Finite Group Databasebeta

← #10845189

Subgroups — #10845189

Order 768.

824 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2566C211yesmaximal, characteristic
2C2×#5069C211yesmaximal, characteristic, complemented
3C2×#5019C211yesmaximal, characteristic
4#73651C211yesmaximal, characteristic, complemented
5#71738C211yesmaximal, characteristic, complemented
6#71609C211yesmaximal, characteristic
7#69660C211yesmaximal, characteristic
8#69660C211yesmaximal, characteristic
9#69651C211yesmaximal, characteristic
10#69651C211yesmaximal, characteristic
11#69589C211yesmaximal, characteristic, complemented
12#69588C211yesmaximal, characteristic, complemented
13#69560C211yesmaximal, characteristic, complemented
14#69559C211yesmaximal, characteristic, complemented
15#69485C211yesmaximal, characteristic, complemented
16#25331—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C22×#214C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C2×#869C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C4×Dic6C2211yescharacteristic
24C3×#396—21no—
25C3×#396—21no—
26C3×#390C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#385—21no—
29C3×#385—21no—
30C3×#384C2211yescharacteristic
31C3×#382C2211yescharacteristic
32C3×C4×C4⋊C4C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5069—24no—
36#5019—24no—
37Dic3×C4⋊C4—21no—
38#4972—21no—
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41C2×#810C2211yescharacteristic
42C2×#810C2211yescharacteristic
43C2×#810C2211yescharacteristic
44C2×#810C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4929C2211yescharacteristic
47#4929C2211yescharacteristic
48C2×C4⋊(C3⋊C8)C2211yescharacteristic
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50C2×C4⋊(C3⋊C8)C2211yescharacteristic
51C2×#803C2211yescharacteristic
52#4654—21no—
53#4654—21no—
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4524C2211yescharacteristic
61#4524C2211yescharacteristic
62C2×#491—31no—
63C2×#461—31no—
64#2566D311yescharacteristic
65#2365—31no—
66#2074—31no—
67#2072—31no—
68#2055—31no—
69#2055—31no—
70#1989—31no—
71#1940—31no—
72#1880—31no—
73#1880—31no—
74#1805—31no—
75#1803—31no—
76#1762—31no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C3×#125—22no—
82C3×#125—22no—
83C3×C4×D4—22no—
84C3×C4×D4—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
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×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#869—22no—
122#869D414yes—
123C4×Dic6—22no—
124C4×Dic6—22no—
125C4×Dic6—41no—
126C4×Dic6—41no—
127C3×#95—21no—
128C3×#95—21no—
129C3×#95D411yescharacteristic
130C3×#95D411yescharacteristic
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#810—22no—
139#810—22no—
140#810—22no—
141#810—22no—
142#810—22no—
143#810—22no—
144C4⋊(C3⋊C8)—22no—
145C4⋊(C3⋊C8)—41no—
146C4⋊(C3⋊C8)—41no—
147#803—22no—
148C22×SD16—31no—
149C22×SD16—31no—
150C2×C4⋊Q8—31no—
151C2×#125C2×D311yescharacteristic
152C2×C4×Q8—31no—
153C2×C4×Q8—31no—
154C2×C4×D4C2×D311yescharacteristic
155#491—64no—
156#461—64no—
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—
169#396—21no—
170#396—21no—
171#391—31no—
172#391—31no—
173#390C2×D311yescharacteristic
174#390C2×D311yescharacteristic
175#389—61no—
176#387—61no—
177#387—61no—
178#385—21no—
179#385—21no—
180#384C2×D311yescharacteristic
181#382C2×D311yescharacteristic
182C4×C4⋊C4—31no—
183C4×C4⋊C4—61no—
184C4×C4⋊C4C2×D311yescharacteristic
185#340—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#336—31no—
192#336—31no—
193C24×C3—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—42no—
204C22×C3×C4—81no—
205C22×C3×C4—81no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2×D411yescharacteristic
211C22×C3×C4C2×D411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6C2×D412yes—
228C2×Dic6C2×D412yes—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—41no—
237C3×C4⋊C4—42no—
238C3×C4⋊C4—42no—
239C3×C4⋊C4—81no—
240C3×C4⋊C4—81no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—42no—
244C3×C22⋊C4—81no—
245C3×C22⋊C4—81no—
246C3×C42—21no—
247C3×C42—21no—
248C3×C42—21no—
249C3×C42—21no—
250C3×C42C2×D412yes—
251C3×C42#4112yes—
252#214—44no—
253#214—44no—
254C4⋊Dic3—24no—
255C4⋊Dic3—24no—
256C4⋊Dic3—21no—
257C4⋊Dic3—21no—
258C4⋊Dic3—21no—
259C4⋊Dic3#4112yes—
260Dic3⋊C4—42no—
261Dic3⋊C4—42no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264C4×Dic3—42no—
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⋊C8—21no—
273C2×C3⋊C8—21no—
274C2×C3⋊C8—21no—
275C2×C3⋊C8—21no—
276C2×C3⋊C8—21no—
277C22×Q8—31no—
278C22×Q8—31no—
279C22×D4C22×D311yescharacteristic
280C23×C4—21no—
281C2×SD16—62no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—62no—
285C2×SD16—121no—
286C2×SD16—121no—
287C22×C8—31no—
288C22×C8—31no—
289C22×C8—31no—
290C22×C8—31no—
291C4⋊Q8—34no—
292C4⋊Q8—62no—
293#125—22no—
294#125—22no—
295C4×Q8—62no—
296C4×Q8—62no—
297C4×Q8—121no—
298C4×Q8—121no—
299C4×D4—22no—
300C4×D4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—31no—
306C2×C4⋊C4—31no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4—61no—
310C2×C4⋊C4—61no—
311C2×C4⋊C4#6711yescharacteristic
312C2×C4⋊C4#6711yescharacteristic
313C2×C4⋊C4C22×D311yescharacteristic
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C22⋊C4—21no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C42—61no—
322C2×C42—61no—
323C2×C42—61no—
324C2×C42C22×D311yescharacteristic
325C2×C42C22×D311yescharacteristic
326C2×C42C22×D311yescharacteristic
327C4⋊C8—62no—
328C4⋊C8—121no—
329C4⋊C8—121no—
330#103—62no—
331#103—62no—
332#103—62no—
333#103—62no—
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338#97—62no—
339#95—21no—
340#95—21no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95#6711yescharacteristic
347#95#6711yescharacteristic
348C23×C3—22no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3C22×D411yescharacteristic
356C3×D4—22no—
357C3×D4—22no—
358C3×D4—41no—
359C3×D4—41no—
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—21no—
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—41no—
384C2×C3×C4—84no—
385C2×C3×C4#12112yes—
386C2×C3×C4#12112yes—
387C2×C3×C4#13712yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390C2×C3×C4C2×#4111yescharacteristic
391C2×C3×C4C2×#4111yescharacteristic
392C2×C3×C4H(5,𝔽2)11yescharacteristic
393C2×C3×C42−1+411yescharacteristic
394C2×Dic3—22no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—41no—
406Dic6—22no—
407Dic6—22no—
408Dic6—22no—
409Dic6—22no—
410Dic6—41no—
411Dic6—41no—
412Dic6—41no—
413Dic6—41no—
414C3⋊C8—41no—
415C3⋊C8—41no—
416C3⋊C8—41no—
417C3⋊C8—41no—
418C24—21no—
419C2×Q8—32no—
420C2×Q8—32no—
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×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4#23912yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—81no—
446C22×C4—81no—
447C22×C4C2×#6711yescharacteristic
448C22×C4C2×#6711yescharacteristic
449C22×C4C2×#6711yescharacteristic
450C22×C4C2×#6711yescharacteristic
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C23×D311yescharacteristic
454SD16—124no—
455SD16—124no—
456C2×C8—61no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—32no—
476C4⋊C4—41no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—64no—
480C4⋊C4—64no—
481C4⋊C4—61no—
482C4⋊C4—61no—
483C4⋊C4—61no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—122no—
487C4⋊C4—122no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—122no—
498C42—122no—
499C42—122no—
500C42D3×D412yes—
501C42#23712yes—
502C22×C3—42no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3—81no—
509C22×C3C2×#12111yescharacteristic
510C22×C3#54611yescharacteristic
511C22×C3#54811yescharacteristic
512C22×C3C2×#13711yescharacteristic
513C22×C3#57511yescharacteristic
514C22×C3#57711yescharacteristic
515C22×C3#57711yescharacteristic
516C3×C4—22no—
517C3×C4—22no—
518C3×C4—22no—
519C3×C4—41no—
520C3×C4—41no—
521C3×C4—41no—
522C3×C4—81no—
523C3×C4—81no—
524C3×C4—81no—
525C3×C4#52911yescharacteristic
526C3×C4#54211yescharacteristic
527C3×C4#54611yescharacteristic
528C3×C4#54711yescharacteristic
529Dic3—42no—
530Dic3—42no—
531Dic3—42no—
532Dic3—81no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C22×#6711yescharacteristic
541Q8—62no—
542Q8—62no—
543Q8—62no—
544Q8—62no—
545Q8—121no—
546Q8—121no—
547Q8—121no—
548Q8—121no—
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
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—21no—
567C2×C4—21no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—84no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4#93812yes—
591C2×C4#93812yes—
592C2×C4#95112yes—
593C2×C4C2×D3×D411yescharacteristic
594C2×C4C2×#23711yescharacteristic
595C2×C4C2×#23911yescharacteristic
596C2×C4#100711yescharacteristic
597C2×C4D3×#4111yescharacteristic
598C2×C4#100911yescharacteristic
599C8—121no—
600C8—121no—
601C8—121no—
602C8—121no—
603C2×C3—82no—
604C2×C3#242711yescharacteristic
605C2×C3#303711yescharacteristic
606C2×C3#304311yescharacteristic
607C2×C3#330711yescharacteristic
608C2×C3#332011yescharacteristic
609C2×C3#337211yescharacteristic
610C2×C3#337611yescharacteristic
611C22—42no—
612C22—42no—
613C22—42no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22C2×#93811yescharacteristic
619C22#581711yescharacteristic
620C22#582211yescharacteristic
621C22C2×#95111yescharacteristic
622C22#582911yescharacteristic
623C22#583611yescharacteristic
624C22#583611yescharacteristic
625C4—22no—
626C4—22no—
627C4—22no—
628C4—41no—
629C4—41no—
630C4—41no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—122no—
635C4—122no—
636C4—122no—
637C4—241no—
638C4#562411yescharacteristic
639C4#563011yescharacteristic
640C4#563111yescharacteristic
641C4#564311yescharacteristic
642C3#2533111yescharacteristic, complemented, Sylow
643C2—82no—
644C2#8275611yescharacteristic
645C2#8331411yescharacteristic
646C2#8339111yescharacteristic
647C2#8361511yescharacteristic
648C2#8362011yescharacteristic
649C2#8364411yescharacteristic
650C2#8364511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2566C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2566D3
hypocenterC3#25331
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
2. C2×#5069C2semidirect
4. #73651C2semidirect
5. #71738C2semidirect
11. #69589C2semidirect
12. #69588C2semidirect
13. #69560C2semidirect
14. #69559C2semidirect
15. #69485C2semidirect
642. C3#25331semidirect

Maximal subgroups

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

subgroupindex
C3×#25662
C2×#50692
C2×#50192
#736512
#717382
#716092
#696602
#696512
#695892
#695882
#695602
#695592
#694852
#253313