Finite Group Databasebeta

← #10796771

Subgroups — #10796771

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×#5715C211yesmaximal, characteristic
2D3×#518C211yesmaximal, characteristic, complemented
3#78924C212yesmaximal, complemented
4C2×D3×#100C211yesmaximal, characteristic, complemented
5C3×#1887C211yesmaximal, characteristic, complemented
6#71763C211yesmaximal, characteristic, complemented
7#71649C211yesmaximal, characteristic, complemented
8#70323C212yesmaximal, complemented
9#70069C211yesmaximal, characteristic, complemented
10#68759C212yesmaximal, complemented
11#68753C212yesmaximal, complemented
12#20953—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C22×D3×D4C2211yescharacteristic
15C2×C3×#127C2211yescharacteristic
16C3×#518C2211yescharacteristic
17C2×#940C2211yescharacteristic
18C2×#939—22no—
19C2×#937—22no—
20#5798C2211yescharacteristic
21C2×#936C2211yescharacteristic
22C2×D3×M16C2212yes—
23#5715—24no—
24#5715C418yescomplemented
25#5545—24no—
26C4×D3×D4—24no—
27#5521C2212yes—
28D3×#117C2212yes—
29#5486—22no—
30C2×D3×C22⋊C4—22no—
31C2×#875C2211yescharacteristic
32C2×#872C2211yescharacteristic
33C3×#418C2212yescomplemented
34C2×C3×#100C2211yescharacteristic
35C2×#833C2211yescharacteristic
36#5215C2212yescomplemented
37C2×#823C2212yes—
38#5120C2212yescomplemented
39#5004C2212yescomplemented
40#4772—24no—
41#4771—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—32no—
52#1887—31no—
53#1887D311yescharacteristic, complemented
54#1885—31no—
55#1885—32no—
56#1885—31no—
57#1733—32no—
58C24×D3D412yes—
59C22×C3×D4—22no—
60C22×C3×D4C2311yescharacteristic
61C22×C3×D4C2311yescharacteristic
62C22×#67—21no—
63C22×#67—22no—
64C22×#67D412yes—
65C23×Dic3—21no—
66C2×#237—22no—
67C2×#237—22no—
68C2×#237—22no—
69C2×#237—24no—
70C2×#237C2×C414yescomplemented
71C2×D3×D4—22no—
72C2×D3×D4—22no—
73C2×D3×D4—24no—
74C2×D3×D4C2×C414yes—
75C2×D3×D4D414yes—
76C22×D12C2311yescharacteristic
77C22×C4×D3—24no—
78C22×C4×D3C2311yescharacteristic
79C22×Dic6C2311yescharacteristic
80C2×C3×M16C2312yes—
81C3×#127—22no—
82C3×#127C2×C414yescomplemented
83C3×C4×D4—24no—
84C3×#117C2312yes—
85C2×C3×C22⋊C4—22no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—22no—
88C2×#217—22no—
89#940—22no—
90#940C2×C414yescomplemented
91#939—216no—
92#937—216no—
93#936—22no—
94#936C2×C414yescomplemented
95D4×Dic3—24no—
96C4×#67—28no—
97C2×(C2×D3)⋊C4—24no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—22no—
100#926C2312yes—
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3C2311yescharacteristic
103C2×#208C2312yes—
104D3×M16—24no—
105D3×M16—24no—
106C2×ΓL(1,𝔽25)—22no—
107C2×D3×C8—22no—
108#893—24no—
109#892D414yescomplemented
110D3×C4⋊C4D414yescomplemented
111#67⋊C4—28no—
112D3×C22⋊C4—24no—
113D3×C22⋊C4—24no—
114D3×C22⋊C4—22no—
115D3×C22⋊C4—24no—
116#877—22no—
117#875—22no—
118#875C2×C414yescomplemented
119C4×D12—24no—
120#872—22no—
121#872—22no—
122#872C2×C414yescomplemented
123C42×D3—22no—
124C3×#102—24no—
125C3×#100—22no—
126#833—22no—
127#832—24no—
128#823—24no—
129#816—24no—
130#810—24no—
131C22×#41—31no—
132C23×D4—31no—
133C22×M16—32no—
134#535—38no—
135#535—64no—
136C2×#127—31no—
137C2×#127C2×D311yescharacteristic
138C2×#124—31no—
139C2×#124—31no—
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#419—32no—
153#418—32no—
154#418C2×D312yescomplemented
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×C3—21no—
166C24×C3C2×D412yes—
167C23×D3—22no—
168C23×D3—22no—
169C23×D3—24no—
170C23×D3—84no—
171C23×D3C22⋊C418yes—
172C23×D3C2×D412yes—
173C23×D3C2×D411yescharacteristic
174C2×C3×D4—28no—
175C2×C3×D4—22no—
176C2×C3×D4—22no—
177C2×C3×D4—21no—
178C2×C3×D4—22no—
179C2×C3×D4—22no—
180C2×C3×D4—84no—
181C2×C3×D4C22×C412yes—
182C2×C3×D4C22×C412yescomplemented
183C2×C3×D4C2×D412yes—
184C22×C3×C4—21no—
185C22×C3×C4—24no—
186C22×C3×C4C2411yescharacteristic
187C2×#67—24no—
188C2×#67—28no—
189C2×#67—24no—
190C2×#67—24no—
191C2×#67—41no—
192C2×#67—41no—
193C2×#67—42no—
194C2×#67—84no—
195C2×#67C22⋊C418yes—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—24no—
199C22×Dic3—24no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3C2×D411yescharacteristic
203#237—28no—
204#237—28no—
205#237—44no—
206#237—44no—
207D3×D4—28no—
208D3×D4—28no—
209D3×D4—28no—
210D3×D4—28no—
211C2×D12—22no—
212C2×D12—22no—
213C2×D12C22×C412yes—
214C2×D12C2×D412yes—
215C2×C4×D3—24no—
216C2×C4×D3—28no—
217C2×C4×D3—88no—
218C2×C4×D3C22×C412yes—
219C2×C4×D3C2×D412yes—
220C2×C4×D3C2×D412yes—
221C2×C4×D3C2×D412yes—
222C2×C4×D3#4114yes—
223C2×Dic6—22no—
224C2×Dic6—22no—
225C2×Dic6—21no—
226C2×Dic6C22×C412yescomplemented
227C3×M16—22no—
228C3×M16—22no—
229C2×C3×C8—22no—
230C3×C4⋊C4C2×D414yes—
231C3×C22⋊C4—24no—
232C3×C22⋊C4—22no—
233C3×C42—22no—
234C3×C42—21no—
235C3×C42C22×C412yescomplemented
236#217—24no—
237#217—28no—
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×Q8—31no—
262C22×D4—22no—
263C22×D4—32no—
264C22×D4—34no—
265C22×D4—34no—
266C22×D4—31no—
267C22×D4—62no—
268C22×D4—62no—
269C22×D4—64no—
270C22×D4—61no—
271C22×D4—62no—
272C22×D4C22×D311yescharacteristic
273C22×D4C22×D311yescharacteristic
274C23×C4—31no—
275C23×C4—64no—
276C23×C4—61no—
277C2×M16—32no—
278C2×M16—62no—
279C2×M16—64no—
280C2×M16—64no—
281C2×M16C22×D312yes—
282C22×C8—62no—
283#127—22no—
284#127—34no—
285#127—62no—
286#127C4×D314yescomplemented
287#124—34no—
288#124—34no—
289#124—62no—
290#124—62no—
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—62no—
310C2×C22⋊C4—64no—
311C2×C22⋊C4—64no—
312C2×C22⋊C4—64no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—62no—
315C2×C22⋊C4—64no—
316C2×C22⋊C4—62no—
317C2×C22⋊C4—62no—
318C2×C42—31no—
319C2×C42—62no—
320C2×C42C22×D311yescharacteristic
321#103—64no—
322#103—64no—
323#102—24no—
324#102—64no—
325#100—22no—
326#100—64no—
327#100—62no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—22no—
331C23×C3—44no—
332C23×C3—44no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3C2×C22⋊C414yes—
336C23×C3C22×D411yescharacteristic
337C22×D3—22no—
338C22×D3—22no—
339C22×D3—28no—
340C22×D3—28no—
341C22×D3—42no—
342C22×D3—48no—
343C22×D3—44no—
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×D4—24no—
353C3×D4—24no—
354C3×D4—24no—
355C3×D4—42no—
356C3×D4—88no—
357C2×C3×C4—22no—
358C2×C3×C4—24no—
359C2×C3×C4—42no—
360C2×C3×C4—84no—
361C2×C3×C4C23×C411yescharacteristic
362C2×C3×C4C22×D412yes—
363C2×C3×C4C22×D411yescharacteristic
364C2×C3×C4C2×#4112yes—
365#67—216no—
366#67—44no—
367#67—44no—
368#67—88no—
369C2×Dic3—22no—
370C2×Dic3—22no—
371C2×Dic3—24no—
372C2×Dic3—42no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—48no—
376C2×Dic3—84no—
377C2×Dic3C2×C22⋊C412yes—
378C2×Dic3#12012yes—
379C2×Dic3#12312yes—
380D12—24no—
381D12—24no—
382C4×D3—48no—
383C4×D3—88no—
384C4×D3C4×D414yes—
385C4×D3C4×D414yes—
386C4×D3#12114yes—
387C4×D3#12114yes—
388Dic6—24no—
389Dic6—42no—
390C3×C8—42no—
391C3⋊C8—42no—
392C24—21no—
393C24—31no—
394C24—32no—
395C24—38no—
396C24—62no—
397C24—62no—
398C24—64no—
399C24—244no—
400C24D3×D412yes—
401#41—68no—
402#41—68no—
403#41—124no—
404#41—124no—
405C2×Q8—32no—
406C2×Q8—61no—
407C2×Q8—62no—
408C2×Q8—62no—
409C2×D4—28no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—21no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—38no—
416C2×D4—32no—
417C2×D4—32no—
418C2×D4—68no—
419C2×D4—68no—
420C2×D4—68no—
421C2×D4—68no—
422C2×D4—64no—
423C2×D4—64no—
424C2×D4—64no—
425C2×D4—68no—
426C2×D4—62no—
427C2×D4—62no—
428C2×D4—84no—
429C2×D4—122no—
430C2×D4—121no—
431C2×D4—121no—
432C2×D4—244no—
433C2×D4C2×C4×D312yes—
434C2×D4C2×C4×D312yescomplemented
435C2×D4D3×D412yes—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—31no—
439C22×C4—32no—
440C22×C4—32no—
441C22×C4—32no—
442C22×C4—32no—
443C22×C4—34no—
444C22×C4—64no—
445C22×C4—68no—
446C22×C4—64no—
447C22×C4—61no—
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—22no—
468C22⋊C4—64no—
469C22⋊C4—64no—
470C22⋊C4—68no—
471C22⋊C4—68no—
472C22⋊C4—64no—
473C22⋊C4—68no—
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—44no—
487C22×C3—42no—
488C22×C3—82no—
489C22×C3—82no—
490C22×C3—82no—
491C22×C3C22×C22⋊C411yescharacteristic
492C22×C3C2×#12011yescharacteristic
493C22×C3C2×#12311yescharacteristic
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—24no—
497C2×D3—42no—
498C2×D3—42no—
499C2×D3—416no—
500C2×D3—44no—
501C2×D3—44no—
502C2×D3—88no—
503C2×D3—82no—
504C2×D3#38612yes—
505C2×D3#38612yes—
506C3×C4—42no—
507C3×C4—44no—
508C3×C4C2×C4×D412yes—
509C3×C4C2×#12112yes—
510Dic3—42no—
511Dic3—44no—
512Dic3—42no—
513Dic3#38614yes—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—31no—
518C23—31no—
519C23—32no—
520C23—32no—
521C23—34no—
522C23—44no—
523C23—44no—
524C23—41no—
525C23—41no—
526C23—68no—
527C23—68no—
528C23—62no—
529C23—62no—
530C23—122no—
531C23—124no—
532C23—128no—
533C23—244no—
534C23—248no—
535C23—248no—
536C23D3×C22⋊C414yes—
537C23C2×D3×D411yescharacteristic
538Q8—64no—
539Q8—122no—
540D4—24no—
541D4—24no—
542D4—24no—
543D4—42no—
544D4—616no—
545D4—64no—
546D4—64no—
547D4—88no—
548D4—124no—
549D4—124no—
550D4—248no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—32no—
556C2×C4—34no—
557C2×C4—34no—
558C2×C4—34no—
559C2×C4—34no—
560C2×C4—42no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—64no—
564C2×C4—84no—
565C2×C4—128no—
566C2×C4—128no—
567C2×C4—122no—
568C2×C4—121no—
569C2×C4—121no—
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×C3—82no—
582C2×C3C2×#38611yescharacteristic
583C2×C3#301912yes—
584D3—22no—
585D3—22no—
586D3—44no—
587C22—22no—
588C22—24no—
589C22—32no—
590C22—32no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—42no—
595C22—62no—
596C22—62no—
597C22—64no—
598C22—82no—
599C22—82no—
600C22—82no—
601C22—122no—
602C22—122no—
603C22—124no—
604C22—1216no—
605C22—124no—
606C22—242no—
607C22—248no—
608C22C2×D3×C22⋊C411yescharacteristic
609C22D3×#12011yescharacteristic
610C22D3×#12311yescharacteristic
611C4—34no—
612C4—42no—
613C4—44no—
614C4—122no—
615C4—122no—
616C4—124no—
617C4C4×D3×D412yes—
618C4D3×#12112yes—
619C3#2095311yescharacteristic, complemented, Sylow
620C2—21no—
621C2—21no—
622C2—44no—
623C2—62no—
624C2—62no—
625C2—82no—
626C2—124no—
627C2D3×#38611yescharacteristic
628C2#8027812yes—
subgroupG/H
centerC22D3×#120
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1887C2
socleC22×C3C2×#120
prime radicalC2×#5715C2
cosocleC22×C3×C4C24
hypercenter#1887D3
hypocenterC3#20953
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1887

Complements

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

NH ≅ G/Ntype
2. D3×#518C2semidirect
3. #78924C2semidirect
4. C2×D3×#100C2semidirect
5. C3×#1887C2semidirect
6. #71763C2semidirect
7. #71649C2semidirect
8. #70323C2semidirect
9. #70069C2semidirect
10. #68759C2semidirect
11. #68753C2semidirect
24. #5715C4semidirect
33. C3×#418C22semidirect
36. #5215C22semidirect
38. #5120C22semidirect
39. #5004C22semidirect
43. C3×#339C22semidirect
44. #4555C22semidirect
45. #4532C22semidirect
46. #4523C22semidirect
53. #1887D3semidirect
70. C2×#237C2×C4semidirect
82. C3×#127C2×C4semidirect
90. #940C2×C4semidirect
94. #936C2×C4semidirect
109. #892D4semidirect
110. D3×C4⋊C4D4semidirect
118. #875C2×C4semidirect
122. #872C2×C4semidirect
154. #418C2×D3semidirect
164. #339C2×D3semidirect
182. C2×C3×D4C22×C4semidirect
226. C2×Dic6C22×C4semidirect
235. C3×C42C22×C4semidirect
249. C4×Dic3C22×C4semidirect
286. #127C4×D3semidirect
434. C2×D4C2×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×#57152
D3×#5182
#789242
C2×D3×#1002
C3×#18872
#717632
#716492
#703232
#700692
#687592
#687532
#209533