—
457D12—41no—
458C3⋊C8—41no—
459C3⋊C8—41no—
460C3⋊C8—42no—
461C24—21no—
462C24—61no—
463C24C2×#6711yescharacteristic
464#41—82no—
465#41—82no—
466#41—81no—
467#41—82no—
468C2×Q8—22no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—81no—
472C2×Q8C2×#6711yescharacteristic
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—22no—
476C2×D4—22no—
477C2×D4—22no—
478C2×D4—21no—
479C2×D4—22no—
480C2×D4—21no—
481C2×D4—61no—
482C2×D4—61no—
483C2×D4—81no—
484C2×D4—81no—
485C2×D4—81no—
486C2×D4—124no—
487C2×D4—121no—
488C2×D4—121no—
489C2×D4—241no—
490C2×D4C2×#6711yescharacteristic
491C2×D4C2×#6712yes—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—62no—
498C22×C4—61no—
499C22×C4—81no—
500C22×C4—122no—
501C22×C4—241no—
502C22×C4—241no—
503C22×C4C2×#6712yes—
504C22×C4C2×#6712yes—
505C22×C4C2×#6711yescharacteristic
506C22×C4C2×#6711yescharacteristic
507C22×C4C2×#6711yescharacteristic
508C22×C4C2×#6711yescharacteristic
509C22×C4C2×#6711yescharacteristic
510C22×C4C23×D311yescharacteristic
511SD16—241no—
512D8—241no—
513D8—242no—
514M16—241no—
515C2×C8—61no—
516C2×C8—62no—
517C2×C8—61no—
518C2×C8—241no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—61no—
531C4⋊C4—61no—
532C4⋊C4C2×#6712yes—
533C4⋊C4C2×#6711yescharacteristic
534C4⋊C4C2×#6711yescharacteristic
535C22⋊C4—22no—
536C22⋊C4—22no—
537C22⋊C4—22no—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—21no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—21no—
544C22⋊C4—21no—
545C22⋊C4—22no—
546C22⋊C4—21no—
547C22⋊C4—22no—
548C22⋊C4—21no—
549C22⋊C4—121no—
550C22⋊C4—121no—
551C22⋊C4—242no—
552C22⋊C4—242no—
553C42—22no—
554C42—21no—
555C22×C3—21no—
556C22×C3—21no—
557C22×C3—21no—
558C22×C3—21no—
559C22×C3—21no—
560C22×C3—21no—
561C22×C3—21no—
562C22×C3—42no—
563C22×C3—42no—
564C22×C3—41no—
565C22×C3—42no—
566C22×C3—41no—
567C22×C3—42no—
568C22×C3—41no—
569C22×C3—42no—
570C22×C3—162no—
571C22×C3#45412yes—
572C22×C3C2×#12011yescharacteristic
573C22×C3#57611yescharacteristic
574C22×C3#57711yescharacteristic
575C2×D3—82no—
576C2×D3—82no—
577C2×D3—81no—
578C2×D3—81no—
579C2×D3—81no—
580C3×C4—21no—
581C3×C4—21no—
582C3×C4—21no—
583C3×C4—21no—
584C3×C4—42no—
585C3×C4—42no—
586C3×C4—41no—
587C3×C4—41no—
588C3×C4—41no—
589C3×C4—42no—
590Dic3—82no—
591Dic3—81no—
592C23—22no—
593C23—21no—
594C23—21no—
595C23—22no—
596C23—21no—
597C23—41no—
598C23—42no—
599C23—81no—
600C23—81no—
601C23—122no—
602C23—122no—
603C23—121no—
604C23—121no—
605C23—121no—
606C23#95311yescharacteristic
607C23#95311yescharacteristic
608C23C22×#6711yescharacteristic
609Q8—41no—
610Q8—42no—
611Q8—81no—
612Q8—82no—
613D4—42no—
614D4—41no—
615D4—42no—
616D4—42no—
617D4—42no—
618D4—41no—
619D4—42no—
620D4—41no—
621D4—82no—
622D4—81no—
623D4—82no—
624D4—82no—
625D4—122no—
626D4—121no—
627D4—121no—
628D4—242no—
629D4—242no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—21no—
633C2×C4—22no—
634C2×C4—21no—
635C2×C4—22no—
636C2×C4—21no—
637C2×C4—22no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—81no—
647C2×C4—82no—
648C2×C4—82no—
649C2×C4—82no—
650C2×C4—122no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—122no—
654C2×C4—121no—
655C2×C4—244no—
656C2×C4#95311yescharacteristic
657C2×C4#95311yescharacteristic
658C2×C4C22×#6711yescharacteristic
659C2×C4C22×#6711yescharacteristic
660C8—121no—
661C8—121no—
662C8—122no—
663C2×C3—21no—
664C2×C3—21no—
665C2×C3—22no—
666C2×C3—41no—
667C2×C3—81no—
668C2×C3—82no—
669C2×C3#301211yescharacteristic
670C2×C3C2×#45411yescharacteristic
671C2×C3#302611yescharacteristic
672D3—161no—
673C22—21no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—21no—
678C22—21no—
679C22—21no—
680C22—42no—
681C22—42no—
682C22—41no—
683C22—42no—
684C22—41no—
685C22—42no—
686C22—41no—
687C22—42no—
688C22—162no—
689C22—242no—
690C22—242no—
691C22—241no—
692C22—241no—
693C22—241no—
694C22#524512yes—
695C22#583511yescharacteristic
696C22#583611yescharacteristic
697C22C2×#95311yescharacteristic
698C4—21no—
699C4—21no—
700C4—21no—
701C4—21no—
702C4—42no—
703C4—42no—
704C4—41no—
705C4—41no—
706C4—42no—
707C4—41no—
708C4—242no—
709C4—241no—
710C3#1842411yescharacteristic, complemented, Sylow
711C2—21no—
712C2—22no—
713C2—21no—
714C2—41no—
715C2—82no—
716C2—81no—
717C2—481no—
718C2#8410211yescharacteristic
719C2#8411011yescharacteristic
720C2C2×#524511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3536C2
socleC22×C3C2×#120
hypercenter#3536D3
hypocenterC3#18424
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3536

Complements

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

NH ≅ G/Ntype
1. C3×#3536C2semidirect
2. #82929C2semidirect
3. #82824C2semidirect
4. #82627C2semidirect
5. #71888C2semidirect
6. #71887C2semidirect
7. #71878C2semidirect
8. #71875C2semidirect
9. #71708C2semidirect
10. #71691C2semidirect
11. C2×#4537C2semidirect
36. #5046C22semidirect
37. #5044C22semidirect
41. #5036C22semidirect
43. #5034C22semidirect
46. #4991C22semidirect
47. #4986C22semidirect
50. #4538C22semidirect
53. #4537C22semidirect
54. #4528C22semidirect
55. #4528C22semidirect
56. #3536D3semidirect
710. C3#18424semidirect

Maximal subgroups

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

subgroupindex
C3×#35362
#829292
#828242
#826272
#718882
#718872
#718782
#718752
#717082
#716912
C2×#45372
#184243