Finite Group Databasebeta

← #10824494

Subgroups — #10824494

Order 768.

844 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#440C211yesmaximal, characteristic
2#82456C211yesmaximal, characteristic, complemented
3#82325C211yesmaximal, characteristic
4C2×C4×#214C211yesmaximal, characteristic
5#72057C211yesmaximal, characteristic
6#72051C211yesmaximal, characteristic
7C4×#835C211yesmaximal, characteristic, complemented
8#70643C211yesmaximal, characteristic, complemented
9#70620C211yesmaximal, characteristic, complemented
10#70261C211yesmaximal, characteristic, complemented
11#70246C211yesmaximal, characteristic, complemented
12#70217C211yesmaximal, characteristic
13#70166C211yesmaximal, characteristic
14#69832C211yesmaximal, characteristic, complemented
15#69813C211yesmaximal, characteristic, complemented
16#23689—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#440—24no—
24C2×C3×#107C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31#5221C2211yescharacteristic
32#5221C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5212—21no—
35#5211—21no—
36C2×#816C2211yescharacteristic
37C2×#816C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39C2×Dic3⋊C8C2211yescharacteristic
40C4×#214—24no—
41#4984C2211yescharacteristic
42#4984C2211yescharacteristic
43#4978—21no—
44#4978—21no—
45#4977C2211yescharacteristic
46#4973C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#808C2211yescharacteristic
49#4951—21no—
50#4951—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52#4933—21no—
53#4933—21no—
54#4932C2211yescharacteristic
55C2×C4×C3⋊C8C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#440D311yescharacteristic
63C2×C4×SD16—31no—
64#2327—31no—
65#2319—31no—
66#2085—31no—
67#2073—31no—
68#2068—31no—
69#2066—31no—
70#2055—31no—
71#2055—31no—
72#2043—31no—
73#2013—31no—
74#1768—31no—
75#1763—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×Dic6C2311yescharacteristic
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16—22no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×C4×Q8—22no—
87C3×C4×Q8—22no—
88C3×C4×D4—22no—
89C3×C4×D4D414yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100#944—22no—
101#944—22no—
102C2×#214—22no—
103C2×#214—22no—
104C2×#214—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3C2311yescharacteristic
116C2×C4×Dic3C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C4×Dic6—22no—
120C4×Dic6—22no—
121C3×#107—22no—
122C3×#103—22no—
123C3×#102—22no—
124C3×#97—22no—
125#835—22no—
126#835—22no—
127#832—22no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140C3⋊#95D411yescharacteristic
141#816—22no—
142#816—22no—
143Dic3⋊C8—22no—
144Dic3⋊C8—22no—
145#810—22no—
146#808—22no—
147C4×C3⋊C8—22no—
148C22×SD16—31no—
149C22×SD16C2×D311yescharacteristic
150C2×C4⋊Q8—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8C2×D311yescharacteristic
153C2×C4×D4C2×D311yescharacteristic
154#440—24no—
155C4×SD16—64no—
156C2×#107C2×D311yescharacteristic
157C2×#106—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8—31no—
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103C2×D311yescharacteristic
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×#97C2×D311yescharacteristic
169C2×C4×C8—31no—
170#391—31no—
171#391—31no—
172#390—31no—
173#390—31no—
174#389—31no—
175#389—31no—
176#388—61no—
177#388—61no—
178#387—61no—
179#387—61no—
180#387—61no—
181#387—61no—
182#382—31no—
183#381—61no—
184#380—61no—
185C4×C4⋊C4—31no—
186#376—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8#4112yes—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4C2×D412yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—24no—
209C22×C3×C4—21no—
210C22×C3×C4—81no—
211C22×C3×C4—81no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#4112yes—
229C3×SD16—28no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8#4112yes—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—42no—
240C3×C4⋊C4C2×D412yes—
241C3×C22⋊C4—24no—
242C3×C42—42no—
243C3×C42C2×D412yes—
244C3×C42#4112yes—
245#217—81no—
246#217—81no—
247#217—81no—
248#217—81no—
249#214—28no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—81no—
257C4⋊Dic3—81no—
258Dic3⋊C4—24no—
259Dic3⋊C4—24no—
260Dic3⋊C4—42no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8#4112yes—
273C22×Q8—31no—
274C22×Q8C22×D311yescharacteristic
275C22×D4C22×D311yescharacteristic
276C23×C4—21no—
277C2×SD16—22no—
278C2×SD16—22no—
279C2×SD16—22no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C22×C8—31no—
284C22×C8—31no—
285C22×C8C22×D311yescharacteristic
286C22×C8C22×D311yescharacteristic
287C4⋊Q8—62no—
288C4⋊Q8—62no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×D4—22no—
294C4×D4#6714yes—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C42—21no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42C22×D311yescharacteristic
317#107—22no—
318#106—62no—
319C4⋊C8—62no—
320C4⋊C8—62no—
321#103—22no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—62no—
327#102—22no—
328#102—62no—
329#97—22no—
330C4×C8—62no—
331#95—31no—
332#95—31no—
333#95—31no—
334#95—31no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344C23×C3—22no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3C22×D411yescharacteristic
352C3×Q8—22no—
353C3×Q8—22no—
354C3×Q8—41no—
355C3×Q8—41no—
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—41no—
359C3×D4—41no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—22no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—84no—
374C2×C3×C4#12112yes—
375C2×C3×C4#12112yes—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—22no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406Dic6—22no—
407Dic6—22no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—22no—
413C3⋊C8—41no—
414C24—21no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—32no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8#23512yes—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4C2×#6712yes—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—21no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—28no—
451SD16—68no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—32no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8#23512yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—42no—
468C4⋊C4—64no—
469C4⋊C4—64no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—122no—
477C4⋊C4—241no—
478C4⋊C4—241no—
479C4⋊C4C2×#6712yes—
480C22⋊C4—24no—
481C22⋊C4—241no—
482C22⋊C4—241no—
483C22⋊C4—241no—
484C22⋊C4—241no—
485C42—42no—
486C42—61no—
487C42—61no—
488C42—61no—
489C42—61no—
490C42—122no—
491C42—241no—
492C42#23712yes—
493C42C2×#6712yes—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—42no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3C2×#12111yescharacteristic
502C22×C3#52511yescharacteristic
503C22×C3#53211yescharacteristic
504C22×C3#57711yescharacteristic
505C22×C3#57711yescharacteristic
506C22×C3#57711yescharacteristic
507C22×C3#57811yescharacteristic
508C3×C4—22no—
509C3×C4—22no—
510C3×C4—42no—
511C3×C4—42no—
512C3×C4—81no—
513C3×C4#52511yescharacteristic
514C3×C4#52511yescharacteristic
515C3×C4#52911yescharacteristic
516C3×C4#52911yescharacteristic
517Dic3—41no—
518Dic3—41no—
519Dic3—42no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23C2×D3×D411yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537Q8—62no—
538Q8—62no—
539Q8—121no—
540Q8—121no—
541D4—22no—
542D4—22no—
543D4—41no—
544D4—41no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—62no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—84no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4#93912yes—
584C2×C4#93912yes—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×#23511yescharacteristic
587C2×C4C2×#23711yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4D3×#4111yescharacteristic
590C2×C4C22×#6711yescharacteristic
591C8—22no—
592C8—41no—
593C8—62no—
594C8—121no—
595C2×C3—82no—
596C2×C3#239211yescharacteristic
597C2×C3#304211yescharacteristic
598C2×C3#304611yescharacteristic
599C2×C3#312111yescharacteristic
600C2×C3#312611yescharacteristic
601C2×C3#317411yescharacteristic
602C2×C3#317811yescharacteristic
603C22—42no—
604C22—42no—
605C22—42no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22#554711yescharacteristic
611C22#554811yescharacteristic
612C22#577511yescharacteristic
613C22#577511yescharacteristic
614C22#577811yescharacteristic
615C22#577911yescharacteristic
616C22C2×#93911yescharacteristic
617C4—22no—
618C4—22no—
619C4—42no—
620C4—42no—
621C4—81no—
622C4—121no—
623C4—121no—
624C4—122no—
625C4—121no—
626C4—121no—
627C4—241no—
628C4—241no—
629C4—241no—
630C4#553811yescharacteristic
631C4#555711yescharacteristic
632C4#579111yescharacteristic
633C4#582811yescharacteristic
634C3#2368911yescharacteristic, complemented, Sylow
635C2—82no—
636C2#7976711yescharacteristic
637C2#7976911yescharacteristic
638C2#7978911yescharacteristic
639C2#7981411yescharacteristic
640C2#8234511yescharacteristic
641C2#8393511yescharacteristic
642C2#8397111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterC2×#440D3
hypocenterC3#23689
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
2. #82456C2semidirect
7. C4×#835C2semidirect
8. #70643C2semidirect
9. #70620C2semidirect
10. #70261C2semidirect
11. #70246C2semidirect
14. #69832C2semidirect
15. #69813C2semidirect
634. C3#23689semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4402
#824562
#823252
C2×C4×#2142
#720572
#720512
C4×#8352
#706432
#706202
#702612
#702462
#702172
#701662
#698322
#698132
#236893