Finite Group Databasebeta

← #10796768

Subgroups — #10796768

Order 768.

1974 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5674C211yesmaximal, characteristic
2D3×#518C211yesmaximal, characteristic, complemented
3#78956C212yesmaximal, complemented
4C2×D3×#100C211yesmaximal, characteristic, complemented
5C3×#1885C211yesmaximal, characteristic, complemented
6#71723C211yesmaximal, characteristic, complemented
7#71650C211yesmaximal, characteristic, complemented
8#70343C212yesmaximal, complemented
9#70070C211yesmaximal, characteristic, complemented
10#68760C212yesmaximal, complemented
11#68753C212yesmaximal, complemented
12#20953—31nomaximal, Sylow
13C22×#239C2211yescharacteristic
14C22×D3×D4C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C3×#518C2211yescharacteristic
17C2×#947C2211yescharacteristic
18C2×#940C2211yescharacteristic
19#5798C2211yescharacteristic
20C2×D3×M16C2212yes—
21#5674—24no—
22#5674C418yescomplemented
23#5545—24no—
24C4×D3×D4—24no—
25#5521C2212yes—
26D3×#117C2212yes—
27C2×#884—22no—
28C2×#882—22no—
29#5486—22no—
30C2×D3×C22⋊C4—22no—
31C2×#874C2211yescharacteristic
32C2×#872C2211yescharacteristic
33C3×#419C2212yescomplemented
34C2×C3×#100C2211yescharacteristic
35#5225C2212yescomplemented
36C2×#833C2211yescharacteristic
37#5131C2212yescomplemented
38C2×#823C2212yes—
39#5001C2212yescomplemented
40#4804—24no—
41#4803—24no—
42D3×#100—24no—
43C3×#339C2212yescomplemented
44#4555C2212yescomplemented
45#4532C2212yescomplemented
46#4523C2212yescomplemented
47C2×#535—31no—
48C2×#518—31no—
49#2896—32no—
50C22×#100—31no—
51#1887—31no—
52#1887—32no—
53#1887—31no—
54#1885—32no—
55#1885—31no—
56#1885D311yescharacteristic, complemented
57#1733—32no—
58C24×D3D412yes—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4C2311yescharacteristic
61C22×#67—22no—
62C22×#67D412yes—
63C2×#239—22no—
64C2×#239—22no—
65C2×#239—22no—
66C2×#239—24no—
67C2×#239C2×C414yescomplemented
68C2×D3×D4—22no—
69C2×D3×D4—22no—
70C2×D3×D4—24no—
71C2×D3×D4C2×C414yes—
72C2×D3×D4D414yes—
73C22×D12—21no—
74C22×D12—22no—
75C22×D12C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—24no—
79C22×C4×D3C2311yescharacteristic
80C2×C3×M16C2312yes—
81C3×#124—22no—
82C3×#124C2×C414yescomplemented
83C3×C4×D4—24no—
84C3×#117C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42C2311yescharacteristic
88C2×#217—22no—
89#947—22no—
90#947C2×C414yescomplemented
91#940—22no—
92#940C2×C414yescomplemented
93D4×Dic3—24no—
94C4×#67—28no—
95C2×(C2×D3)⋊C4—24no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—22no—
98#926C2312yes—
99C2×Dic3⋊C4—21no—
100C2×C4×Dic3C2311yescharacteristic
101C2×#208C2312yes—
102D3×M16—24no—
103D3×M16—24no—
104C2×ΓL(1,𝔽25)—22no—
105C2×D3×C8—22no—
106#893—24no—
107#892D414yescomplemented
108D3×C4⋊C4D414yescomplemented
109#884—216no—
110#882—216no—
111#67⋊C4—28no—
112D3×C22⋊C4—24no—
113D3×C22⋊C4—24no—
114D3×C22⋊C4—22no—
115D3×C22⋊C4—24no—
116#877—22no—
117#874—22no—
118#874C2×C414yescomplemented
119C4×D12—24no—
120#872—22no—
121#872—22no—
122#872C2×C414yescomplemented
123C42×D3—22no—
124C3×#103—24no—
125C3×#100—22no—
126#835—24no—
127#833—22no—
128#823—24no—
129#821—24no—
130#809—24no—
131C22×#41—31no—
132C23×D4—31no—
133C22×M16—32no—
134#535—38no—
135#535—64no—
136C2×#127—31no—
137C2×#127—31no—
138C2×#124—31no—
139C2×#124C2×D311yescharacteristic
140C2×#121—62no—
141C2×#120—62no—
142#518—31no—
143#518—32no—
144#518—62no—
145#518—64no—
146#518C2×D311yescharacteristic
147C2×C4×D4—64no—
148C2×#117—31no—
149C2×#117—32no—
150C22×C22⋊C4—62no—
151#419—32no—
152#419C2×D312yescomplemented
153#418—32no—
154#418—32no—
155#417—64no—
156#416—64no—
157C2×#100—31no—
158C2×#100—32no—
159C2×#100—64no—
160C2×#100C2×D311yescharacteristic
161#339—32no—
162#339—32no—
163#339—32no—
164#339C2×D312yescomplemented
165C24×C3C2×D412yes—
166C23×D3—22no—
167C23×D3—22no—
168C23×D3—24no—
169C23×D3—21no—
170C23×D3—84no—
171C23×D3C22⋊C418yes—
172C23×D3C2×D412yes—
173C23×D3C2×D411yescharacteristic
174C2×C3×Q8—21no—
175C2×C3×Q8—22no—
176C2×C3×Q8—22no—
177C2×C3×Q8C22×C412yescomplemented
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4C22×C412yes—
181C2×C3×D4C2×D412yes—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—24no—
185C22×C3×C4C2411yescharacteristic
186C2×#67—28no—
187C2×#67—24no—
188C2×#67—24no—
189C2×#67—84no—
190C2×#67C22⋊C418yes—
191C22×Dic3—21no—
192C22×Dic3—24no—
193C22×Dic3C2×D411yescharacteristic
194#239—28no—
195#239—28no—
196#239—44no—
197#239—44no—
198D3×D4—28no—
199D3×D4—28no—
200D3×D4—28no—
201D3×D4—28no—
202C2×D12—22no—
203C2×D12—22no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—24no—
207C2×D12—21no—
208C2×D12—28no—
209C2×D12—41no—
210C2×D12—41no—
211C2×D12—42no—
212C2×D12—84no—
213C2×D12C22×C412yes—
214C2×D12C22×C412yescomplemented
215C2×D12C2×D412yes—
216C2×C4×D3—24no—
217C2×C4×D3—24no—
218C2×C4×D3—28no—
219C2×C4×D3—42no—
220C2×C4×D3—42no—
221C2×C4×D3—88no—
222C2×C4×D3C22×C412yes—
223C2×C4×D3C2×D412yes—
224C2×C4×D3C2×D412yes—
225C2×C4×D3C2×D412yes—
226C2×C4×D3#4114yes—
227C3×M16—22no—
228C3×M16—22no—
229C2×C3×C8—22no—
230C3×C4⋊C4C2×D414yes—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—28no—
233C3×C22⋊C4—22no—
234C3×C42—22no—
235C3×C42—21no—
236C3×C42C22×C412yescomplemented
237#217—24no—
238#217—22no—
239(C2×D3)⋊C4—28no—
240(C2×D3)⋊C4—24no—
241(C2×D3)⋊C4—28no—
242(C2×D3)⋊C4—24no—
243C4⋊Dic3C2×D414yes—
244Dic3⋊C4—24no—
245Dic3⋊C4—24no—
246C4×Dic3—22no—
247C4×Dic3—21no—
248C4×Dic3—24no—
249C4×Dic3C22×C412yescomplemented
250#208—22no—
251#208—22no—
252C2×C3⋊C8—22no—
253ΓL(1,𝔽25)—44no—
254D3×C8—44no—
255C25—32no—
256C2×#41—34no—
257C2×#41—62no—
258C2×#41—62no—
259C2×#41—62no—
260C2×#41—64no—
261C22×Q8C22×D311yescharacteristic
262C22×D4—32no—
263C22×D4—34no—
264C22×D4—34no—
265C22×D4—31no—
266C22×D4—31no—
267C22×D4—61no—
268C22×D4—62no—
269C22×D4—62no—
270C22×D4—62no—
271C22×D4—64no—
272C22×D4—62no—
273C22×D4C22×D311yescharacteristic
274C23×C4—31no—
275C23×C4—61no—
276C23×C4—64no—
277C2×M16—32no—
278C2×M16—62no—
279C2×M16—64no—
280C2×M16—64no—
281C2×M16C22×D312yes—
282C22×C8—62no—
283#127—34no—
284#127—34no—
285#127—62no—
286#127—62no—
287#124—22no—
288#124—34no—
289#124—62no—
290#124C4×D314yescomplemented
291#121—616no—
292#120—616no—
293C4×D4—24no—
294C4×D4—64no—
295C4×D4—68no—
296C4×D4—64no—
297C4×D4—68no—
298C4×D4—64no—
299#117—34no—
300#117—32no—
301#117—34no—
302#117—62no—
303#117—62no—
304#117—62no—
305#117C22×D312yes—
306C2×C4⋊C4—34no—
307C2×C4⋊C4—61no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—62no—
311C2×C22⋊C4—64no—
312C2×C22⋊C4—64no—
313C2×C22⋊C4—64no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—62no—
316C2×C22⋊C4—64no—
317C2×C22⋊C4—62no—
318C2×C42—31no—
319C2×C42—62no—
320C2×C42C22×D311yescharacteristic
321#103—24no—
322#103—64no—
323#102—64no—
324#102—64no—
325#100—22no—
326#100—64no—
327#100—62no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—44no—
331C23×C3C2×C22⋊C414yes—
332C23×C3C22×D411yescharacteristic
333C22×D3—22no—
334C22×D3—22no—
335C22×D3—28no—
336C22×D3—28no—
337C22×D3—22no—
338C22×D3—42no—
339C22×D3—48no—
340C22×D3—44no—
341C22×D3—44no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—84no—
345C22×D3—88no—
346C22×D3—88no—
347C22×D3C2×C22⋊C414yes—
348C22×D3C2×C22⋊C412yes—
349C22×D3#12011yescharacteristic
350C22×D3#12011yescharacteristic
351C22×D3#12312yes—
352C3×Q8—24no—
353C3×Q8—42no—
354C3×D4—24no—
355C3×D4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—42no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—84no—
363C2×C3×C4C23×C411yescharacteristic
364C2×C3×C4C22×D412yes—
365C2×C3×C4C22×D411yescharacteristic
366C2×C3×C4C2×#4112yes—
367#67—216no—
368#67—88no—
369C2×Dic3—22no—
370C2×Dic3—24no—
371C2×Dic3—42no—
372C2×Dic3—84no—
373C2×Dic3C2×C22⋊C412yes—
374C2×Dic3#12012yes—
375C2×Dic3#12312yes—
376D12—24no—
377D12—24no—
378D12—24no—
379D12—44no—
380D12—44no—
381D12—42no—
382D12—88no—
383C4×D3—48no—
384C4×D3—48no—
385C4×D3—88no—
386C4×D3C4×D414yes—
387C4×D3C4×D414yes—
388C4×D3#12114yes—
389C4×D3#12114yes—
390C3×C8—42no—
391C3⋊C8—42no—
392C24—31no—
393C24—32no—
394C24—38no—
395C24—62no—
396C24—62no—
397C24—64no—
398C24—61no—
399C24—244no—
400C24D3×D412yes—
401#41—68no—
402#41—68no—
403#41—124no—
404#41—124no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C2×C4×D312yescomplemented
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—38no—
412C2×D4—32no—
413C2×D4—32no—
414C2×D4—32no—
415C2×D4—64no—
416C2×D4—68no—
417C2×D4—68no—
418C2×D4—68no—
419C2×D4—68no—
420C2×D4—68no—
421C2×D4—64no—
422C2×D4—64no—
423C2×D4—68no—
424C2×D4—62no—
425C2×D4—62no—
426C2×D4—61no—
427C2×D4—62no—
428C2×D4—62no—
429C2×D4—122no—
430C2×D4—121no—
431C2×D4—121no—
432C2×D4—244no—
433C2×D4—244no—
434C2×D4C2×C4×D312yes—
435C2×D4D3×D412yes—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—31no—
440C22×C4—32no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—32no—
444C22×C4—34no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4—68no—
448C22×C4—61no—
449C22×C4—64no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—248no—
453C22×C4C23×D311yescharacteristic
454M16—22no—
455M16—22no—
456M16—62no—
457M16—62no—
458M16—124no—
459C2×C8—22no—
460C2×C8—62no—
461C2×C8—124no—
462C4⋊C4—34no—
463C4⋊C4—64no—
464C4⋊C4—64no—
465C4⋊C4D3×D414yes—
466C22⋊C4—24no—
467C22⋊C4—28no—
468C22⋊C4—22no—
469C22⋊C4—64no—
470C22⋊C4—64no—
471C22⋊C4—68no—
472C22⋊C4—68no—
473C22⋊C4—64no—
474C22⋊C4—62no—
475C42—22no—
476C42—21no—
477C42—32no—
478C42—64no—
479C42—61no—
480C42—62no—
481C42C2×C4×D312yescomplemented
482C22×C3—22no—
483C22×C3—24no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—82no—
487C22×C3C22×C22⋊C411yescharacteristic
488C22×C3C2×#12011yescharacteristic
489C22×C3C2×#12311yescharacteristic
490C2×D3—22no—
491C2×D3—22no—
492C2×D3—24no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—416no—
496C2×D3—44no—
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—42no—
500C2×D3—88no—
501C2×D3—82no—
502C2×D3—82no—
503C2×D3—82no—
504C2×D3#38612yes—
505C2×D3#38612yes—
506C3×C4—42no—
507C3×C4—44no—
508C3×C4—42no—
509C3×C4C2×C4×D412yes—
510C3×C4C2×#12112yes—
511Dic3—42no—
512Dic3—44no—
513Dic3#38614yes—
514C23—22no—
515C23—22no—
516C23—31no—
517C23—31no—
518C23—32no—
519C23—32no—
520C23—34no—
521C23—44no—
522C23—68no—
523C23—68no—
524C23—62no—
525C23—62no—
526C23—62no—
527C23—122no—
528C23—124no—
529C23—124no—
530C23—128no—
531C23—121no—
532C23—121no—
533C23—244no—
534C23—248no—
535C23—248no—
536C23D3×C22⋊C414yes—
537C23C2×D3×D411yescharacteristic
538Q8—24no—
539Q8—42no—
540D4—24no—
541D4—24no—
542D4—616no—
543D4—64no—
544D4—64no—
545D4—64no—
546D4—124no—
547D4—124no—
548D4—122no—
549D4—248no—
550D4—248no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—32no—
557C2×C4—34no—
558C2×C4—34no—
559C2×C4—34no—
560C2×C4—34no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—62no—
565C2×C4—64no—
566C2×C4—84no—
567C2×C4—128no—
568C2×C4—128no—
569C2×C4—122no—
570C2×C4—248no—
571C2×C4—244no—
572C2×C4C22×C4×D311yescharacteristic
573C2×C4C2×D3×D412yes—
574C2×C4C2×D3×D411yescharacteristic
575C2×C4D3×#4112yes—
576C8—42no—
577C8—122no—
578C2×C3—21no—
579C2×C3—21no—
580C2×C3—44no—
581C2×C3C2×#38611yescharacteristic
582C2×C3#301912yes—
583D3—22no—
584D3—22no—
585D3—44no—
586D3—82no—
587C22—22no—
588C22—24no—
589C22—32no—
590C22—32no—
591C22—44no—
592C22—44no—
593C22—62no—
594C22—62no—
595C22—64no—
596C22—82no—
597C22—122no—
598C22—122no—
599C22—124no—
600C22—124no—
601C22—1216no—
602C22—124no—
603C22—122no—
604C22—242no—
605C22—242no—
606C22—248no—
607C22—242no—
608C22C2×D3×C22⋊C411yescharacteristic
609C22D3×#12011yescharacteristic
610C22D3×#12311yescharacteristic
611C4—34no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—122no—
616C4—124no—
617C4C4×D3×D412yes—
618C4D3×#12112yes—
619C3#2095311yescharacteristic, complemented, Sylow
620C2—21no—
621C2—21no—
622C2—44no—
623C2—62no—
624C2—62no—
625C2—124no—
626C2—242no—
627C2D3×#38611yescharacteristic
628C2#8029212yes—
subgroupG/H
centerC22D3×#120
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1885C2
socleC22×C3C2×#120
prime radicalC2×#5674C2
cosocleC22×C3×C4C24
hypercenter#1885D3
hypocenterC3#20953
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1885

Complements

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

NH ≅ G/Ntype
2. D3×#518C2semidirect
3. #78956C2semidirect
4. C2×D3×#100C2semidirect
5. C3×#1885C2semidirect
6. #71723C2semidirect
7. #71650C2semidirect
8. #70343C2semidirect
9. #70070C2semidirect
10. #68760C2semidirect
11. #68753C2semidirect
22. #5674C4semidirect
33. C3×#419C22semidirect
35. #5225C22semidirect
37. #5131C22semidirect
39. #5001C22semidirect
43. C3×#339C22semidirect
44. #4555C22semidirect
45. #4532C22semidirect
46. #4523C22semidirect
56. #1885D3semidirect
67. C2×#239C2×C4semidirect
82. C3×#124C2×C4semidirect
90. #947C2×C4semidirect
92. #940C2×C4semidirect
107. #892D4semidirect
108. D3×C4⋊C4D4semidirect
118. #874C2×C4semidirect
122. #872C2×C4semidirect
152. #419C2×D3semidirect
164. #339C2×D3semidirect
177. C2×C3×Q8C22×C4semidirect
214. C2×D12C22×C4semidirect
236. C3×C42C22×C4semidirect
249. C4×Dic3C22×C4semidirect
290. #124C4×D3semidirect
408. C2×Q8C2×C4×D3semidirect
481. C42C2×C4×D3semidirect
619. C3#20953semidirect

Maximal subgroups

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

subgroupindex
C2×#56742
D3×#5182
#789562
C2×D3×#1002
C3×#18852
#717232
#716502
#703432
#700702
#687602
#687532
#209533