Finite Group Databasebeta

← #10822722

Subgroups — #10822722

Order 768.

1482 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#537C211yesmaximal, characteristic
2#84087C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5232C211yesmaximal, characteristic, complemented
5C2×#5233C211yesmaximal, characteristic, complemented
6C2×#4964C211yesmaximal, characteristic, complemented
7#71695C212yesmaximal, complemented
8#71694C212yesmaximal, complemented
9#71680C212yesmaximal, complemented
10#71679C212yesmaximal, complemented
11C2×#4529C211yesmaximal, characteristic, complemented
12#23485—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×C3×#41C2211yescharacteristic
15C3×#537—24no—
16C3×#537—24no—
17C2×C3×C22⋊Q8—21no—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×#121—21no—
20C2×C3×#121—22no—
21C2×C3×#121C2212yes—
22C22×C3×C4⋊C4C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C22×#208C2211yescharacteristic
26C23×C3⋊C8C2211yescharacteristic
27#5234—24no—
28#5233—24no—
29#5232—24no—
30#5229—21no—
31C2×#835C2211yescharacteristic
32C2×#835C2212yes—
33C2×#835C2211yescharacteristic
34#5221—21no—
35#5220—22no—
36C2×#832C2211yescharacteristic
37C2×#832C2212yes—
38C2×#832C2211yescharacteristic
39#5212C2212yes—
40#5212C2212yes—
41C2×#830C2212yes—
42C2×#830C2212yes—
43#4964—24no—
44C2×#808C2212yes—
45C2×#807C2212yes—
46#4529C2212yes—
47#4529C2212yes—
48C2×#537D311yescharacteristic
49C2×#428—31no—
50C2×#417—31no—
51C2×#416—31no—
52C2×#408—31no—
53#2311—31no—
54#2036—32no—
55#2035—32no—
56#2034—32no—
57#2033—32no—
58C2×#340—31no—
59C2×C3×#41—22no—
60C2×C3×#41—22no—
61C2×C3×#41—22no—
62C2×C3×#41—22no—
63C2×C3×#41—22no—
64C2×C3×#41—22no—
65C2×C3×#41—22no—
66C2×C3×#41—22no—
67C2×C3×#41—22no—
68C2×C3×#41—22no—
69C2×C3×#41—22no—
70C2×C3×#41D414yes—
71C22×C3×Q8D411yescharacteristic
72C22×C3×Q8C2311yescharacteristic
73C22×C3×D4—21no—
74C22×C3×D4—22no—
75C22×C3×D4D412yes—
76C22×C3×D4D411yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×C3×C4—21no—
80C23×C3×C4C2312yes—
81C23×C3×C4C2311yescharacteristic
82C23×Dic3—21no—
83C3×C22⋊Q8—28no—
84C3×C22⋊Q8—24no—
85C3×C22⋊Q8—24no—
86C3×#121—216no—
87C3×#121—28no—
88C3×#121—24no—
89C3×#121—24no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4D414yes—
94C2×C3×C4⋊C4C2312yes—
95C2×C3×C22⋊C4—22no—
96C2×C3×C22⋊C4—22no—
97C2×#217—21no—
98C2×#217—22no—
99C2×#217—22no—
100#926—22no—
101#926—22no—
102C2×C4⋊Dic3—22no—
103C2×C4⋊Dic3—22no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2312yes—
108C2×C4×Dic3—21no—
109C2×#208—22no—
110C2×#208—22no—
111C2×#208D414yescomplemented
112C22×C3⋊C8—22no—
113C22×C3⋊C8—22no—
114C22×C3⋊C8—22no—
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2312yes—
118#835—22no—
119#835—22no—
120#835—24no—
121#832—22no—
122#832—22no—
123#832—24no—
124C3⋊#95—22no—
125C3⋊#95—22no—
126#830—24no—
127#830—24no—
128#808—24no—
129#807—24no—
130C22×#41C2×D311yescharacteristic
131C22×#41C2×D311yescharacteristic
132C22×M16—31no—
133C23×C8—31no—
134#537—24no—
135#537—24no—
136C2×C22⋊Q8—21no—
137C2×C22⋊Q8C2×D312yes—
138C2×#121—21no—
139C2×#121—22no—
140C2×#121C2×D312yes—
141C2×#117—31no—
142C22×C4⋊C4—31no—
143C22×C4⋊C4C2×D311yescharacteristic
144#428—64no—
145C2×#107—32no—
146C2×#106—32no—
147#417—64no—
148#416—64no—
149C2×#103—31no—
150C2×#103—31no—
151C2×#103—32no—
152C2×#102—31no—
153C2×#102—31no—
154C2×#102—32no—
155#408—64no—
156C2×C22⋊C8—32no—
157C2×C22⋊C8—32no—
158#391—61no—
159#390—61no—
160#386—62no—
161#380—32no—
162#380—32no—
163#340—32no—
164#340—32no—
165C24×C3—21no—
166C24×C3C2×D412yes—
167C24×C3C2×D411yescharacteristic
168C3×#41—28no—
169C3×#41—28no—
170C3×#41—44no—
171C3×#41—44no—
172C3×#41—44no—
173C3×#41—44no—
174C3×#41—44no—
175C3×#41—44no—
176C2×C3×Q8—22no—
177C2×C3×Q8—22no—
178C2×C3×Q8—22no—
179C2×C3×Q8—21no—
180C2×C3×Q8—21no—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8—21no—
184C2×C3×Q8C2×D412yes—
185C2×C3×D4—24no—
186C2×C3×D4—24no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—28no—
190C2×C3×D4—22no—
191C2×C3×D4—22no—
192C2×C3×D4—22no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—42no—
199C2×C3×D4—42no—
200C2×C3×D4—82no—
201C2×C3×D4—82no—
202C2×C3×D4C2×D412yes—
203C22×C3×C4—22no—
204C22×C3×C4—21no—
205C22×C3×C4—24no—
206C22×C3×C4—24no—
207C22×C3×C4—22no—
208C22×C3×C4—22no—
209C22×C3×C4—22no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—42no—
214C22×C3×C4—42no—
215C22×C3×C4—81no—
216C22×C3×C4—81no—
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2×D412yes—
219C22×C3×C4C2×D411yescharacteristic
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D412yes—
222C22×C3×C4#4112yes—
223C22×C3×C4#4112yes—
224C22×C3×C4#4112yes—
225C22×C3×C4C2411yescharacteristic
226C22×Dic3—21no—
227C22×Dic3—21no—
228C22×Dic3—22no—
229C22×Dic3—42no—
230C22×Dic3—81no—
231C22×Dic3—81no—
232C3×C4⋊C4—28no—
233C3×C4⋊C4—24no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4C2×D414yes—
236C3×C22⋊C4—28no—
237C3×C22⋊C4—28no—
238#217—42no—
239#217—82no—
240#217—82no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—22no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—22no—
247C4⋊Dic3—42no—
248C4×Dic3—42no—
249#208—28no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—44no—
259C2×C3⋊C8C2×D414yes—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41—22no—
266C2×#41—22no—
267C2×#41—22no—
268C2×#41—22no—
269C2×#41—22no—
270C2×#41—22no—
271C2×#41#6714yes—
272C22×Q8#6711yescharacteristic
273C22×Q8C22×D311yescharacteristic
274C22×D4—21no—
275C22×D4—22no—
276C22×D4#6712yes—
277C22×D4#6711yescharacteristic
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C23×C4—21no—
281C23×C4—61no—
282C23×C4C22×D312yes—
283C23×C4C22×D311yescharacteristic
284C2×M16—34no—
285C2×M16—62no—
286C2×M16—62no—
287C22×C8—32no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8—62no—
291C22×C8—62no—
292C22×C8—62no—
293C22⋊Q8—28no—
294C22⋊Q8—24no—
295C22⋊Q8—24no—
296#121—216no—
297#121—28no—
298#121—24no—
299#121—24no—
300#117—62no—
301#117—62no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—62no—
309C2×C4⋊C4—62no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4#6714yes—
312C2×C4⋊C4C22×D312yes—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C42—61no—
319#107—64no—
320#106—64no—
321#103—64no—
322#103—62no—
323#103—62no—
324#102—64no—
325#102—62no—
326#102—62no—
327C22⋊C8—64no—
328C22⋊C8—64no—
329#95—62no—
330#95—62no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—22no—
336C23×C3—22no—
337C23×C3—22no—
338C23×C3—22no—
339C23×C3—22no—
340C23×C3—22no—
341C23×C3—22no—
342C23×C3—22no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3#12012yes—
350C23×C3#12112yes—
351C23×C3#12312yes—
352C23×C3C22×D411yescharacteristic
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—44no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—48no—
361C3×D4—48no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—44no—
365C3×D4—88no—
366C2×C3×C4—24no—
367C2×C3×C4—22no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—22no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×C3×C4—42no—
379C2×C3×C4—48no—
380C2×C3×C4—84no—
381C2×C3×C4#12012yes—
382C2×C3×C4#12114yes—
383C2×C3×C4#12114yes—
384C2×C3×C4#12112yes—
385C2×C3×C4#12314yes—
386C2×C3×C4#12312yes—
387C2×C3×C4#12414yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—42no—
397C2×Dic3—41no—
398C2×Dic3—42no—
399C2×Dic3—41no—
400C2×Dic3—42no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—84no—
404C3⋊C8—24no—
405C3⋊C8—41no—
406C3⋊C8—41no—
407C24—21no—
408C24C2×#6712yes—
409C24C2×#6711yescharacteristic
410#41—28no—
411#41—28no—
412#41—44no—
413#41—44no—
414#41—44no—
415#41—44no—
416#41—44no—
417#41—44no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8C2×#6712yes—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—28no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—42no—
441C2×D4—42no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4C2×#6712yes—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4—81no—
461C22×C4—81no—
462C22×C4—122no—
463C22×C4—241no—
464C22×C4—241no—
465C22×C4D3×D412yes—
466C22×C4#23712yes—
467C22×C4#23712yes—
468C22×C4#23712yes—
469C22×C4C2×#6712yes—
470C22×C4C2×#6711yescharacteristic
471C22×C4C2×#6712yes—
472C22×C4C2×#6712yes—
473C22×C4C23×D311yescharacteristic
474M16—68no—
475C2×C8—34no—
476C2×C8—62no—
477C2×C8—62no—
478C2×C8—61no—
479C2×C8—61no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—124no—
485C4⋊C4—28no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—62no—
489C4⋊C4—62no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—122no—
495C4⋊C4C2×#6714yes—
496C22⋊C4—28no—
497C22⋊C4—28no—
498C22⋊C4—122no—
499C22⋊C4—242no—
500C22⋊C4—242no—
501C42—122no—
502C22×C3—22no—
503C22×C3—22no—
504C22×C3—22no—
505C22×C3—22no—
506C22×C3—22no—
507C22×C3—22no—
508C22×C3—42no—
509C22×C3—42no—
510C22×C3—42no—
511C22×C3—42no—
512C22×C3—42no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—81no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3—81no—
522C22×C3#39414yes—
523C22×C3C2×#12011yescharacteristic
524C22×C3C2×#12111yescharacteristic
525C22×C3C2×#12311yescharacteristic
526C22×C3#57611yescharacteristic
527C22×C3#57711yescharacteristic
528C22×C3#57711yescharacteristic
529C22×C3#57811yescharacteristic
530C3×C4—42no—
531C3×C4—42no—
532C3×C4—44no—
533C3×C4#39414yes—
534C3×C4C2×#12111yescharacteristic
535C3×C4C2×#12111yescharacteristic
536C3×C4C2×#12311yescharacteristic
537C3×C4C2×#12411yescharacteristic
538Dic3—81no—
539Dic3—81no—
540Dic3—82no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23#93512yes—
560C23#93912yes—
561C23#95312yes—
562C23C22×#6711yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—44no—
566Q8—41no—
567Q8—41no—
568D4—22no—
569D4—22no—
570D4—48no—
571D4—48no—
572D4—44no—
573D4—41no—
574D4—41no—
575D4—88no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—84no—
591C2×C4—122no—
592C2×C4—121no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—122no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—244no—
601C2×C4#93512yes—
602C2×C4#93514yes—
603C2×C4#93614yes—
604C2×C4#93814yes—
605C2×C4#93912yes—
606C2×C4#93914yes—
607C2×C4#95312yes—
608C2×C4C2×D3×D411yescharacteristic
609C2×C4C2×#23711yescharacteristic
610C2×C4C2×#23711yescharacteristic
611C2×C4C2×#23711yescharacteristic
612C2×C4C22×#6711yescharacteristic
613C2×C4C22×#6711yescharacteristic
614C8—64no—
615C8—121no—
616C8—121no—
617C2×C3—24no—
618C2×C3—82no—
619C2×C3—82no—
620C2×C3C2×#39411yescharacteristic
621C2×C3#301211yescharacteristic
622C2×C3#301311yescharacteristic
623C2×C3#306211yescharacteristic
624C2×C3#306311yescharacteristic
625C2×C3#309811yescharacteristic
626C2×C3#309911yescharacteristic
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22#522214yes—
648C22C2×#93511yescharacteristic
649C22C2×#93911yescharacteristic
650C22#583511yescharacteristic
651C22#583611yescharacteristic
652C22#583711yescharacteristic
653C22#583811yescharacteristic
654C22C2×#95311yescharacteristic
655C4—42no—
656C4—42no—
657C4—44no—
658C4—241no—
659C4—241no—
660C4—242no—
661C4#522214yes—
662C4C2×#93511yescharacteristic
663C4C2×#93611yescharacteristic
664C4C2×#93811yescharacteristic
665C4C2×#93911yescharacteristic
666C3#2348511yescharacteristic, complemented, Sylow
667C2—24no—
668C2—82no—
669C2—82no—
670C2#8341811yescharacteristic
671C2#8341911yescharacteristic
672C2#8342211yescharacteristic
673C2#8342311yescharacteristic
674C2C2×#522211yescharacteristic
675C2#8410211yescharacteristic
676C2#8411411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#537C2
socleC23×C3C22×D4
hypercenterC2×#537D3
hypocenterC3#23485
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×#41 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
2. #84087C2semidirect
3. C2×#5234C2semidirect
4. C2×#5232C2semidirect
5. C2×#5233C2semidirect
6. C2×#4964C2semidirect
7. #71695C2semidirect
8. #71694C2semidirect
9. #71680C2semidirect
10. #71679C2semidirect
11. C2×#4529C2semidirect
111. C2×#208D4semidirect
666. C3#23485semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5372
#840872
C2×#52342
C2×#52322
C2×#52332
C2×#49642
#716952
#716942
#716802
#716792
C2×#45292
#234853