Finite Group Databasebeta

← #10844900

Subgroups — #10844900

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2580C211yesmaximal, characteristic
2#82479C211yesmaximal, characteristic, complemented
3C2×#5024C211yesmaximal, characteristic, complemented
4C2×#5019C211yesmaximal, characteristic
5#71609C211yesmaximal, characteristic
6#69651C211yesmaximal, characteristic
7#69651C211yesmaximal, characteristic
8#69649C211yesmaximal, characteristic
9#69646C211yesmaximal, characteristic
10#69635C211yesmaximal, characteristic, complemented
11#69634C211yesmaximal, characteristic, complemented
12#69616C211yesmaximal, characteristic, complemented
13#69614C211yesmaximal, characteristic, complemented
14#69552C211yesmaximal, characteristic, complemented
15#69540C211yesmaximal, characteristic, complemented
16#25308—31nomaximal, Sylow
17C2×C3×C4×Q8C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C22×#214C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C3×#399—21no—
23C3×#393—21no—
24C3×#390C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#385—21no—
29C3×#385—21no—
30C3×#382C2211yescharacteristic
31#5229—21no—
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34#5024—24no—
35#5019—24no—
36#4986—21no—
37#4986—21no—
38#4977—21no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C2×#808C2211yescharacteristic
46C2×#807C2211yescharacteristic
47C4×C4⋊Dic3C2211yescharacteristic
48#4929C2211yescharacteristic
49#4929C2211yescharacteristic
50#4929C2211yescharacteristic
51#4929C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54C2×C4⋊(C3⋊C8)C2211yescharacteristic
55C2×#803C2211yescharacteristic
56C2×#803C2211yescharacteristic
57C2×C4×C3⋊C8C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#461—31no—
63C2×#446—31no—
64#2580D311yescharacteristic
65#2568—31no—
66#2068—31no—
67#2066—31no—
68#2055—31no—
69#2055—31no—
70#1973—31no—
71#1968—31no—
72#1960—31no—
73#1949—31no—
74#1935—31no—
75#1933—31no—
76#1762—31no—
77C22×C3×Q8D411yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×C3×C4—21no—
80C22×Dic6C2311yescharacteristic
81C22×Dic6C2311yescharacteristic
82C3×C4×Q8—22no—
83C3×C4×Q8—22no—
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4D411yescharacteristic
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#214—22no—
103C2×#214—22no—
104C2×#214—22no—
105C2×#214—22no—
106C2×#214—41no—
107C2×#214—41no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117#869—22no—
118#869—22no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123#832—22no—
124#832—22no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#810—22no—
135#810—22no—
136#810—22no—
137#810—22no—
138#810—41no—
139#810—41no—
140#808—41no—
141#807—41no—
142C4⋊(C3⋊C8)—22no—
143C4⋊(C3⋊C8)—22no—
144C4⋊(C3⋊C8)D414yes—
145#803—22no—
146#803—22no—
147C4×C3⋊C8—22no—
148C22×SD16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8—31no—
151C2×C4×Q8C2×D311yescharacteristic
152C2×C4×D4C2×D311yescharacteristic
153#461—64no—
154#446—64no—
155C2×#107—31no—
156C2×#106—31no—
157C2×C4⋊C8—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×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#400—61no—
172#400—61no—
173#399—21no—
174#393—21no—
175#391—31no—
176#391—31no—
177#391—31no—
178#391—31no—
179#391—61no—
180#390C2×D311yescharacteristic
181#390C2×D311yescharacteristic
182#389C2×D311yescharacteristic
183#389C2×D311yescharacteristic
184#385—21no—
185#385—21no—
186#382—61no—
187#382C2×D311yescharacteristic
188C4×C4⋊C4—31no—
189#340—31no—
190#340—31no—
191#340—31no—
192#340—31no—
193C24×C3—21no—
194C2×C3×Q8—22no—
195C2×C3×Q8—22no—
196C2×C3×Q8—41no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4#4112yes—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—81no—
208C22×C3×C4—81no—
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2411yescharacteristic
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—21no—
228C2×Dic6—21no—
229C2×Dic6—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C3×C4⋊C4—22no—
233C3×C4⋊C4—22no—
234C3×C4⋊C4—22no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—81no—
242C3×C4⋊C4—81no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—42no—
245C3×C22⋊C4—42no—
246C3×C22⋊C4—81no—
247C3×C22⋊C4—81no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42—21no—
251C3×C42—21no—
252C3×C42C2×D412yes—
253C3×C42#4112yes—
254#214—44no—
255#214—44no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3—21no—
260C4⋊Dic3—42no—
261C4⋊Dic3—42no—
262C4×Dic3—42no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8C2×D412yes—
273C2×C3⋊C8C2×D412yes—
274C2×C3⋊C8#4112yes—
275C22×Q8—31no—
276C22×Q8—31no—
277C22×Q8#6711yescharacteristic
278C22×D4C22×D311yescharacteristic
279C23×C4—21no—
280C2×SD16—62no—
281C2×SD16—62no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—121no—
285C2×SD16—121no—
286C22×C8—31no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290C4⋊Q8—62no—
291C4⋊Q8—62no—
292C4×Q8—22no—
293C4×Q8—22no—
294C4×D4—22no—
295C4×D4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4#6711yescharacteristic
303C2×C4⋊C4#6711yescharacteristic
304C2×C4⋊C4#6711yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315C2×C42C22×D311yescharacteristic
316C2×C42C22×D311yescharacteristic
317#107—121no—
318#106—121no—
319C4⋊C8—34no—
320C4⋊C8—62no—
321C4⋊C8—62no—
322#103—62no—
323#103—62no—
324#103—62no—
325#103—62no—
326#103—121no—
327#103—121no—
328#102—62no—
329#102—62no—
330#97—62no—
331#97—62no—
332C4×C8—62no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—61no—
338#95—61no—
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—44no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—41no—
358C3×D4—41no—
359C2×C3×C4—21no—
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—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
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—84no—
384C2×C3×C4#12112yes—
385C2×C3×C4#12112yes—
386C2×C3×C4#13712yes—
387C2×C3×C4C22×D411yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4H(5,𝔽2)11yescharacteristic
392C2×C3×C42−1+411yescharacteristic
393C2×Dic3—41no—
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—41no—
405Dic6—41no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412Dic6—41no—
413C3⋊C8—22no—
414C3⋊C8—22no—
415C3⋊C8—22no—
416C3⋊C8—41no—
417C24—21no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—61no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4#23912yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—42no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—81no—
448C22×C4—81no—
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C23×D311yescharacteristic
456SD16—124no—
457SD16—124no—
458C2×C8—32no—
459C2×C8—32no—
460C2×C8—32no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C2×C8—61no—
469C2×C8—61no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—122no—
486C4⋊C4—122no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—122no—
497C42D3×D412yes—
498C42#23712yes—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3C2×#12111yescharacteristic
507C22×C3#54811yescharacteristic
508C22×C3#55011yescharacteristic
509C22×C3C2×#13711yescharacteristic
510C22×C3#57511yescharacteristic
511C22×C3#57711yescharacteristic
512C22×C3#57711yescharacteristic
513C3×C4—22no—
514C3×C4—22no—
515C3×C4—22no—
516C3×C4—41no—
517C3×C4—41no—
518C3×C4—41no—
519C3×C4—81no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4#52911yescharacteristic
523C3×C4#54211yescharacteristic
524C3×C4#54611yescharacteristic
525C3×C4#54811yescharacteristic
526Dic3—81no—
527Dic3—81no—
528Dic3—81no—
529Dic3—81no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23C22×#6711yescharacteristic
538Q8—44no—
539Q8—121no—
540Q8—121no—
541Q8—121no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545Q8—121no—
546Q8—121no—
547D4—22no—
548D4—22no—
549D4—41no—
550D4—41no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—84no—
576C2×C4—121no—
577C2×C4—121no—
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#93912yes—
589C2×C4#93912yes—
590C2×C4#95112yes—
591C2×C4C2×D3×D411yescharacteristic
592C2×C4C2×#23711yescharacteristic
593C2×C4C2×#23911yescharacteristic
594C2×C4#100711yescharacteristic
595C2×C4D3×#4111yescharacteristic
596C2×C4#100911yescharacteristic
597C8—62no—
598C8—62no—
599C8—62no—
600C8—121no—
601C2×C3—82no—
602C2×C3#242611yescharacteristic
603C2×C3#305611yescharacteristic
604C2×C3#305711yescharacteristic
605C2×C3#337211yescharacteristic
606C2×C3#337611yescharacteristic
607C2×C3#340011yescharacteristic
608C2×C3#340111yescharacteristic
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×#93911yescharacteristic
617C22#581711yescharacteristic
618C22#582211yescharacteristic
619C22C2×#95111yescharacteristic
620C22#583411yescharacteristic
621C22#583611yescharacteristic
622C22#583611yescharacteristic
623C4—22no—
624C4—22no—
625C4—22no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—241no—
633C4—241no—
634C4—241no—
635C4—241no—
636C4#562411yescharacteristic
637C4#563011yescharacteristic
638C4#563311yescharacteristic
639C4#564311yescharacteristic
640C3#2530811yescharacteristic, complemented, Sylow
641C2—82no—
642C2#8275511yescharacteristic
643C2#8292511yescharacteristic
644C2#8337711yescharacteristic
645C2#8364411yescharacteristic
646C2#8364511yescharacteristic
647C2#8371211yescharacteristic
648C2#8372111yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2580C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2580D3
hypocenterC3#25308
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
2. #82479C2semidirect
3. C2×#5024C2semidirect
10. #69635C2semidirect
11. #69634C2semidirect
12. #69616C2semidirect
13. #69614C2semidirect
14. #69552C2semidirect
15. #69540C2semidirect
640. C3#25308semidirect

Maximal subgroups

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

subgroupindex
C3×#25802
#824792
C2×#50242
C2×#50192
#716092
#696512
#696492
#696462
#696352
#696342
#696162
#696142
#695522
#695402
#253083