Finite Group Databasebeta

← #10770221

Subgroups — #10770221

Order 768.

960 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3538C211yesmaximal, characteristic, complemented
2#82929C211yesmaximal, characteristic
3#82824C211yesmaximal, characteristic, complemented
4#82738C211yesmaximal, characteristic, complemented
5#71902C212yesmaximal, complemented
6#71899C212yesmaximal, complemented
7#71888C212yesmaximal, complemented
8#71887C212yesmaximal, complemented
9#71708C211yesmaximal, characteristic, complemented
10#71691C211yesmaximal, characteristic, complemented
11C2×#4538C211yesmaximal, characteristic, complemented
12#18422—31nomaximal, Sylow
13C3×#563—22no—
14C3×#552—21no—
15C3×#552—21no—
16C3×#551—22no—
17C3×D4×Q8—21no—
18C3×D4×Q8—21no—
19C3×#544—21no—
20C3×#544C2212yes—
21C3×#541—21no—
22C3×#539—21no—
23C3×#539C2212yes—
24C3×#536—21no—
25C3×#525C2211yescharacteristic
26C2×C3×C22⋊Q8C2211yescharacteristic
27C2×C3×C22⋊Q8C2211yescharacteristic
28C2×#930C2211yescharacteristic
29#5238—21no—
30#5231—21no—
31#5227—21no—
32C2×C3⋊#95C2211yescharacteristic
33#5207C2211yescharacteristic
34C2×#830C2211yescharacteristic
35#5049—22no—
36#5048—21no—
37#5046C2212yes—
38#5046C2211yescharacteristic
39#5044C2212yes—
40#5044C2211yescharacteristic
41#5038—21no—
42#5036C2211yescharacteristic
43#5034C2211yescharacteristic
44#5003—21no—
45#4992—21no—
46#4991C2212yes—
47#4986C2212yes—
48#4967—21no—
49#4958—22no—
50#4538C2212yes—
51#4538C2212yes—
52#4538C2212yes—
53#4537C2212yes—
54#4528C2212yes—
55#4528C2212yes—
56#3538D311yescharacteristic, complemented
57#3193—31no—
58#3188—31no—
59#2639—31no—
60#1617—32no—
61#1615—32no—
62#1599—32no—
63#1597—32no—
64#1504—31no—
65C2×#328—31no—
66#1472—31no—
67C2×C3×#41D411yescharacteristic
68C22×C3×Q8—21no—
69C22×C3×Q8D411yescharacteristic
70C22×C3×Q8D411yescharacteristic
71C23×C3×C4C2311yescharacteristic
72C22×#67—21no—
73C23×Dic3—21no—
74C22×D12C2311yescharacteristic
75C3×C4⋊Q8—22no—
76C3×C4⋊Q8—21no—
77C3×C4⋊Q8—22no—
78C3×C4⋊Q8—22no—
79C3×#126—22no—
80C3×#126—22no—
81C3×#126—22no—
82C3×#126—22no—
83C3×#125—22no—
84C3×#125—21no—
85C3×#125—22no—
86C3×#125—22no—
87C3×#125—22no—
88C3×#124—22no—
89C3×#123—22no—
90C3×#123—22no—
91C3×#123—22no—
92C3×#123—21no—
93C3×#123—22no—
94C3×C22⋊Q8—22no—
95C3×C22⋊Q8—21no—
96C3×C22⋊Q8—21no—
97C3×C22⋊Q8—22no—
98C3×C22⋊Q8—22no—
99C3×C22⋊Q8—21no—
100C3×C22⋊Q8—21no—
101C3×C22⋊Q8—21no—
102C3×C22⋊Q8—21no—
103C3×C22⋊Q8—21no—
104C3×C22⋊Q8—21no—
105C3×C22⋊Q8—22no—
106C3×C22⋊Q8D412yes—
107C3×C22⋊Q8D411yescharacteristic
108C3×C22⋊Q8C2312yes—
109C3×C22⋊Q8C2311yescharacteristic
110C3×#121D411yescharacteristic
111C3×#121C2311yescharacteristic
112C3×#120—21no—
113C3×C4×Q8—21no—
114C3×C4×Q8—21no—
115C3×C4×D4—22no—
116C3×C4×D4—21no—
117C3×C4×D4—21no—
118C3×C4×D4—22no—
119C3×#117—21no—
120C3×#117—22no—
121C3×#117—22no—
122C3×#117D411yescharacteristic
123C2×C3×C4⋊C4—21no—
124C2×C3×C4⋊C4D411yescharacteristic
125C2×C3×C4⋊C4D411yescharacteristic
126C2×C3×C4⋊C4C2312yes—
127C2×C3×C22⋊C4—21no—
128C2×C3×C22⋊C4D412yes—
129C2×#215—22no—
130C2×#215—21no—
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—22no—
143#835—21no—
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—21no—
158#807—22no—
159#807—21no—
160#563—22no—
161#552—21no—
162#552—21no—
163#551—22no—
164D4×Q8—21no—
165D4×Q8—21no—
166#544—21no—
167#544C2×D312yes—
168#541—21no—
169#539—21no—
170#539C2×D312yes—
171#536—21no—
172#525C2×D311yescharacteristic
173C2×C22⋊Q8C2×D311yescharacteristic
174C2×C22⋊Q8C2×D311yescharacteristic
175C2×#121—31no—
176#484—32no—
177#484—31no—
178#482—61no—
179#481—31no—
180#480—61no—
181#468—62no—
182#466—61no—
183#465—61no—
184#450—32no—
185#450—31no—
186#449—61no—
187#448—61no—
188#447—31no—
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#328—32no—
199#328—32no—
200#327—32no—
201#324—32no—
202#324—32no—
203C24×C3C2×D411yescharacteristic
204C23×D3—21no—
205C3×#41—81no—
206C2×C3×Q8—22no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—81no—
211C2×C3×Q8—82no—
212C2×C3×Q8—81no—
213C2×C3×Q8C2×D412yes—
214C2×C3×Q8C2×D411yescharacteristic
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4C2×D411yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4—22no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—81no—
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×D412yes—
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2×D411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C2×#67—44no—
233C2×#67—41no—
234C2×#67—41no—
235C22×Dic3—21no—
236C22×Dic3—22no—
237C22×Dic3—42no—
238C22×Dic3—81no—
239C22×Dic3—81no—
240C2×D12—21no—
241C2×D12—21no—
242C2×D12—81no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—22no—
246C3×C4⋊C4—22no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—22no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—22no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—22no—
258C3×C4⋊C4—22no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4C2×D412yes—
261C3×C4⋊C4C2×D411yescharacteristic
262C3×C4⋊C4C2×D411yescharacteristic
263C3×C22⋊C4—22no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—22no—
266C3×C22⋊C4—22no—
267C3×C22⋊C4—22no—
268C3×C22⋊C4—22no—
269C3×C22⋊C4—21no—
270C3×C22⋊C4—21no—
271C3×C22⋊C4—22no—
272C3×C22⋊C4—21no—
273C3×C22⋊C4—21no—
274C3×C22⋊C4—21no—
275C3×C42—22no—
276C3×C42—22no—
277C3×C42—22no—
278C3×C42—21no—
279#215—81no—
280#215—82no—
281#213—81no—
282(C2×D3)⋊C4—41no—
283(C2×D3)⋊C4—41no—
284(C2×D3)⋊C4—82no—
285(C2×D3)⋊C4—82no—
286C4⋊Dic3—21no—
287C4⋊Dic3—21no—
288#208—81no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—22no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—81no—
293C2×#41#6711yescharacteristic
294C22×Q8—21no—
295C22×Q8#6711yescharacteristic
296C22×Q8#6711yescharacteristic
297C22×D4—31no—
298C22×D4—61no—
299C23×C4—61no—
300C23×C4C22×D311yescharacteristic
301C2×SD16—62no—
302C2×SD16—61no—
303C2×D8—61no—
304C2×M16—61no—
305C22×C8—61no—
306C4⋊Q8—21no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310#126—22no—
311#126—22no—
312#126—22no—
313#126—22no—
314#125—21no—
315#125—22no—
316#125—22no—
317#125—22no—
318#125—22no—
319#124—22no—
320#123—22no—
321#123—22no—
322#123—22no—
323#123—21no—
324#123—22no—
325C22⋊Q8—21no—
326C22⋊Q8—21no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330C22⋊Q8—21no—
331C22⋊Q8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—22no—
337C22⋊Q8#6712yes—
338C22⋊Q8#6711yescharacteristic
339C22⋊Q8C22×D312yes—
340C22⋊Q8C22×D311yescharacteristic
341#121—61no—
342#121—61no—
343#121—61no—
344#121—61no—
345#121#6711yescharacteristic
346#121C22×D311yescharacteristic
347#120—21no—
348C4×Q8—21no—
349C4×Q8—21no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—21no—
353C4×D4—22no—
354#117—21no—
355#117—22no—
356#117—22no—
357#117#6711yescharacteristic
358C2×C4⋊C4—21no—
359C2×C4⋊C4—31no—
360C2×C4⋊C4#6711yescharacteristic
361C2×C4⋊C4#6711yescharacteristic
362C2×C4⋊C4C22×D312yes—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—61no—
365C2×C22⋊C4—62no—
366C2×C22⋊C4—62no—
367C2×C22⋊C4#6712yes—
368#107—61no—
369#107—62no—
370#106—61no—
371#103—62no—
372#103—61no—
373#102—61no—
374#102—61no—
375#102—61no—
376#102—62no—
377C22⋊C8—32no—
378C22⋊C8—31no—
379C22⋊C8—31no—
380C22⋊C8—61no—
381C22⋊C8—61no—
382#95—32no—
383#95—61no—
384#95—62no—
385#95—62no—
386C23×C3—21no—
387C23×C3—21no—
388C23×C3—22no—
389C23×C3—21no—
390C23×C3—41no—
391C23×C3#12011yescharacteristic
392C23×C3#12011yescharacteristic
393C23×C3C22×D411yescharacteristic
394C22×D3—42no—
395C22×D3—42no—
396C22×D3—41no—
397C22×D3—41no—
398C22×D3—41no—
399C3×Q8—42no—
400C3×Q8—41no—
401C3×Q8—42no—
402C3×Q8—42no—
403C3×Q8—81no—
404C3×Q8—82no—
405C3×D4—41no—
406C3×D4—41no—
407C3×D4—41no—
408C3×D4—42no—
409C3×D4—81no—
410C2×C3×C4—21no—
411C2×C3×C4—21no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—21no—
415C2×C3×C4—22no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—22no—
419C2×C3×C4—41no—
420C2×C3×C4—41no—
421C2×C3×C4—41no—
422C2×C3×C4—41no—
423C2×C3×C4—41no—
424C2×C3×C4—41no—
425C2×C3×C4—42no—
426C2×C3×C4—42no—
427C2×C3×C4—42no—
428C2×C3×C4—81no—
429C2×C3×C4—82no—
430C2×C3×C4#12011yescharacteristic
431C2×C3×C4#12011yescharacteristic
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434#67—82no—
435#67—82no—
436C2×Dic3—42no—
437C2×Dic3—42no—
438C2×Dic3—42no—
439C2×Dic3—41no—
440C2×Dic3—42no—
441C2×Dic3—84no—
442D12—41no—
443D12—41no—
444D12—42no—
445C3⋊C8—41no—
446C3⋊C8—41no—
447C3⋊C8—42no—
448C24—61no—
449C24C2×#6711yescharacteristic
450#41—81no—
451C2×Q8—22no—
452C2×Q8—21no—
453C2×Q8—21no—
454C2×Q8—21no—
455C2×Q8—81no—
456C2×Q8—82no—
457C2×Q8—81no—
458C2×Q8C2×#6712yes—
459C2×Q8C2×#6711yescharacteristic
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—61no—
464C2×D4—61no—
465C2×D4—124no—
466C2×D4—121no—
467C2×D4—121no—
468C2×D4—241no—
469C2×D4C2×#6711yescharacteristic
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—62no—
475C22×C4—61no—
476C22×C4—81no—
477C22×C4—122no—
478C22×C4—241no—
479C22×C4—241no—
480C22×C4C2×#6712yes—
481C22×C4C2×#6712yes—
482C22×C4C2×#6711yescharacteristic
483C22×C4C2×#6711yescharacteristic
484C22×C4C2×#6711yescharacteristic
485C22×C4C2×#6711yescharacteristic
486C22×C4C2×#6711yescharacteristic
487C22×C4C23×D311yescharacteristic
488SD16—241no—
489SD16—242no—
490D8—241no—
491M16—241no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—62no—
495C2×C8—241no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4C2×#6711yescharacteristic
516C4⋊C4C2×#6712yes—
517C4⋊C4C2×#6711yescharacteristic
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C22⋊C4—21no—
525C22⋊C4—21no—
526C22⋊C4—22no—
527C22⋊C4—21no—
528C22⋊C4—21no—
529C22⋊C4—21no—
530C22⋊C4—121no—
531C22⋊C4—121no—
532C22⋊C4—242no—
533C22⋊C4—242no—
534C42—22no—
535C42—22no—
536C42—22no—
537C42—21no—
538C22×C3—21no—
539C22×C3—21no—
540C22×C3—21no—
541C22×C3—21no—
542C22×C3—21no—
543C22×C3—21no—
544C22×C3—21no—
545C22×C3—42no—
546C22×C3—42no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3#45412yes—
551C22×C3C2×#12011yescharacteristic
552C22×C3#57711yescharacteristic
553C22×C3#57811yescharacteristic
554C2×D3—82no—
555C2×D3—82no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C3×C4—21no—
560C3×C4—21no—
561C3×C4—21no—
562C3×C4—21no—
563C3×C4—42no—
564C3×C4—41no—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—42no—
569C3×C4—42no—
570Dic3—82no—
571Dic3—81no—
572C23—22no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—41no—
577C23—122no—
578C23—122no—
579C23—121no—
580C23—121no—
581C23—121no—
582C23#95311yescharacteristic
583C23#95311yescharacteristic
584C23C22×#6711yescharacteristic
585Q8—42no—
586Q8—41no—
587Q8—42no—
588Q8—42no—
589Q8—81no—
590Q8—82no—
591D4—41no—
592D4—42no—
593D4—41no—
594D4—41no—
595D4—81no—
596D4—121no—
597D4—121no—
598D4—122no—
599D4—242no—
600D4—242no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—81no—
620C2×C4—82no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—121no—
626C2×C4—244no—
627C2×C4#95311yescharacteristic
628C2×C4#95311yescharacteristic
629C2×C4C22×#6711yescharacteristic
630C2×C4C22×#6711yescharacteristic
631C8—121no—
632C8—121no—
633C8—122no—
634C2×C3—21no—
635C2×C3—21no—
636C2×C3—22no—
637C2×C3—41no—
638C2×C3—81no—
639C2×C3#301311yescharacteristic
640C2×C3C2×#45411yescharacteristic
641C2×C3#302611yescharacteristic
642D3—161no—
643C22—21no—
644C22—21no—
645C22—21no—
646C22—21no—
647C22—21no—
648C22—21no—
649C22—21no—
650C22—42no—
651C22—42no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—242no—
656C22—242no—
657C22—241no—
658C22—241no—
659C22—241no—
660C22#524512yes—
661C22#583711yescharacteristic
662C22#583811yescharacteristic
663C22C2×#95311yescharacteristic
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—41no—
669C4—42no—
670C4—42no—
671C4—42no—
672C4—41no—
673C4—42no—
674C4—41no—
675C4—242no—
676C4—241no—
677C3#1842211yescharacteristic, complemented, Sylow
678C2—21no—
679C2—21no—
680C2—22no—
681C2—41no—
682C2—81no—
683C2—481no—
684C2#8411011yescharacteristic
685C2#8411411yescharacteristic
686C2C2×#524511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3538C2
socleC22×C3C2×#120
prime radical#82929C2
hypercenter#3538D3
hypocenterC3#18422
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3538

Complements

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

NH ≅ G/Ntype
1. C3×#3538C2semidirect
3. #82824C2semidirect
4. #82738C2semidirect
5. #71902C2semidirect
6. #71899C2semidirect
7. #71888C2semidirect
8. #71887C2semidirect
9. #71708C2semidirect
10. #71691C2semidirect
11. C2×#4538C2semidirect
56. #3538D3semidirect
677. C3#18422semidirect

Maximal subgroups

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

subgroupindex
C3×#35382
#829292
#828242
#827382
#719022
#718992
#718882
#718872
#717082
#716912
C2×#45382
#184223