Finite Group Databasebeta

← #10845170

Subgroups — #10845170

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
1#82466C211yesmaximal, characteristic
2C2×#4798C211yesmaximal, characteristic
3C2×#4800C211yesmaximal, characteristic
4#74183C211yesmaximal, characteristic, complemented
5C3×#1805C211yesmaximal, characteristic, complemented
6#71632C211yesmaximal, characteristic, complemented
7#70482C211yesmaximal, characteristic
8#70451C211yesmaximal, characteristic, complemented
9#70285C211yesmaximal, characteristic, complemented
10#70233C211yesmaximal, characteristic, complemented
11#69557C211yesmaximal, characteristic
12#68896C211yesmaximal, characteristic
13#68892C211yesmaximal, characteristic
14#68801C211yesmaximal, characteristic, complemented
15#68781C211yesmaximal, characteristic, complemented
16#25329—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×#947C2211yescharacteristic
19C22×#216C2211yescharacteristic
20C22×#204C2211yescharacteristic
21C2×#893C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×#887C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#103C2211yescharacteristic
26C2×C3×#103C2211yescharacteristic
27C3×#382C2211yescharacteristic
28#5229C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×Dic3⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×C8×Dic3C2211yescharacteristic
35#4990—21no—
36#4989—21no—
37#4975C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4973C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43#4940—21no—
44#4939C2211yescharacteristic
45C4×(C2×D3)⋊C4—21no—
46C4×Dic3⋊C4C2211yescharacteristic
47#4929C2211yescharacteristic
48#4800—24no—
49#4798—24no—
50#4657—21no—
51#4657—21no—
52#4652—21no—
53#4652—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
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#2055—31no—
70#1989—31no—
71#1939—31no—
72#1878—31no—
73#1877—31no—
74#1805D311yescharacteristic, complemented
75#1802—31no—
76C4×#103—31no—
77C22×C3×Q8C2311yescharacteristic
78C22×D12C2311yescharacteristic
79C22×C4×D3—21no—
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×C4×Q8—22no—
84C3×C4×Q8—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91#947—22no—
92#947D414yes—
93C2×#216—22no—
94C2×#216—22no—
95C2×#216—41no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C2×#204—22no—
117C2×#204—22no—
118C2×#204—41no—
119#893—22no—
120#893—41no—
121#890—22no—
122#890—22no—
123#887—22no—
124#887—22no—
125C3×C4⋊C8—41no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#95—21no—
129C3×#95—21no—
130#835—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95D411yescharacteristic
139C3⋊#95D411yescharacteristic
140#821—22no—
141#816—22no—
142Dic3⋊C8—22no—
143Dic3⋊C8—41no—
144C8×Dic3—22no—
145#810—22no—
146#809—22no—
147#809—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#125—31no—
151C2×#124—31no—
152C2×C4×Q8—31no—
153C2×C4×Q8C2×D311yescharacteristic
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×#103C2×D311yescharacteristic
164C2×#103C2×D311yescharacteristic
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C4×C8—31no—
169#398—61no—
170#398—61no—
171#391—31no—
172#391—31no—
173#391—31no—
174#390—31no—
175#388—61no—
176#388—61no—
177#387—61no—
178#387—61no—
179#385—61no—
180#384—31no—
181#382C2×D311yescharacteristic
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#340C2×D311yescharacteristic
190#340C2×D311yescharacteristic
191#336—31no—
192#336—31no—
193C23×D3—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8—21no—
198C2×C3×Q8—21no—
199C2×C3×Q8C2×D412yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2411yescharacteristic
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C2×D12—21no—
215C2×D12—21no—
216C2×D12—21no—
217C2×D12—21no—
218C2×D12—21no—
219C2×D12C2×D412yes—
220C2×C4×D3—42no—
221C2×C4×D3—42no—
222C2×C4×D3—81no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—42no—
241C3×C4⋊C4#4112yes—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—42no—
245#216—44no—
246(C2×D3)⋊C4—24no—
247(C2×D3)⋊C4—24no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—42no—
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—81no—
266C4×Dic3C2×D412yes—
267C4×Dic3#4112yes—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—21no—
273C2×C3⋊C8—21no—
274#204—44no—
275C22×Q8—31no—
276C22×Q8C22×D311yescharacteristic
277C22×D4—31no—
278C23×C4—61no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—121no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—121no—
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—22no—
294C4×Q8—41no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×D4—62no—
298C4×D4—121no—
299C2×C4⋊C4—21no—
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⋊C4C22×D311yescharacteristic
310C2×C4⋊C4C22×D311yescharacteristic
311C2×C4⋊C4C22×D311yescharacteristic
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4—61no—
317C2×C42—21no—
318C2×C42—31no—
319C2×C42—31no—
320C2×C42—61no—
321C2×C42—61no—
322C2×C42C22×D311yescharacteristic
323C4⋊C8—41no—
324C4⋊C8—62no—
325C4⋊C8—121no—
326#103—22no—
327#103—22no—
328#103—62no—
329#103—62no—
330#103—62no—
331#102—62no—
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×C3C22×D411yescharacteristic
347C22×D3—22no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—41no—
357C3×Q8—41no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
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—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4C22×D411yescharacteristic
377C2×C3×C4C2×#4111yescharacteristic
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4H(5,𝔽2)11yescharacteristic
381C2×C3×C42−1+411yescharacteristic
382C2×Dic3—21no—
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—21no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3#12112yes—
401C2×Dic3#12112yes—
402C2×Dic3#13512yes—
403D12—22no—
404D12—22no—
405D12—41no—
406D12—41no—
407C4×D3—84no—
408Dic6—22no—
409Dic6—22no—
410Dic6—41no—
411Dic6—41no—
412C3×C8—41no—
413C3×C8—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C24—61no—
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×Q8D3×D412yes—
429C2×D4—32no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—31no—
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—124no—
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—21no—
471C4⋊C4—42no—
472C4⋊C4—62no—
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#23512yes—
484C22⋊C4—64no—
485C22⋊C4—64no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—241no—
489C42—21no—
490C42—21no—
491C42—32no—
492C42—32no—
493C42—42no—
494C42—61no—
495C42—61no—
496C42—122no—
497C42—241no—
498C22×C3C2×#12111yescharacteristic
499C22×C3#54611yescharacteristic
500C22×C3#54811yescharacteristic
501C22×C3C2×#13511yescharacteristic
502C22×C3#57511yescharacteristic
503C22×C3#57711yescharacteristic
504C22×C3#57811yescharacteristic
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—42no—
513C3×C4—42no—
514C3×C4—41no—
515C3×C4—41no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4#52911yescharacteristic
519C3×C4#54211yescharacteristic
520C3×C4#54611yescharacteristic
521C3×C4#54711yescharacteristic
522Dic3—22no—
523Dic3—22no—
524Dic3—22no—
525Dic3—42no—
526Dic3—41no—
527Dic3—81no—
528Dic3—81no—
529C23—62no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C2×D3×D411yescharacteristic
537Q8—22no—
538Q8—22no—
539Q8—41no—
540Q8—41no—
541Q8—62no—
542Q8—62no—
543Q8—121no—
544Q8—121no—
545D4—62no—
546D4—62no—
547D4—121no—
548D4—121no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—32no—
558C2×C4—32no—
559C2×C4—32no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—62no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—61no—
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×#23511yescharacteristic
590C2×C4C2×D3×D411yescharacteristic
591C2×C4#100411yescharacteristic
592C2×C4#100711yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C8—41no—
596C8—41no—
597C8—121no—
598C8—121no—
599C2×C3#242711yescharacteristic
600C2×C3#303511yescharacteristic
601C2×C3#304611yescharacteristic
602C2×C3#329611yescharacteristic
603C2×C3#332711yescharacteristic
604C2×C3#336411yescharacteristic
605C2×C3#337111yescharacteristic
606D3—82no—
607C22—122no—
608C22—122no—
609C22—122no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22#549011yescharacteristic
615C22D3×#12111yescharacteristic
616C22#563311yescharacteristic
617C22D3×#13511yescharacteristic
618C22#577011yescharacteristic
619C22#577711yescharacteristic
620C22#577911yescharacteristic
621C4—41no—
622C4—42no—
623C4—41no—
624C4—42no—
625C4—62no—
626C4—62no—
627C4—62no—
628C4—81no—
629C4—81no—
630C4—121no—
631C4—122no—
632C4—241no—
633C4—241no—
634C4#550711yescharacteristic
635C4#552811yescharacteristic
636C4#561011yescharacteristic
637C4#564011yescharacteristic
638C3#2532911yescharacteristic, complemented, Sylow
639C2—242no—
640C2#7389511yescharacteristic
641C2#7884511yescharacteristic
642C2#7897111yescharacteristic
643C2#8059711yescharacteristic
644C2#8060311yescharacteristic
645C2#8121611yescharacteristic
646C2#8121811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1805D3
hypocenterC3#25329
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #74183C2semidirect
5. C3×#1805C2semidirect
6. #71632C2semidirect
8. #70451C2semidirect
9. #70285C2semidirect
10. #70233C2semidirect
14. #68801C2semidirect
15. #68781C2semidirect
74. #1805D3semidirect
638. C3#25329semidirect

Maximal subgroups

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

subgroupindex
#824662
C2×#47982
C2×#48002
#741832
C3×#18052
#716322
#704822
#704512
#702852
#702332
#695572
#688962
#688922
#688012
#687812
#253293