Finite Group Databasebeta

← #11017264

Subgroups — #11017264

Order 768.

3114 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×#5607C212yesmaximal, complemented
2C3×#2856C211yesmaximal, characteristic, complemented
3#83175C212yesmaximal, complemented
4#82714C218yesmaximal, complemented
5#82676C212yesmaximal, complemented
6#82081C211yesmaximal, characteristic, complemented
7#82070C211yesmaximal, characteristic, complemented
8#76464C214yesmaximal, complemented
9#76453C212yesmaximal, complemented
10#76402C214yesmaximal, complemented
11#76314C212yesmaximal, complemented
12#74367C212yesmaximal, complemented
13#37297—31nomaximal, Sylow
14C22×D3×D4C2212yescomplemented
15C22×#235C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#121C2212yescomplemented
18C2×C3×#121C2214yescomplemented
19C22×C3×C42C2211yescharacteristic, complemented
20C2×#940C2214yes—
21C2×#940C2212yes—
22C2×#938C2212yes—
23C2×#937C2214yescomplemented
24C2×#937C2218yescomplemented
25C2×#930C2212yescomplemented
26C2×#930C2214yescomplemented
27C2×#926C2211yescharacteristic, complemented
28#5607—216no—
29C2×#895C2212yes—
30C2×#892C2212yescomplemented
31C2×#885C2214yescomplemented
32C2×#882C2214yes—
33C2×#882C2218yescomplemented
34C2×#874C2211yescharacteristic
35C3×#390C2212yescomplemented
36C3×#390C2214yescomplemented
37C3×#384C2211yescharacteristic, complemented
38#5223C2218yescomplemented
39#5223C2218yescomplemented
40#5221C2212yescomplemented
41#5213C2214yescomplemented
42#4991C2218yescomplemented
43#4990C2218yescomplemented
44#4988C2212yescomplemented
45#4977C2212yescomplemented
46#4961C2218yescomplemented
47#4961C2218yescomplemented
48#4960C2218yescomplemented
49#4951C2218yescomplemented
50#4939C2212yes—
51#4939C2214yescomplemented
52#4939C2212yescomplemented
53C4×(C2×D3)⋊C4C2212yescomplemented
54C4×(C2×D3)⋊C4C2212yescomplemented
55#4935C2211yescharacteristic, complemented
56#4929C2212yescomplemented
57C2×#535—32no—
58#2856—31no—
59#2856D311yescharacteristic, complemented
60#2733—34no—
61#2679—34no—
62#2669—38no—
63#2619—32no—
64#2500—32no—
65#2500—32no—
66#2433—32no—
67#2433—32no—
68#2297—31no—
69C24×D3C2314yes—
70C22×C3×D4C2312yes—
71C22×C3×D4C2314yes—
72C22×C3×D4C2314yes—
73C23×C3×C4C2312yes—
74C23×C3×C4C2311yescharacteristic
75C22×#67C2314yes—
76C22×#67C2314yes—
77C22×#67C2314yes—
78C22×#67C2318yes—
79C22×#67C2312yes—
80C2×D3×D4—28no—
81C2×D3×D4—44no—
82C2×D3×D4—44no—
83C2×#235—42no—
84C2×#235—42no—
85C2×#235D418yescomplemented
86C22×D12C2312yes—
87C22×D12C2314yes—
88C22×D12C2314yes—
89C22×D12C2311yescharacteristic
90C22×C4×D3C2312yes—
91C22×Dic6C2311yescharacteristic
92C3×#127—42no—
93C3×#121—28no—
94C3×#121—48no—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2314yes—
97C2×C3×C22⋊C4C2318yes—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C42—42no—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×C3×C42C2312yes—
103C2×#217C2314yes—
104C2×#217C2318yes—
105C2×#217C2312yes—
106#940—216no—
107#940—44no—
108#938—28no—
109#937—216no—
110#937—416no—
111#930—28no—
112#930—48no—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2318yes—
117C2×(C2×D3)⋊C4C2314yes—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2318yes—
120#926—42no—
121C2×C4⋊Dic3C2312yes—
122C2×C4⋊Dic3C2312yes—
123C2×C4⋊Dic3C2314yes—
124C2×C4×Dic3C2312yes—
125C2×C4×Dic3C2314yes—
126#895—28no—
127#892—28no—
128#885—216no—
129#882—216no—
130#882—416no—
131#874—42no—
132C3⋊#95C2314yes—
133C3⋊#95C2314yes—
134C3⋊#95C2318yes—
135C22×#41—31no—
136C23×D4—32no—
137#535—616no—
138C2×#127—34no—
139C2×#127—32no—
140C2×#127—31no—
141C2×#127C2×D311yescharacteristic
142C2×#124—34no—
143C2×#121—32no—
144C2×#121—32no—
145C2×#121—34no—
146C2×#121—32no—
147C2×#121C2×D312yescomplemented
148C2×#121C2×D314yescomplemented
149C2×#120—34no—
150C2×#120—38no—
151C2×#120—34no—
152C2×#120—38no—
153C2×#117—31no—
154C2×#117—32no—
155C22×C42C2×D311yescharacteristic, complemented
156#394—38no—
157#394—38no—
158#394—38no—
159#392—38no—
160#392—38no—
161#392—38no—
162#391—32no—
163#390—32no—
164#390—32no—
165#390—34no—
166#390—32no—
167#390—32no—
168#390C2×D314yescomplemented
169#390C2×D312yescomplemented
170#388—38no—
171#388—38no—
172#386—34no—
173#384—31no—
174#384C2×D311yescharacteristic, complemented
175#382—32no—
176C4×C22⋊C4—32no—
177C4×C22⋊C4—32no—
178C24×C3C2414yes—
179C24×C3C2411yescharacteristic
180C23×D3—28no—
181C23×D3—44no—
182C23×D3—44no—
183C23×D3—48no—
184C23×D3—48no—
185C23×D3C2414yes—
186C23×D3C2412yes—
187C2×C3×D4—28no—
188C2×C3×D4—24no—
189C2×C3×D4—24no—
190C2×C3×D4—24no—
191C2×C3×D4—28no—
192C2×C3×D4—44no—
193C2×C3×D4—48no—
194C2×C3×D4—44no—
195C22×C3×C4—22no—
196C22×C3×C4—22no—
197C22×C3×C4—24no—
198C22×C3×C4—24no—
199C22×C3×C4—42no—
200C22×C3×C4C2×D414yes—
201C22×C3×C4C2414yes—
202C22×C3×C4C2411yescharacteristic
203C22×C3×C4C2412yes—
204C22×C3×C4C2412yes—
205C22×C3×C4C2411yescharacteristic
206C2×#67—28no—
207C2×#67—28no—
208C2×#67—28no—
209C2×#67—42no—
210C2×#67—42no—
211C2×#67—44no—
212C2×#67—44no—
213C2×#67—48no—
214C2×#67—48no—
215C2×#67—48no—
216C2×#67—48no—
217C2×#67—48no—
218C2×#67C2×D418yes—
219C22×Dic3C2414yes—
220C22×Dic3C2414yes—
221C22×Dic3C2412yes—
222D3×D4—432no—
223#235—416no—
224C2×D12—24no—
225C2×D12—24no—
226C2×D12—24no—
227C2×D12—28no—
228C2×D12—28no—
229C2×D12—41no—
230C2×D12—41no—
231C2×D12—44no—
232C2×D12—48no—
233C2×D12—44no—
234C2×D12C2×D414yes—
235C2×C4×D3—24no—
236C2×C4×D3—24no—
237C2×C4×D3C2×D418yes—
238C2×Dic6—41no—
239C2×Dic6—41no—
240C2×Dic6C2×D414yes—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—44no—
243C3×C22⋊C4—28no—
244C3×C22⋊C4—216no—
245C3×C42—22no—
246C3×C42—22no—
247C3×C42—42no—
248#217—28no—
249#217—42no—
250#217—48no—
251(C2×D3)⋊C4—28no—
252(C2×D3)⋊C4—28no—
253(C2×D3)⋊C4—28no—
254(C2×D3)⋊C4—216no—
255(C2×D3)⋊C4—28no—
256(C2×D3)⋊C4—28no—
257(C2×D3)⋊C4—48no—
258C4⋊Dic3—24no—
259C4⋊Dic3—42no—
260C4⋊Dic3—44no—
261C4×Dic3—24no—
262C4×Dic3—28no—
263C25—34no—
264C2×#41—38no—
265C2×#41—122no—
266C2×#41—122no—
267C22×Q8—31no—
268C22×D4—32no—
269C22×D4—34no—
270C22×D4—34no—
271C22×D4—34no—
272C22×D4—38no—
273C22×D4—31no—
274C22×D4—32no—
275C22×D4—34no—
276C22×D4—34no—
277C22×D4—68no—
278C22×D4—124no—
279C22×D4—124no—
280C22×D4C22×D312yes—
281C22×D4C22×D314yes—
282C22×D4C22×D314yes—
283C23×C4—32no—
284C23×C4C22×D312yes—
285C23×C4C22×D311yescharacteristic
286#127—42no—
287#127—616no—
288#127—122no—
289#127—124no—
290#124—616no—
291#121—28no—
292#121—48no—
293#121—68no—
294#121—68no—
295#121—68no—
296#121—128no—
297#120—616no—
298#120—616no—
299#120—1216no—
300#120—1216no—
301#117—68no—
302#117—122no—
303C2×C4⋊C4—32no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—34no—
306C2×C4⋊C4C22×D312yes—
307C2×C4⋊C4C22×D314yes—
308C2×C22⋊C4—34no—
309C2×C22⋊C4—34no—
310C2×C22⋊C4—34no—
311C2×C22⋊C4—34no—
312C2×C22⋊C4—34no—
313C2×C22⋊C4—34no—
314C2×C22⋊C4—38no—
315C2×C22⋊C4—32no—
316C2×C22⋊C4—38no—
317C2×C22⋊C4—38no—
318C2×C22⋊C4C22×D314yes—
319C2×C22⋊C4C22×D318yes—
320C2×C42—34no—
321C2×C42—32no—
322C2×C42—42no—
323C2×C42C22×D311yescharacteristic
324C2×C42C22×D311yescharacteristic
325C2×C42C22×D312yes—
326#95—34no—
327#95—34no—
328#95—38no—
329C23×C3—21no—
330C23×C3—22no—
331C23×C3—21no—
332C23×C3—22no—
333C23×C3—24no—
334C23×C3—28no—
335C23×C3—28no—
336C23×C3—24no—
337C23×C3—24no—
338C23×C3C22×D412yes—
339C23×C3C2511yescharacteristic
340C22×D3—24no—
341C22×D3—24no—
342C22×D3—22no—
343C22×D3—22no—
344C22×D3—24no—
345C22×D3—28no—
346C22×D3—28no—
347C22×D3—24no—
348C22×D3—24no—
349C22×D3—416no—
350C22×D3—88no—
351C22×D3—88no—
352C22×D3—816no—
353C22×D3—816no—
354C22×D3C22×D414yes—
355C3×D4—48no—
356C3×D4—88no—
357C3×D4—88no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—22no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—28no—
365C2×C3×C4—24no—
366C2×C3×C4—28no—
367C2×C3×C4—48no—
368C2×C3×C4C22×D412yes—
369C2×C3×C4C22×D412yes—
370C2×C3×C4H(5,𝔽2)14yes—
371C2×C3×C4H(5,𝔽2)12yes—
372C2×C3×C4H(5,𝔽2)14yes—
373C2×C3×C4H(5,𝔽2)12yes—
374#67—48no—
375#67—416no—
376#67—88no—
377#67—88no—
378#67—816no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—28no—
384C2×Dic3—24no—
385C2×Dic3—24no—
386C2×Dic3C22×D414yes—
387D12—44no—
388D12—48no—
389D12—88no—
390D12—88no—
391C4×D3—216no—
392Dic6—44no—
393C24—32no—
394C24—34no—
395C24—68no—
396C24—124no—
397C24—124no—
398C24—128no—
399C24—128no—
400C24C23×D314yes—
401C24C23×D311yescharacteristic
402#41—1216no—
403C2×Q8—34no—
404C2×Q8—121no—
405C2×Q8—121no—
406C2×D4—28no—
407C2×D4—24no—
408C2×D4—24no—
409C2×D4—24no—
410C2×D4—28no—
411C2×D4—38no—
412C2×D4—34no—
413C2×D4—44no—
414C2×D4—48no—
415C2×D4—44no—
416C2×D4—68no—
417C2×D4—68no—
418C2×D4—68no—
419C2×D4—68no—
420C2×D4—64no—
421C2×D4—64no—
422C2×D4—68no—
423C2×D4—64no—
424C2×D4—121no—
425C2×D4—121no—
426C2×D4—128no—
427C2×D4—124no—
428C2×D4—124no—
429C2×D4—1232no—
430C2×D4—124no—
431C2×D4—124no—
432C2×D4—122no—
433C2×D4—122no—
434C2×D4—128no—
435C2×D4—128no—
436C2×D4—128no—
437C2×D4—128no—
438C2×D4—128no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—32no—
444C22×C4—38no—
445C22×C4—34no—
446C22×C4—34no—
447C22×C4—42no—
448C22×C4—64no—
449C22×C4—64no—
450C22×C4D3×D414yes—
451C22×C4C23×D314yes—
452C22×C4C23×D311yescharacteristic
453C22×C4C23×D312yes—
454C22×C4C23×D312yes—
455C22×C4C23×D311yescharacteristic
456C4⋊C4—24no—
457C4⋊C4—44no—
458C4⋊C4—64no—
459C4⋊C4—122no—
460C4⋊C4—124no—
461C22⋊C4—28no—
462C22⋊C4—216no—
463C22⋊C4—68no—
464C22⋊C4—68no—
465C22⋊C4—68no—
466C22⋊C4—68no—
467C22⋊C4—68no—
468C22⋊C4—68no—
469C22⋊C4—616no—
470C22⋊C4—128no—
471C22⋊C4—122no—
472C22⋊C4—128no—
473C42—22no—
474C42—22no—
475C42—42no—
476C42—68no—
477C42—64no—
478C22×C3—22no—
479C22×C3—22no—
480C22×C3—22no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—44no—
484C22×C3—44no—
485C22×C3—44no—
486C22×C3—48no—
487C22×C3—48no—
488C22×C3C23×D411yescharacteristic
489C22×C3C2×H(5,𝔽2)11yescharacteristic
490C22×C3C2×H(5,𝔽2)12yes—
491C22×C3C2×H(5,𝔽2)11yescharacteristic
492C22×C3C2×H(5,𝔽2)12yes—
493C2×D3—24no—
494C2×D3—24no—
495C2×D3—24no—
496C2×D3—44no—
497C2×D3—44no—
498C2×D3—44no—
499C2×D3—44no—
500C2×D3—44no—
501C2×D3—48no—
502C2×D3—48no—
503C2×D3—832no—
504C3×C4—22no—
505C3×C4—24no—
506C3×C4—24no—
507C3×C4—44no—
508C3×C4#53414yes—
509Dic3—24no—
510Dic3—44no—
511Dic3—44no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—22no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—28no—
520C23—24no—
521C23—34no—
522C23—64no—
523C23—62no—
524C23—64no—
525C23—62no—
526C23—64no—
527C23—68no—
528C23—64no—
529C23—68no—
530C23—64no—
531C23—1216no—
532C23—248no—
533C23—248no—
534C23—2416no—
535C23—2416no—
536C23C2×D3×D412yes—
537C23C24×D311yescharacteristic
538Q8—124no—
539D4—48no—
540D4—88no—
541D4—88no—
542D4—124no—
543D4—128no—
544D4—1216no—
545D4—128no—
546D4—248no—
547D4—248no—
548D4—248no—
549D4—248no—
550D4—2416no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—34no—
561C2×C4—48no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—616no—
565C2×C4—64no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—68no—
570C2×C4C2×D3×D412yes—
571C2×C4C2×D3×D412yes—
572C2×C4#100414yes—
573C2×C4#100412yes—
574C2×C4#100914yes—
575C2×C4#100912yes—
576C2×C3—42no—
577C2×C3—84no—
578C2×C3C2×#53411yescharacteristic
579C2×C3#349811yescharacteristic
580C2×C3#349811yescharacteristic
581C2×C3#353712yes—
582C2×C3#354912yes—
583D3—44no—
584D3—84no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—42no—
589C22—42no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—64no—
596C22—64no—
597C22—64no—
598C22—124no—
599C22—124no—
600C22—128no—
601C22—124no—
602C22—124no—
603C22—124no—
604C22—128no—
605C22—2432no—
606C22C22×D3×D411yescharacteristic
607C22C2×#100411yescharacteristic
608C22C2×#100412yes—
609C22C2×#100912yes—
610C22C2×#100911yescharacteristic
611C4—24no—
612C4—24no—
613C4—22no—
614C4—44no—
615C4—64no—
616C4—124no—
617C4—124no—
618C4#560514yes—
619C3#3729711yescharacteristic, complemented, Sylow
620C2—42no—
621C2—84no—
622C2—124no—
623C2—244no—
624C2#8593812yes—
625C2C2×#560511yescharacteristic
626C2#8619511yescharacteristic
627C2#8624211yescharacteristic
628C2#8625212yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2856C2
hypercenter#2856D3
hypocenterC3#37297
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2856

Complements

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

NH ≅ G/Ntype
1. C2×#5607C2semidirect
2. C3×#2856C2semidirect
3. #83175C2semidirect
4. #82714C2semidirect
5. #82676C2semidirect
6. #82081C2semidirect
7. #82070C2semidirect
8. #76464C2semidirect
9. #76453C2semidirect
10. #76402C2semidirect
11. #76314C2semidirect
12. #74367C2semidirect
14. C22×D3×D4C22semidirect
17. C2×C3×#121C22semidirect
18. C2×C3×#121C22semidirect
19. C22×C3×C42C22semidirect
23. C2×#937C22semidirect
24. C2×#937C22semidirect
25. C2×#930C22semidirect
26. C2×#930C22semidirect
27. C2×#926C22semidirect
30. C2×#892C22semidirect
31. C2×#885C22semidirect
33. C2×#882C22semidirect
35. C3×#390C22semidirect
36. C3×#390C22semidirect
37. C3×#384C22semidirect
38. #5223C22semidirect
39. #5223C22semidirect
40. #5221C22semidirect
41. #5213C22semidirect
42. #4991C22semidirect
43. #4990C22semidirect
44. #4988C22semidirect
45. #4977C22semidirect
46. #4961C22semidirect
47. #4961C22semidirect
48. #4960C22semidirect
49. #4951C22semidirect
51. #4939C22semidirect
52. #4939C22semidirect
53. C4×(C2×D3)⋊C4C22semidirect
54. C4×(C2×D3)⋊C4C22semidirect
55. #4935C22semidirect
56. #4929C22semidirect
59. #2856D3semidirect
85. C2×#235D4semidirect
147. C2×#121C2×D3semidirect
148. C2×#121C2×D3semidirect
155. C22×C42C2×D3semidirect
168. #390C2×D3semidirect
169. #390C2×D3semidirect
174. #384C2×D3semidirect
619. C3#37297semidirect

Maximal subgroups

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

subgroupindex
C2×#56072
C3×#28562
#831752
#827142
#826762
#820812
#820702
#764642
#764532
#764022
#763142
#743672
#372973