Finite Group Databasebeta

← #10868435

Subgroups — #10868435

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76492C211yesmaximal, characteristic, complemented
2#76341C211yesmaximal, characteristic
3C3×#1968C211yesmaximal, characteristic
4#72057C211yesmaximal, characteristic
5#72048C211yesmaximal, characteristic
6#70591C211yesmaximal, characteristic, complemented
7#70558C211yesmaximal, characteristic, complemented
8#70246C211yesmaximal, characteristic, complemented
9#70222C211yesmaximal, characteristic, complemented
10#70217C211yesmaximal, characteristic
11#70173C211yesmaximal, characteristic
12#69784C211yesmaximal, characteristic, complemented
13C4×#816C211yesmaximal, characteristic, complemented
14#69667C211yesmaximal, characteristic, complemented
15#69660C211yesmaximal, characteristic
16#27069—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×C4×C8C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31#5221C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×Dic3⋊C8C2211yescharacteristic
36#4984C2211yescharacteristic
37#4977C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4973C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4953—21no—
43#4952—21no—
44#4951—21no—
45#4945—21no—
46C4×C4⋊Dic3C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48#4655—21no—
49#4652—21no—
50#4649—21no—
51#4647—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513—31no—
61#2365—31no—
62#2097—31no—
63#2055—31no—
64#2043—31no—
65#2013—31no—
66#1968D311yescharacteristic
67#1928—31no—
68#1925—31no—
69#1920—31no—
70#1916—31no—
71#1880—31no—
72#1880—31no—
73#1805—31no—
74C4×#103—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C2×C3×SD16—22no—
79C2×C3×SD16—22no—
80C2×C3×SD16—41no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—21no—
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92Q8×Dic3—22no—
93Q8×Dic3—41no—
94#944—22no—
95#944—22no—
96C2×#214—22no—
97C2×#214—41no—
98C2×#214—41no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3C2311yescharacteristic
114C2×C4×Dic3C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C4×Dic6—22no—
118C4×Dic6—41no—
119C3×#106—22no—
120C3×#103—41no—
121C3×#102—41no—
122C3×C4×C8—41no—
123C3×#95—21no—
124C3×#95—21no—
125#835—22no—
126#835—41no—
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⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#816—22no—
138#816—22no—
139Dic3⋊C8—41no—
140#810—41no—
141#807—41no—
142C22×SD16—31no—
143C22×SD16C2×D311yescharacteristic
144C2×C4⋊Q8—31no—
145C2×C4×Q8—31no—
146C2×C4×Q8—31no—
147C2×#107—31no—
148C2×#106C2×D311yescharacteristic
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#103C2×D311yescharacteristic
156C2×#102—31no—
157C2×#102C2×D311yescharacteristic
158C2×C4×C8C2×D311yescharacteristic
159#397—61no—
160#397—61no—
161#391—31no—
162#391C2×D311yescharacteristic
163#390—31no—
164#390—31no—
165#390C2×D311yescharacteristic
166#389—31no—
167#389—61no—
168#388—61no—
169#388—61no—
170#387—61no—
171#387—61no—
172#382—31no—
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—61no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185C24×C3—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8#4112yes—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6#4112yes—
218C3×SD16—44no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8#4112yes—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—81no—
231C3×C42—21no—
232C3×C42—21no—
233#217—42no—
234#217—81no—
235#217—81no—
236#217—81no—
237#214—44no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—81no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—24no—
248Dic3⋊C4—22no—
249Dic3⋊C4—22no—
250Dic3⋊C4—42no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—42no—
256C4×Dic3—42no—
257C4×Dic3—42no—
258C4×Dic3—81no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C22×Q8—31no—
266C22×Q8C22×D311yescharacteristic
267C22×D4C22×D311yescharacteristic
268C2×SD16—22no—
269C2×SD16—22no—
270C2×SD16—41no—
271C2×SD16—62no—
272C2×SD16—121no—
273C2×SD16—121no—
274C22×C8—31no—
275C22×C8—31no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4⋊Q8—62no—
279C4⋊Q8—62no—
280C4×Q8—62no—
281C4×Q8—62no—
282C4×Q8—121no—
283C4×Q8—121no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4C22×D311yescharacteristic
295C2×C4⋊C4C22×D311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42C22×D311yescharacteristic
309#107—121no—
310#106—22no—
311C4⋊C8—121no—
312#103—41no—
313#103—62no—
314#103—62no—
315#103—62no—
316#103—121no—
317#103—121no—
318#102—41no—
319#102—62no—
320C4×C8—41no—
321#95—21no—
322#95—21no—
323#95—31no—
324#95—31no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3C22×D411yescharacteristic
340C3×Q8—22no—
341C3×Q8—22no—
342C3×Q8—41no—
343C3×Q8—41no—
344C3×D4—41no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4—41no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4#13712yes—
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4H(5,𝔽2)11yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—22no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3#13512yes—
393Dic6—22no—
394Dic6—22no—
395Dic6—41no—
396Dic6—41no—
397C3×C8—22no—
398C3×C8—41no—
399C3⋊C8—41no—
400C3⋊C8—41no—
401C24—21no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—32no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×Q8#23512yes—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4D3×D411yescharacteristic
432C22×C4D3×D411yescharacteristic
433C22×C4C23×D311yescharacteristic
434SD16—44no—
435SD16—124no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8#23712yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—32no—
453C4⋊C4—62no—
454C4⋊C4—64no—
455C4⋊C4—64no—
456C4⋊C4—62no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—241no—
465C22⋊C4—42no—
466C22⋊C4—81no—
467C22⋊C4—122no—
468C22⋊C4—241no—
469C22⋊C4—241no—
470C22⋊C4—241no—
471C42—21no—
472C42—21no—
473C42—61no—
474C42—61no—
475C42—61no—
476C42—61no—
477C42—122no—
478C42—122no—
479C42—122no—
480C42—241no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3#52511yescharacteristic
489C22×C3#54611yescharacteristic
490C22×C3#55011yescharacteristic
491C22×C3C2×#13511yescharacteristic
492C22×C3C2×#13711yescharacteristic
493C22×C3#57711yescharacteristic
494C22×C3#57711yescharacteristic
495C3×C4—22no—
496C3×C4—42no—
497C3×C4—41no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4#52911yescharacteristic
501C3×C4#53811yescharacteristic
502C3×C4#55111yescharacteristic
503C3×C4#55311yescharacteristic
504Dic3—22no—
505Dic3—41no—
506Dic3—42no—
507Dic3—41no—
508Dic3—42no—
509Dic3—41no—
510Dic3—81no—
511Dic3—81no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C2×D3×D411yescharacteristic
520Q8—22no—
521Q8—22no—
522Q8—41no—
523Q8—41no—
524Q8—62no—
525Q8—62no—
526Q8—121no—
527Q8—121no—
528D4—41no—
529D4—41no—
530D4—41no—
531D4—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—32no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—62no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4#91512yes—
571C2×C4C2×#23511yescharacteristic
572C2×C4C2×#23711yescharacteristic
573C2×C4#100411yescharacteristic
574C2×C4#100411yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4D3×#4111yescharacteristic
577C8—22no—
578C8—41no—
579C8—121no—
580C8—121no—
581C2×C3—161no—
582C2×C3#246711yescharacteristic
583C2×C3#310811yescharacteristic
584C2×C3#312611yescharacteristic
585C2×C3#330311yescharacteristic
586C2×C3#332011yescharacteristic
587C2×C3#339411yescharacteristic
588C2×C3#340011yescharacteristic
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22—81no—
595C22—81no—
596C22#554711yescharacteristic
597C22#561011yescharacteristic
598C22#571811yescharacteristic
599C22C2×#91511yescharacteristic
600C22#576211yescharacteristic
601C22D3×#13511yescharacteristic
602C22#577011yescharacteristic
603C4—22no—
604C4—42no—
605C4—41no—
606C4—62no—
607C4—81no—
608C4—81no—
609C4—122no—
610C4—121no—
611C4—122no—
612C4—121no—
613C4—121no—
614C4—241no—
615C4—241no—
616C4#549511yescharacteristic
617C4#554011yescharacteristic
618C4#561311yescharacteristic
619C4#568011yescharacteristic
620C3#2706911yescharacteristic, complemented, Sylow
621C2—161no—
622C2#7632811yescharacteristic
623C2#7965611yescharacteristic
624C2#7966111yescharacteristic
625C2#8068211yescharacteristic
626C2#8069411yescharacteristic
627C2#8175811yescharacteristic
628C2#8181411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1968C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1968D3
hypocenterC3#27069
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1968

Complements

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

NH ≅ G/Ntype
1. #76492C2semidirect
6. #70591C2semidirect
7. #70558C2semidirect
8. #70246C2semidirect
9. #70222C2semidirect
12. #69784C2semidirect
13. C4×#816C2semidirect
14. #69667C2semidirect
620. C3#27069semidirect

Maximal subgroups

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

subgroupindex
#764922
#763412
C3×#19682
#720572
#720482
#705912
#705582
#702462
#702222
#702172
#701732
#697842
C4×#8162
#696672
#696602
#270693