Finite Group Databasebeta

← #10767446

Subgroups — #10767446

Order 768.

990 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2#82913C212yesmaximal, complemented
3#82346C211yesmaximal, characteristic, complemented
4#71899C211yesmaximal, characteristic, complemented
5#71898C211yesmaximal, characteristic, complemented
6#71883C212yesmaximal, complemented
7#71882C212yesmaximal, complemented
8#71875C211yesmaximal, characteristic, complemented
9#71874C211yesmaximal, characteristic, complemented
10#71707C212yesmaximal, complemented
11#71600C211yesmaximal, characteristic, complemented
12#18121—31nomaximal, Sylow
13C3×#544C2211yescharacteristic
14C3×#542—22no—
15C3×#540—22no—
16C3×#538—21no—
17C3×#538C2212yes—
18C3×#537C2211yescharacteristic
19C3×#535—21no—
20C3×#532—22no—
21C3×#529—22no—
22C3×#525—21no—
23C3×#525C2212yes—
24C3×C4×#41—21no—
25C2×C3×C4×D4C2211yescharacteristic
26C2×#926C2211yescharacteristic
27#5233—22no—
28#5222—21no—
29C4×#217—21no—
30C4×#217C2211yescharacteristic
31C2×#830C2212yes—
32#5045—22no—
33#5044C2212yes—
34#5043C2212yes—
35#5035—22no—
36#5034C2212yes—
37#5033C2212yes—
38#4998—22no—
39#4986C2211yescharacteristic
40#4981C2211yescharacteristic
41#4953C2211yescharacteristic
42#4952—21no—
43#4949C2211yescharacteristic
44C4×Dic3⋊C4—21no—
45#4538C2212yes—
46#4538C2212yes—
47#4537C2212yes—
48#4537C2212yes—
49#4528C2211yescharacteristic
50#4528C2211yescharacteristic
51#4525C2212yes—
52#3460D311yescharacteristic
53#3091—32no—
54#2395—31no—
55#1618—32no—
56#1617—31no—
57#1616—31no—
58#1612—32no—
59#1611—31no—
60#1610—31no—
61#1501—32no—
62#1459—31no—
63C2×C3×#41—21no—
64C2×C3×#41—21no—
65C2×C3×#41D412yes—
66C22×C3×D4D411yescharacteristic
67C23×C3×C4D411yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C3×C4⋊Q8—22no—
70C3×#127—22no—
71C3×#126—22no—
72C3×#126—22no—
73C3×#125—22no—
74C3×#124—22no—
75C3×#124—22no—
76C3×#124—22no—
77C3×#123—22no—
78C3×#123—22no—
79C3×#123—22no—
80C3×#123—22no—
81C3×C22⋊Q8—21no—
82C3×C22⋊Q8—21no—
83C3×C22⋊Q8—22no—
84C3×C22⋊Q8D412yes—
85C3×C22⋊Q8C2312yes—
86C3×#121—21no—
87C3×#121—22no—
88C3×#121—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—21no—
93C3×#121D412yes—
94C3×#121C2312yes—
95C3×#120—22no—
96C3×#120—22no—
97C3×C4×Q8—21no—
98C3×C4×Q8—21no—
99C3×C4×D4—21no—
100C3×C4×D4—21no—
101C3×C4×D4—22no—
102C3×C4×D4—22no—
103C3×C4×D4—22no—
104C3×C4×D4—22no—
105C3×C4×D4—22no—
106C3×C4×D4—22no—
107C3×#117—22no—
108C3×#117—21no—
109C3×#117—21no—
110C3×#117D412yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C22⋊C4D411yescharacteristic
113C2×C3×C22⋊C4C2311yescharacteristic
114C2×C3×C42—22no—
115C2×C3×C42D411yescharacteristic
116C2×#217—21no—
117C2×#217—21no—
118C2×#217—21no—
119C2×#217—21no—
120#926—22no—
121#926—22no—
122C2×C4⋊Dic3C2311yescharacteristic
123C2×C4⋊Dic3C2311yescharacteristic
124C2×Dic3⋊C4—21no—
125C2×Dic3⋊C4—21no—
126C2×C4×Dic3—21no—
127C2×C4×Dic3—21no—
128C22×C3⋊C8—22no—
129#835—22no—
130#835—22no—
131#832—22no—
132#832—22no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95C2311yescharacteristic
138C3⋊#95C2311yescharacteristic
139#830—22no—
140#830C2312yes—
141#830C2312yes—
142#808—22no—
143#808—22no—
144#807—22no—
145#807—22no—
146#544C2×D311yescharacteristic
147#542—22no—
148#540—22no—
149#538—21no—
150#538C2×D312yes—
151#537C2×D311yescharacteristic
152#535—21no—
153#532—22no—
154#529—22no—
155#525—21no—
156#525C2×D312yes—
157C4×#41—21no—
158C2×C4×D4C2×D311yescharacteristic
159C2×#117—31no—
160#485—62no—
161#484—32no—
162#483—32no—
163#482—62no—
164#481—32no—
165#480—32no—
166#427—62no—
167#416—62no—
168C2×C22⋊C8—32no—
169#400—31no—
170#399—31no—
171#398—31no—
172#397—31no—
173#397—61no—
174#394—61no—
175C4×C4⋊C4—61no—
176C4×C22⋊C4—31no—
177C4×C22⋊C4—61no—
178#328—32no—
179#328—32no—
180#327—32no—
181#327—32no—
182#324—31no—
183#324—31no—
184#323—32no—
185C24×C3C2×D411yescharacteristic
186C24×C3C2×D411yescharacteristic
187C3×#41—42no—
188C3×#41—44no—
189C3×#41—42no—
190C3×#41—82no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8C2×D412yes—
194C2×C3×D4—21no—
195C2×C3×D4—22no—
196C2×C3×D4—22no—
197C2×C3×D4—22no—
198C2×C3×D4—22no—
199C2×C3×D4—21no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—22no—
203C2×C3×D4—41no—
204C2×C3×D4C2×D412yes—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—22no—
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—42no—
211C22×C3×C4—42no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D412yes—
214C22×C3×C4C2×D411yescharacteristic
215C22×C3×C4C2×D411yescharacteristic
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D412yes—
218C22×C3×C4C2411yescharacteristic
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—22no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—22no—
231C3×C4⋊C4—22no—
232C3×C4⋊C4C2×D412yes—
233C3×C4⋊C4C2×D412yes—
234C3×C22⋊C4—22no—
235C3×C22⋊C4—22no—
236C3×C22⋊C4—22no—
237C3×C22⋊C4—22no—
238C3×C22⋊C4—22no—
239C3×C22⋊C4—22no—
240C3×C22⋊C4—22no—
241C3×C22⋊C4—22no—
242C3×C22⋊C4—22no—
243C3×C22⋊C4—22no—
244C3×C22⋊C4—22no—
245C3×C42—22no—
246C3×C42—22no—
247C3×C42—21no—
248C3×C42—22no—
249C3×C42—21no—
250#217—42no—
251#217—42no—
252#217—42no—
253#217—42no—
254C4⋊Dic3—22no—
255C4⋊Dic3—22no—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258C4×Dic3—42no—
259C4×Dic3—41no—
260C4×Dic3—41no—
261C2×C3⋊C8—22no—
262C2×C3⋊C8—22no—
263C2×C3⋊C8—44no—
264C2×#41—21no—
265C2×#41—21no—
266C2×#41#6712yes—
267C22×D4#6711yescharacteristic
268C23×C4#6711yescharacteristic
269C23×C4C22×D311yescharacteristic
270C22×C8—62no—
271C4⋊Q8—22no—
272#127—22no—
273#126—22no—
274#126—22no—
275#125—22no—
276#124—22no—
277#124—22no—
278#124—22no—
279#123—22no—
280#123—22no—
281#123—22no—
282#123—22no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8—22no—
286C22⋊Q8#6712yes—
287C22⋊Q8C22×D312yes—
288#121—21no—
289#121—22no—
290#121—22no—
291#121—22no—
292#121—22no—
293#121—22no—
294#121—21no—
295#121#6712yes—
296#121C22×D312yes—
297#120—22no—
298#120—22no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—22no—
308C4×D4—22no—
309#117—22no—
310#117—21no—
311#117—21no—
312#117—62no—
313#117—62no—
314#117#6712yes—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—31no—
317C2×C4⋊C4—61no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4C22×D311yescharacteristic
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4#6711yescharacteristic
325C2×C22⋊C4C22×D311yescharacteristic
326C2×C42—22no—
327C2×C42—61no—
328C2×C42—61no—
329C2×C42#6711yescharacteristic
330#107—62no—
331#107—62no—
332#106—62no—
333#106—62no—
334#103—62no—
335#103—62no—
336#102—62no—
337#102—62no—
338C22⋊C8—32no—
339C22⋊C8—32no—
340C22⋊C8—62no—
341#95—31no—
342#95—31no—
343#95—61no—
344#95—61no—
345#95—61no—
346#95—61no—
347C23×C3—21no—
348C23×C3—21no—
349C23×C3—21no—
350C23×C3—22no—
351C23×C3—21no—
352C23×C3—21no—
353C23×C3—41no—
354C23×C3—41no—
355C23×C3—41no—
356C23×C3—41no—
357C23×C3#12012yes—
358C23×C3C22×D411yescharacteristic
359C3×Q8—42no—
360C3×Q8—42no—
361C3×Q8—81no—
362C3×D4—44no—
363C3×D4—42no—
364C3×D4—42no—
365C3×D4—42no—
366C3×D4—42no—
367C3×D4—82no—
368C3×D4—82no—
369C3×D4—82no—
370C3×D4—81no—
371C3×D4—82no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—24no—
375C2×C3×C4—22no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—21no—
381C2×C3×C4—42no—
382C2×C3×C4—41no—
383C2×C3×C4—44no—
384C2×C3×C4—42no—
385C2×C3×C4—44no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—81no—
389C2×C3×C4—82no—
390C2×C3×C4#12011yescharacteristic
391C2×C3×C4#12011yescharacteristic
392C2×C3×C4#13512yes—
393C2×C3×C4C22×D412yes—
394C2×Dic3—22no—
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—
405C2×Dic3—41no—
406C3⋊C8—42no—
407C3⋊C8—42no—
408C24C2×#6711yescharacteristic
409C24C2×#6711yescharacteristic
410#41—42no—
411#41—44no—
412#41—42no—
413#41—82no—
414C2×Q8—21no—
415C2×Q8—21no—
416C2×Q8C2×#6712yes—
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—21no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—41no—
427C2×D4C2×#6712yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4C2×#6711yescharacteristic
440C22×C4C2×#6711yescharacteristic
441C22×C4C2×#6712yes—
442C22×C4C2×#6711yescharacteristic
443C22×C4C2×#6711yescharacteristic
444C22×C4C2×#6712yes—
445C22×C4C23×D311yescharacteristic
446C2×C8—62no—
447C2×C8—62no—
448C2×C8—124no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—62no—
459C4⋊C4—62no—
460C4⋊C4—122no—
461C4⋊C4—122no—
462C4⋊C4C2×#6712yes—
463C4⋊C4C2×#6712yes—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—122no—
476C22⋊C4—122no—
477C22⋊C4—122no—
478C22⋊C4—122no—
479C42—22no—
480C42—22no—
481C42—21no—
482C42—21no—
483C42—22no—
484C42—122no—
485C42—121no—
486C42—121no—
487C22×C3—21no—
488C22×C3—21no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—41no—
495C22×C3—42no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—42no—
499C22×C3—81no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3C2×#12011yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3#57511yescharacteristic
505C3×C4—22no—
506C3×C4—22no—
507C3×C4—21no—
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—42no—
511C3×C4—41no—
512C3×C4—42no—
513C3×C4#45812yes—
514Dic3—42no—
515Dic3—81no—
516Dic3—81no—
517Dic3—81no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—22no—
523C23—21no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23#95312yes—
529C23C22×#6711yescharacteristic
530Q8—42no—
531Q8—42no—
532Q8—81no—
533D4—44no—
534D4—42no—
535D4—42no—
536D4—42no—
537D4—42no—
538D4—82no—
539D4—82no—
540D4—81no—
541D4—82no—
542D4—82no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—62no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4#95012yes—
574C2×C4#95311yescharacteristic
575C2×C4#95311yescharacteristic
576C2×C4C22×#6712yes—
577C8—122no—
578C8—122no—
579C2×C3—21no—
580C2×C3—21no—
581C2×C3—42no—
582C2×C3—82no—
583C2×C3—81no—
584C2×C3#300211yescharacteristic
585C2×C3#301811yescharacteristic
586C2×C3C2×#45811yescharacteristic
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—41no—
595C22—41no—
596C22—42no—
597C22—42no—
598C22—41no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C2×#95011yescharacteristic
603C22#582211yescharacteristic
604C22C2×#95311yescharacteristic
605C4—22no—
606C4—22no—
607C4—21no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—122no—
614C4—241no—
615C4—241no—
616C4—241no—
617C4#524312yes—
618C3#1812111yescharacteristic, complemented, Sylow
619C2—21no—
620C2—21no—
621C2—42no—
622C2—81no—
623C2—82no—
624C2#8408011yescharacteristic
625C2C2×#524311yescharacteristic
626C2#8412111yescharacteristic
subgroupG/H
centerC2×C4#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3460C2
socleC22×C3C2×#120
hypercenter#3460D3
hypocenterC3#18121
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #3460

Complements

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

NH ≅ G/Ntype
2. #82913C2semidirect
3. #82346C2semidirect
4. #71899C2semidirect
5. #71898C2semidirect
6. #71883C2semidirect
7. #71882C2semidirect
8. #71875C2semidirect
9. #71874C2semidirect
10. #71707C2semidirect
11. #71600C2semidirect
618. C3#18121semidirect

Maximal subgroups

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

subgroupindex
C3×#34602
#829132
#823462
#718992
#718982
#718832
#718822
#718752
#718742
#717072
#716002
#181213