Finite Group Databasebeta

← #10868447

Subgroups — #10868447

Order 768.

740 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
1#76307C211yesmaximal, characteristic
2#74173C211yesmaximal, characteristic, complemented
3C3×#2092C211yesmaximal, characteristic
4#70627C211yesmaximal, characteristic, complemented
5#70610C211yesmaximal, characteristic, complemented
6#70570C211yesmaximal, characteristic, complemented
7#70570C211yesmaximal, characteristic, complemented
8#70215C211yesmaximal, characteristic
9#70178C211yesmaximal, characteristic
10#70171C211yesmaximal, characteristic
11Dic3×#102C211yesmaximal, characteristic
12#69651C211yesmaximal, characteristic
13#69559C211yesmaximal, characteristic, complemented
14#68685C211yesmaximal, characteristic, complemented
15#68611C211yesmaximal, characteristic, complemented
16#27070—31nomaximal, Sylow
17C2×#936C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C22×#214C2211yescharacteristic
20C22×Dic12C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#107C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#390C2211yescharacteristic
27C3×#382C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4984C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34Dic3×C4⋊C4C2211yescharacteristic
35#4973C2211yescharacteristic
36#4973C2211yescharacteristic
37#4973C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#810C2211yescharacteristic
40C2×#810C2211yescharacteristic
41#4946—21no—
42#4945—21no—
43#4944—21no—
44Dic3×C22⋊C4—21no—
45C4×Dic3⋊C4C2211yescharacteristic
46#4929C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48C3⋊#401—21no—
49#4660—21no—
50#4654—21no—
51#4651—21no—
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2514—31no—
61#2367—31no—
62#2092D311yescharacteristic
63#2055—31no—
64#2042—31no—
65#2015—31no—
66#1969—31no—
67#1927—31no—
68#1927—31no—
69#1921—31no—
70#1916—31no—
71#1878—31no—
72#1877—31no—
73#1805—31no—
74C4×#102—31no—
75C22×C3×D4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6C2311yescharacteristic
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#217—21no—
89C2×#217—21no—
90#936—22no—
91#936—22no—
92D4×Dic3—22no—
93D4×Dic3—41no—
94C2×#214—22no—
95C2×#214—41no—
96C2×#214—41no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4D411yescharacteristic
102C2×Dic3⋊C4D411yescharacteristic
103C2×C4×Dic3—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C2×Dic12—22no—
111C2×Dic12—22no—
112C2×Dic12—41no—
113C4×Dic6—22no—
114C4×Dic6—41no—
115C3×#107—22no—
116C3×#106—41no—
117C3×#102—22no—
118C3×#102—41no—
119C3×#95—21no—
120C3×#95—21no—
121#832—22no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
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⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#816—22no—
136#816—22no—
137C8×Dic3—41no—
138#810—41no—
139#810—41no—
140#810—41no—
141C4⋊(C3⋊C8)—41no—
142C22×Q16—31no—
143C22×SD16—31no—
144C2×#124—31no—
145C2×C4×Q8—31no—
146C2×C4×D4—31no—
147C2×#107C2×D311yescharacteristic
148C2×#106C2×D311yescharacteristic
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#102—31no—
156C2×#102C2×D311yescharacteristic
157C2×#102C2×D311yescharacteristic
158C2×C4×C8—31no—
159#401—61no—
160#400—61no—
161#391—31no—
162#391—31no—
163#391—31no—
164#391—31no—
165#390C2×D311yescharacteristic
166#389—31no—
167#388—61no—
168#388—61no—
169#387—61no—
170#385—61no—
171#383—61no—
172#382C2×D311yescharacteristic
173C4×C4⋊C4—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C22⋊C4—61no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183#336—31no—
184#336C2×D311yescharacteristic
185C24×C3—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4#4112yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3—81no—
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6#4112yes—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8#4112yes—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C22⋊C4—42no—
234C3×C22⋊C4—81no—
235C3×C42—21no—
236C3×C42—21no—
237#217—24no—
238#217—24no—
239#217—42no—
240#214—44no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—22no—
247Dic3⋊C4—41no—
248Dic3⋊C4—42no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
253C4×Dic3—42no—
254C4×Dic3—42no—
255C4×Dic3—81no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262Dic12—44no—
263C22×Q8—31no—
264C22×Q8—31no—
265C22×D4C22×D311yescharacteristic
266C23×C4—61no—
267C2×Q16—62no—
268C2×Q16—62no—
269C2×Q16—121no—
270C2×SD16—62no—
271C2×SD16—121no—
272C2×SD16—121no—
273C22×C8—31no—
274C22×C8—31no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277#124—62no—
278#124—62no—
279C4×Q8—62no—
280C4×Q8—121no—
281C4×D4—62no—
282C4×D4—121no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—61no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—61no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42C22×D311yescharacteristic
303#107—22no—
304#106—41no—
305C4⋊C8—121no—
306#103—62no—
307#103—62no—
308#103—121no—
309#103—121no—
310#103—121no—
311#102—22no—
312#102—41no—
313#102—62no—
314C4×C8—121no—
315#95—21no—
316#95—21no—
317#95—31no—
318#95—31no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95—61no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3—22no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C22×D411yescharacteristic
338C3×D4—22no—
339C3×D4—22no—
340C3×D4—41no—
341C3×D4—41no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—21no—
369C2×Dic3—22no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—84no—
387C2×Dic3#13512yes—
388Dic6—22no—
389Dic6—22no—
390Dic6—41no—
391Dic6—41no—
392Dic6—41no—
393Dic6—41no—
394Dic6—41no—
395Dic6—41no—
396C3×C8—22no—
397C3×C8—41no—
398C3⋊C8—41no—
399C3⋊C8—41no—
400C24—21no—
401C2×Q8—32no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C2×Q8—61no—
408C2×Q8—61no—
409C2×Q8—61no—
410C2×Q8—61no—
411C2×Q8—61no—
412C2×Q8—61no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4#23912yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—241no—
433C22×C4D3×D411yescharacteristic
434C22×C4D3×D411yescharacteristic
435C22×C4C23×D311yescharacteristic
436Q16—124no—
437SD16—124no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8#23712yes—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—32no—
457C4⋊C4—62no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—121no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C22⋊C4—42no—
465C22⋊C4—64no—
466C22⋊C4—64no—
467C22⋊C4—81no—
468C22⋊C4—122no—
469C42—21no—
470C42—21no—
471C42—61no—
472C42—61no—
473C42—61no—
474C42—61no—
475C42—122no—
476C42—122no—
477C42—241no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3#52511yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#55011yescharacteristic
488C22×C3C2×#13511yescharacteristic
489C22×C3#57511yescharacteristic
490C22×C3#57711yescharacteristic
491C22×C3#57811yescharacteristic
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—81no—
495C3×C4—81no—
496C3×C4—81no—
497C3×C4#52911yescharacteristic
498C3×C4#53811yescharacteristic
499C3×C4#55111yescharacteristic
500C3×C4#55311yescharacteristic
501Dic3—22no—
502Dic3—41no—
503Dic3—41no—
504Dic3—42no—
505Dic3—42no—
506Dic3—41no—
507Dic3—81no—
508Dic3—81no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C2×D3×D411yescharacteristic
517Q8—62no—
518Q8—62no—
519Q8—121no—
520Q8—121no—
521Q8—121no—
522Q8—121no—
523Q8—121no—
524Q8—121no—
525D4—22no—
526D4—22no—
527D4—41no—
528D4—41no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—32no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—62no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—62no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—244no—
569C2×C4C2×#23711yescharacteristic
570C2×C4#100411yescharacteristic
571C2×C4C2×#23911yescharacteristic
572C2×C4D3×#4111yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4#100911yescharacteristic
575C8—22no—
576C8—41no—
577C8—121no—
578C8—121no—
579C2×C3—82no—
580C2×C3#246711yescharacteristic
581C2×C3#310211yescharacteristic
582C2×C3#312011yescharacteristic
583C2×C3#329611yescharacteristic
584C2×C3#331311yescharacteristic
585C2×C3#339311yescharacteristic
586C2×C3#340111yescharacteristic
587C22—42no—
588C22—42no—
589C22—42no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—81no—
594C22#562711yescharacteristic
595C22#565411yescharacteristic
596C22#571811yescharacteristic
597C22D3×#13511yescharacteristic
598C22#577111yescharacteristic
599C22#577311yescharacteristic
600C22#577511yescharacteristic
601C4—41no—
602C4—41no—
603C4—62no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—122no—
608C4—122no—
609C4—121no—
610C4—121no—
611C4—121no—
612C4—241no—
613C4—241no—
614C4#562411yescharacteristic
615C4#564411yescharacteristic
616C4#565811yescharacteristic
617C4#566111yescharacteristic
618C3#2707011yescharacteristic, complemented, Sylow
619C2—82no—
620C2#7802511yescharacteristic
621C2#8114011yescharacteristic
622C2#8114511yescharacteristic
623C2#8138111yescharacteristic
624C2#8138411yescharacteristic
625C2#8181311yescharacteristic
626C2#8184111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2092C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2092D3
hypocenterC3#27070
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2092

Complements

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

NH ≅ G/Ntype
2. #74173C2semidirect
4. #70627C2semidirect
5. #70610C2semidirect
6. #70570C2semidirect
7. #70570C2semidirect
13. #69559C2semidirect
14. #68685C2semidirect
15. #68611C2semidirect
618. C3#27070semidirect

Maximal subgroups

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

subgroupindex
#763072
#741732
C3×#20922
#706272
#706102
#705702
#702152
#701782
#701712
Dic3×#1022
#696512
#695592
#686852
#686112
#270703