Finite Group Databasebeta

← #10845164

Subgroups — #10845164

Order 768.

822 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5166C211yesmaximal, characteristic
2C2×#4768C211yesmaximal, characteristic
3#76542C211yesmaximal, characteristic, complemented
4#76307C211yesmaximal, characteristic
5C3×#2055C211yesmaximal, characteristic
6#72047C211yesmaximal, characteristic
7#70486C211yesmaximal, characteristic, complemented
8#70451C211yesmaximal, characteristic, complemented
9#70285C211yesmaximal, characteristic, complemented
10Dic3×#103C211yesmaximal, characteristic, complemented
11#70153C211yesmaximal, characteristic
12#69690C211yesmaximal, characteristic, complemented
13#69641C211yesmaximal, characteristic
14#68914C211yesmaximal, characteristic, complemented
15#68802C211yesmaximal, characteristic, complemented
16#25329—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C2×#936C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#391C2211yescharacteristic
28C3×#390C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33#5166—24no—
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43#4946—21no—
44#4945—21no—
45#4944—21no—
46Dic3×C22⋊C4—21no—
47#4936C2211yescharacteristic
48C4×Dic3⋊C4C2211yescharacteristic
49#4768—24no—
50#4657—21no—
51#4657—21no—
52#4655—21no—
53#4655—21no—
54C3×#340C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#488—31no—
63C2×#464—31no—
64#2565—31no—
65#2367—31no—
66#2073—31no—
67#2072—31no—
68#2056—31no—
69#2055D311yescharacteristic
70#1989—31no—
71#1939—31no—
72#1878—31no—
73#1877—31no—
74#1805—31no—
75#1802—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C22×Dic6C2311yescharacteristic
81C2×C3×SD16—22no—
82C2×C3×SD16—22no—
83C2×C3×SD16—41no—
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94Q8×Dic3—22no—
95Q8×Dic3—22no—
96C2×#216—22no—
97C2×#216—22no—
98C2×#216—41no—
99#936—22no—
100#936D414yes—
101D4×Dic3—22no—
102D4×Dic3—41no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×Dic3⋊C4D411yescharacteristic
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3C2311yescharacteristic
118C2×C4×Dic3C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#890—22no—
122#890—22no—
123C4×Dic6—22no—
124C4×Dic6—41no—
125C3×C4⋊C8—41no—
126C3×#103—22no—
127C3×#102—22no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95D411yescharacteristic
141C3⋊#95D411yescharacteristic
142#816—22no—
143#816—22no—
144Dic3⋊C8—22no—
145Dic3⋊C8—41no—
146C8×Dic3—22no—
147#810—22no—
148C22×Q16—31no—
149C22×SD16C2×D311yescharacteristic
150C2×#125—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4—31no—
155#488—64no—
156#464—64no—
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102C2×D311yescharacteristic
168C2×C4×C8—31no—
169#398—61no—
170#398—61no—
171#391—31no—
172#391—31no—
173#391C2×D311yescharacteristic
174#390C2×D311yescharacteristic
175#388—61no—
176#388—61no—
177#387—61no—
178#387—61no—
179#385—61no—
180#384—31no—
181#382—31no—
182C4×C4⋊C4—31no—
183C4×C4⋊C4—31no—
184C4×C22⋊C4—61no—
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
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—21no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2411yescharacteristic
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—81no—
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×Dic6C2×D412yes—
229C3×SD16—44no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C22⋊C4—42no—
241C3×C22⋊C4—81no—
242C3×C42—21no—
243C3×C42—21no—
244#217—24no—
245#217—24no—
246#217—42no—
247#216—44no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—42no—
254C4⋊Dic3—42no—
255C4⋊Dic3#4112yes—
256Dic3⋊C4—22no—
257Dic3⋊C4—41no—
258Dic3⋊C4—42no—
259Dic3⋊C4—42no—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—42no—
265C4×Dic3—42no—
266C4×Dic3—81no—
267C4×Dic3C2×D412yes—
268C4×Dic3#4112yes—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C22×Q8—31no—
276C22×Q8C22×D311yescharacteristic
277C22×D4C22×D311yescharacteristic
278C23×C4—61no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—121no—
282C2×SD16—22no—
283C2×SD16—22no—
284C2×SD16—41no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#125—62no—
290#125—62no—
291#124—34no—
292#124—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C4×Q8—62no—
296C4×Q8—121no—
297C4×D4—62no—
298C4×D4—121no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—61no—
305C2×C4⋊C4—61no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—31no—
318C2×C42—31no—
319C2×C42—61no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323C4⋊C8—41no—
324C4⋊C8—62no—
325C4⋊C8—121no—
326#103—22no—
327#103—62no—
328#103—62no—
329#103—62no—
330#103—62no—
331#102—22no—
332#102—62no—
333#102—62no—
334C4×C8—62no—
335#95—21no—
336#95—21no—
337#95—31no—
338#95—31no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3—22no—
347C23×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3C22×D411yescharacteristic
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×D4—22no—
359C3×D4—22no—
360C3×D4—41no—
361C3×D4—41no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—22no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4H(5,𝔽2)11yescharacteristic
382C2×C3×C42−1+411yescharacteristic
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—22no—
390C2×Dic3—21no—
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—41no—
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—84no—
405C2×Dic3#12112yes—
406C2×Dic3#12112yes—
407C2×Dic3#13512yes—
408Dic6—22no—
409Dic6—22no—
410Dic6—41no—
411Dic6—41no—
412C3×C8—41no—
413C3×C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—32no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8#23712yes—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4D3×D412yes—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—241no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—124no—
453SD16—44no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—32no—
471C4⋊C4—62no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—121no—
478C4⋊C4—122no—
479C4⋊C4—122no—
480C4⋊C4—122no—
481C4⋊C4—122no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C22⋊C4—42no—
485C22⋊C4—64no—
486C22⋊C4—64no—
487C22⋊C4—81no—
488C22⋊C4—122no—
489C42—21no—
490C42—21no—
491C42—32no—
492C42—32no—
493C42—61no—
494C42—61no—
495C42—122no—
496C42—122no—
497C42—241no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3C2×#12111yescharacteristic
506C22×C3#54611yescharacteristic
507C22×C3#54811yescharacteristic
508C22×C3C2×#13511yescharacteristic
509C22×C3#57511yescharacteristic
510C22×C3#57711yescharacteristic
511C22×C3#57811yescharacteristic
512C3×C4—42no—
513C3×C4—41no—
514C3×C4—41no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4#52911yescharacteristic
518C3×C4#54211yescharacteristic
519C3×C4#54611yescharacteristic
520C3×C4#54711yescharacteristic
521Dic3—22no—
522Dic3—22no—
523Dic3—22no—
524Dic3—42no—
525Dic3—42no—
526Dic3—41no—
527Dic3—81no—
528Dic3—81no—
529C23—22no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23C2×D3×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—62no—
542Q8—62no—
543Q8—121no—
544Q8—121no—
545D4—22no—
546D4—22no—
547D4—41no—
548D4—41no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—32no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—62no—
576C2×C4—61no—
577C2×C4—62no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—244no—
589C2×C4C2×D3×D411yescharacteristic
590C2×C4C2×#23711yescharacteristic
591C2×C4#100411yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4#101011yescharacteristic
595C8—41no—
596C8—41no—
597C8—121no—
598C8—121no—
599C2×C3—82no—
600C2×C3#242711yescharacteristic
601C2×C3#303511yescharacteristic
602C2×C3#304611yescharacteristic
603C2×C3#329611yescharacteristic
604C2×C3#332711yescharacteristic
605C2×C3#336411yescharacteristic
606C2×C3#337111yescharacteristic
607C22—42no—
608C22—42no—
609C22—42no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22#559811yescharacteristic
615C22D3×#12111yescharacteristic
616C22#561011yescharacteristic
617C22#576211yescharacteristic
618C22D3×#13511yescharacteristic
619C22#577311yescharacteristic
620C22#577911yescharacteristic
621C4—41no—
622C4—42no—
623C4—41no—
624C4—62no—
625C4—62no—
626C4—62no—
627C4—81no—
628C4—81no—
629C4—122no—
630C4—122no—
631C4—121no—
632C4—241no—
633C4—241no—
634C4#559111yescharacteristic
635C4#564911yescharacteristic
636C4#565511yescharacteristic
637C4#566911yescharacteristic
638C3#2532911yescharacteristic, complemented, Sylow
639C2—82no—
640C2#7690211yescharacteristic
641C2#8037311yescharacteristic
642C2#8043111yescharacteristic
643C2#8053311yescharacteristic
644C2#8058011yescharacteristic
645C2#8067111yescharacteristic
646C2#8084511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2055C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2055D3
hypocenterC3#25329
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
3. #76542C2semidirect
7. #70486C2semidirect
8. #70451C2semidirect
9. #70285C2semidirect
10. Dic3×#103C2semidirect
12. #69690C2semidirect
14. #68914C2semidirect
15. #68802C2semidirect
638. C3#25329semidirect

Maximal subgroups

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

subgroupindex
C2×#51662
C2×#47682
#765422
#763072
C3×#20552
#720472
#704862
#704512
#702852
Dic3×#1032
#701532
#696902
#696412
#689142
#688022
#253293