Finite Group Databasebeta

← #10945107

Subgroups — #10945107

Order 768.

974 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5162C211yesmaximal, characteristic
2C4×D4×Dic3C211yesmaximal, characteristic
3C2×#4761C211yesmaximal, characteristic
4#76607C211yesmaximal, characteristic, complemented
5C3×#1877C211yesmaximal, characteristic
6#72047C211yesmaximal, characteristic
7#71612C211yesmaximal, characteristic
8#70229C211yesmaximal, characteristic, complemented
9#70223C211yesmaximal, characteristic, complemented
10#70171C211yesmaximal, characteristic
11#69666C211yesmaximal, characteristic, complemented
12#68799C211yesmaximal, characteristic, complemented
13#68793C211yesmaximal, characteristic, complemented
14#68627C211yesmaximal, characteristic, complemented
15#68605C211yesmaximal, characteristic, complemented
16#33026—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×#944C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C22×#214C2211yescharacteristic
23C22×C4×Dic3—21no—
24C2×#888C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#391C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C4×#217—21no—
33#5162—24no—
34C2×#817C2211yescharacteristic
35C2×#816C2211yescharacteristic
36C2×#816C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×Dic3⋊C8C2211yescharacteristic
39Dic3×C4⋊C4C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41#4973C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#808C2211yescharacteristic
44Dic3×C22⋊C4—21no—
45Dic3×C22⋊C4—21no—
46C4×C4⋊Dic3C2211yescharacteristic
47C42×Dic3C2211yescharacteristic
48#4929C2211yescharacteristic
49#4761—24no—
50#4650—21no—
51#4650—21no—
52#4646—21no—
53#4646—21no—
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4551C2211yescharacteristic
57#4550C2211yescharacteristic
58#4550C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#476—31no—
63C2×#461—31no—
64#2862—31no—
65C42×D4—31no—
66#1987—31no—
67#1981—31no—
68#1927—31no—
69#1927—31no—
70#1927—31no—
71#1927—31no—
72#1877—31no—
73#1877—31no—
74#1877—31no—
75#1877D311yescharacteristic
76#1831—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C23×Dic3—21no—
81C23×Dic3—21no—
82C22×Dic6—21no—
83C22×Dic6C2311yescharacteristic
84C2×C3×SD16—22no—
85C2×C3×SD16—22no—
86C2×C3×SD16—22no—
87C22×C3×C8C2311yescharacteristic
88C22×C3×C8C2311yescharacteristic
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97#944—22no—
98#944—22no—
99D4×Dic3—22no—
100D4×Dic3—22no—
101D4×Dic3—22no—
102D4×Dic3D414yes—
103C2×#214—22no—
104C2×#214—22no—
105C2×#214—22no—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—44no—
116C2×C4×Dic3D411yescharacteristic
117C2×C4×Dic3D411yescharacteristic
118C2×C4×Dic3D411yescharacteristic
119C2×C4×Dic3D411yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C2×C4×Dic3C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123C22×C3⋊C8C2311yescharacteristic
124#888—22no—
125#888—22no—
126C3×#103—22no—
127C3×#102—22no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131#835—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⋊#95—21no—
141C3⋊#95—21no—
142#817—22no—
143#816—22no—
144#816—22no—
145Dic3⋊C8—22no—
146Dic3⋊C8—22no—
147#810—22no—
148#808—22no—
149C22×SD16—31no—
150C22×SD16C2×D311yescharacteristic
151C2×C4⋊Q8—31no—
152C2×C4⋊Q8—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4C2×D311yescharacteristic
156C22×C42—61no—
157#476—64no—
158#461—64no—
159C2×#106—31no—
160C2×#106—31no—
161C2×C4⋊C8—31no—
162C2×C4⋊C8—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#103—31no—
167C2×#103—31no—
168C2×#103C2×D311yescharacteristic
169C2×#102—31no—
170C2×#102C2×D311yescharacteristic
171#391—31no—
172#391—31no—
173#391—31no—
174#391—61no—
175#391—61no—
176#391C2×D311yescharacteristic
177#382—61no—
178#382—61no—
179C4×C4⋊C4—31no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4—31no—
182C4×C22⋊C4—61no—
183C4×C22⋊C4—61no—
184C4×C22⋊C4—61no—
185C43—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340C2×D311yescharacteristic
190#336—31no—
191#336—31no—
192#336—31no—
193#336C2×D311yescharacteristic
194C24×C3—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4C2×D412yes—
205C2×C3×D4#4112yes—
206C2×C3×D4#4112yes—
207C22×C3×C4—24no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—42no—
212C22×C3×C4—42no—
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—24no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—24no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—42no—
223C22×Dic3—42no—
224C22×Dic3—42no—
225C22×Dic3—42no—
226C22×Dic3C2×D411yescharacteristic
227C22×Dic3C2×D411yescharacteristic
228C22×Dic3C2×D411yescharacteristic
229C22×Dic3C2×D411yescharacteristic
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—42no—
237C2×Dic6—42no—
238C2×Dic6—42no—
239C3×SD16—44no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4#4112yes—
250C3×C22⋊C4—24no—
251C3×C42—21no—
252C3×C42—21no—
253#217—24no—
254#217—24no—
255#214—44no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3C2×D412yes—
261C4⋊Dic3#4112yes—
262Dic3⋊C4—42no—
263Dic3⋊C4—42no—
264Dic3⋊C4—42no—
265C4×Dic3—21no—
266C4×Dic3—21no—
267C4×Dic3—22no—
268C4×Dic3—41no—
269C4×Dic3—41no—
270C4×Dic3—42no—
271C4×Dic3—42no—
272C4×Dic3—42no—
273C4×Dic3SD1614yescomplemented
274C4×Dic3C2×D412yes—
275C4×Dic3#4112yes—
276C2×C3⋊C8—21no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C22×Q8—31no—
283C22×Q8—61no—
284C22×Q8C22×D311yescharacteristic
285C22×D4C22×D311yescharacteristic
286C23×C4—21no—
287C23×C4—61no—
288C23×C4—61no—
289C2×SD16—22no—
290C2×SD16—22no—
291C2×SD16—22no—
292C2×SD16—62no—
293C2×SD16—62no—
294C2×SD16—62no—
295C22×C8—31no—
296C22×C8—31no—
297C22×C8C22×D311yescharacteristic
298C22×C8C22×D311yescharacteristic
299C4⋊Q8—62no—
300C4⋊Q8—62no—
301C4⋊Q8—62no—
302C4⋊Q8—62no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—34no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C2×C4⋊C4—31no—
310C2×C4⋊C4—31no—
311C2×C4⋊C4—31no—
312C2×C4⋊C4—61no—
313C2×C4⋊C4—61no—
314C2×C4⋊C4—61no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C42—31no—
321C2×C42—31no—
322C2×C42—31no—
323C2×C42—31no—
324C2×C42—31no—
325C2×C42—31no—
326C2×C42—61no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42—124no—
330C2×C42C22×D311yescharacteristic
331#106—62no—
332#106—62no—
333C4⋊C8—62no—
334C4⋊C8—62no—
335#103—22no—
336#103—62no—
337#103—62no—
338#103—62no—
339#103—62no—
340#103—62no—
341#102—22no—
342#102—62no—
343#95—21no—
344#95—21no—
345#95—61no—
346#95—61no—
347#95—61no—
348#95—61no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354C23×C3—22no—
355C23×C3—22no—
356C23×C3—22no—
357C23×C3—41no—
358C23×C3—41no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3C22×D411yescharacteristic
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×D4—22no—
367C3×D4—22no—
368C3×D4—22no—
369C3×D4#12414yes—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—22no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—48no—
377C2×C3×C4—41no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4—41no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4C2×SD1612yes—
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—48no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406C2×Dic3—48no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409C2×Dic3—41no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3#12112yes—
414C2×Dic3#12112yes—
415C2×Dic3C2×SD1612yes—
416C2×Dic3C2×SD1612yes—
417C2×Dic3#13712yes—
418Dic6—41no—
419Dic6—41no—
420Dic6—41no—
421Dic6—41no—
422Dic6—84no—
423C3×C8—41no—
424C3×C8—41no—
425C3⋊C8—41no—
426C3⋊C8—41no—
427C24—21no—
428C2×Q8—21no—
429C2×Q8—21no—
430C2×Q8—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×Q8—122no—
441C2×Q8—122no—
442C2×Q8—122no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4#23512yes—
447C2×D4D3×D412yes—
448C2×D4#23712yes—
449C22×C4—24no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—31no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—31no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—122no—
466C22×C4—122no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4D3×D411yescharacteristic
470C22×C4D3×D411yescharacteristic
471C22×C4C23×D311yescharacteristic
472SD16—44no—
473SD16—124no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—61no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—32no—
490C4⋊C4—32no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—122no—
496C4⋊C4—122no—
497C4⋊C4—122no—
498C4⋊C4#23712yes—
499C22⋊C4—24no—
500C22⋊C4—64no—
501C22⋊C4—64no—
502C42—21no—
503C42—21no—
504C42—34no—
505C42—32no—
506C42—32no—
507C42—62no—
508C42—61no—
509C42—61no—
510C42—121no—
511C42—121no—
512C42—122no—
513C42—122no—
514C42—122no—
515C22×C3—24no—
516C22×C3—42no—
517C22×C3—42no—
518C22×C3—42no—
519C22×C3—42no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3C2×#12111yescharacteristic
523C22×C3#52511yescharacteristic
524C22×C3#52511yescharacteristic
525C22×C3C22×SD1611yescharacteristic
526C22×C3C2×#13711yescharacteristic
527C22×C3#57511yescharacteristic
528C22×C3#57511yescharacteristic
529C3×C4—22no—
530C3×C4—42no—
531C3×C4—41no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4C2×#12411yescharacteristic
535C3×C4#52911yescharacteristic
536C3×C4#52911yescharacteristic
537C3×C4#53211yescharacteristic
538Dic3—22no—
539Dic3—22no—
540Dic3—42no—
541Dic3—41no—
542Dic3—42no—
543Dic3—81no—
544Dic3—81no—
545Dic3#46114yes—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23C2×D3×D411yescharacteristic
554Q8—41no—
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—121no—
559Q8—121no—
560Q8—121no—
561Q8—121no—
562Q8—244no—
563D4—22no—
564D4—22no—
565D4—22no—
566D4#88514yes—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—32no—
574C2×C4—32no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—32no—
578C2×C4—48no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—128no—
597C2×C4—128no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—121no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4D3×SD1612yes—
609C2×C4C2×#23511yescharacteristic
610C2×C4C2×D3×D411yescharacteristic
611C2×C4C2×#23711yescharacteristic
612C2×C4C2×#23711yescharacteristic
613C2×C4D3×#4111yescharacteristic
614C2×C4D3×#4111yescharacteristic
615C8—41no—
616C8—41no—
617C8—121no—
618C8—121no—
619C2×C3—44no—
620C2×C3#285011yescharacteristic
621C2×C3C2×#46111yescharacteristic
622C2×C3#304011yescharacteristic
623C2×C3#310111yescharacteristic
624C2×C3#310111yescharacteristic
625C2×C3#311611yescharacteristic
626C2×C3#311611yescharacteristic
627C22—24no—
628C22—42no—
629C22—42no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22#554711yescharacteristic
635C22#558611yescharacteristic
636C22D3×#12111yescharacteristic
637C22C2×D3×SD1611yescharacteristic
638C22#576811yescharacteristic
639C22#577311yescharacteristic
640C22D3×#13711yescharacteristic
641C4—22no—
642C4—34no—
643C4—42no—
644C4—41no—
645C4—62no—
646C4—62no—
647C4—81no—
648C4—81no—
649C4—122no—
650C4—122no—
651C4—121no—
652C4—241no—
653C4—241no—
654C4C2×#88511yescharacteristic
655C4#553811yescharacteristic
656C4#559611yescharacteristic
657C4#564911yescharacteristic
658C3#3302611yescharacteristic, complemented, Sylow
659C2—44no—
660C2#7486911yescharacteristic
661C2#7965111yescharacteristic
662C2#7978311yescharacteristic
663C2#8013211yescharacteristic
664C2#8021511yescharacteristic
665C2D3×#46111yescharacteristic
666C2#8053511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1877C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1877D3
hypocenterC3#33026
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1877

Complements

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

NH ≅ G/Ntype
4. #76607C2semidirect
8. #70229C2semidirect
9. #70223C2semidirect
11. #69666C2semidirect
12. #68799C2semidirect
13. #68793C2semidirect
14. #68627C2semidirect
15. #68605C2semidirect
273. C4×Dic3SD16semidirect
658. C3#33026semidirect

Maximal subgroups

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

subgroupindex
C2×#51622
C4×D4×Dic32
C2×#47612
#766072
C3×#18772
#720472
#716122
#702292
#702232
#701712
#696662
#687992
#687932
#686272
#686052
#330263