Finite Group Databasebeta

← #11652968

Subgroups — #11652968

Order 768.

4828 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D42C211yesmaximal, characteristic, complemented
2C2×D4×#67C214yesmaximal, complemented
3C2×#5717C211yesmaximal, characteristic, complemented
4#84053C214yesmaximal, complemented
5Dic3×#127C211yesmaximal, characteristic, complemented
6#82667C214yesmaximal, complemented
7#82640C214yesmaximal, complemented
8#82597C214yesmaximal, complemented
9#82582C214yesmaximal, complemented
10#82417C212yesmaximal, complemented
11C4×#940C212yesmaximal, complemented
12#59816—31nomaximal, Sylow
13C23×C3×D4C2214yescomplemented
14C23×#67C2214yescomplemented
15C22×D3×D4C2214yescomplemented
16C3×D42—28no—
17C2×C3×#127C2211yescharacteristic
18C2×C3×#121C2214yescomplemented
19C2×C3×#120C2214yescomplemented
20C2×C3×C4×D4C2212yescomplemented
21C2×#953C2218yescomplemented
22C2×#940C2212yes—
23C2×#939C2218yescomplemented
24D4×#67—232no—
25C2×#938C2214yescomplemented
26C2×#938C2214yescomplemented
27C2×#937C2214yescomplemented
28C2×#937C2218yescomplemented
29C2×D4×Dic3C2214yescomplemented
30C2×D4×Dic3C2212yescomplemented
31C2×#930C2214yescomplemented
32C2×C4×#67C2214yescomplemented
33C2×C4×#67C2214yescomplemented
34#5717—28no—
35C2×C4×D12C2212yescomplemented
36#5247C2214yescomplemented
37#5223C2218yescomplemented
38#5222C2218yescomplemented
39#5221C2212yescomplemented
40#5221C2214yescomplemented
41#5221C2212yescomplemented
42#5212C2214yescomplemented
43#4988C2212yescomplemented
44#4987C2214yescomplemented
45#4975C2212yescomplemented
46#4961C2218yescomplemented
47#4959C2218yescomplemented
48#4957C2214yescomplemented
49#4957C2214yescomplemented
50#4955C2214yescomplemented
51#4942C2214yescomplemented
52#4935C2211yescharacteristic, complemented
53C42×Dic3C2211yescharacteristic, complemented
54C2×D42—34no—
55C2×D42—31no—
56C2×D42D311yescharacteristic, complemented
57#2679—34no—
58#2537—34no—
59#2537—32no—
60#2408—34no—
61#2408—34no—
62#2408—34no—
63C4×#127—31no—
64C4×#127—32no—
65C25×C3C2314yes—
66C24×D3C2314yes—
67C22×C3×D4—216no—
68C22×C3×D4—216no—
69C22×C3×D4—48no—
70C22×C3×D4C2318yes—
71C22×C3×D4C2314yes—
72C22×C3×D4C2312yes—
73C22×C3×D4C2312yes—
74C23×C3×C4C2314yes—
75C22×#67—232no—
76C22×#67—48no—
77C22×#67C2318yes—
78C22×#67C2318yes—
79C22×#67C2314yes—
80C22×#67C2314yes—
81C23×Dic3C2314yes—
82C2×D3×D4—216no—
83C2×D3×D4—216no—
84C2×D3×D4—48no—
85C22×D12C2312yes—
86C22×C4×D3C2314yes—
87C3×#127D418yescomplemented
88C3×#121—216no—
89C3×#120—216no—
90C3×C4×D4—28no—
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C22⋊C4C2314yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C42C2311yescharacteristic
95#953—232no—
96C2×#217C2314yes—
97C2×#217C2318yes—
98C2×#217C2314yes—
99C2×#217C2318yes—
100C2×#217C2314yes—
101C2×#217C2314yes—
102#940D4116yescomplemented
103#939—232no—
104#938—216no—
105#938—216no—
106#937—216no—
107#937—232no—
108D4×Dic3—216no—
109D4×Dic3—44no—
110#930—216no—
111C4×#67—216no—
112C4×#67—48no—
113C2×(C2×D3)⋊C4C2318yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×C4⋊Dic3C2314yes—
118C2×C4⋊Dic3C2312yes—
119C2×C4⋊Dic3C2312yes—
120C2×Dic3⋊C4C2314yes—
121C2×Dic3⋊C4C2314yes—
122C2×C4×Dic3C2311yescharacteristic
123C2×C4×Dic3C2312yes—
124C2×C4×Dic3C2311yescharacteristic
125C2×C4×Dic3C2312yes—
126C4×D12—28no—
127C3⋊#95C2318yes—
128C23×D4—34no—
129C23×D4—34no—
130C23×D4C2×D314yescomplemented
131D42—28no—
132D42—632no—
133D42—68no—
134C2×#127—32no—
135C2×#127C2×D311yescharacteristic
136C2×#121—34no—
137C2×#121—34no—
138C2×#121—38no—
139C2×#121—34no—
140C2×#121C2×D314yescomplemented
141C2×#120—38no—
142C2×#120—38no—
143C2×#120—34no—
144C2×#120C2×D314yescomplemented
145C2×C4×D4—34no—
146C2×C4×D4—34no—
147C2×C4×D4—32no—
148C2×C4×D4—34no—
149C2×C4×D4—32no—
150C2×C4×D4C2×D312yescomplemented
151#394—38no—
152#394—38no—
153#394—38no—
154#392—38no—
155#390—32no—
156#390—34no—
157#390—32no—
158#390—32no—
159#390—34no—
160#390—34no—
161#384—32no—
162#384—31no—
163#380—34no—
164#380—34no—
165#380—34no—
166#379—34no—
167#379—34no—
168C43—31no—
169C24×C3—216no—
170C24×C3—28no—
171C24×C3—44no—
172C24×C3—48no—
173C24×C3—44no—
174C24×C3C2414yes—
175C24×C3C2414yes—
176C23×D3—28no—
177C23×D3—216no—
178C23×D3—48no—
179C23×D3—44no—
180C23×D3—44no—
181C23×D3C2414yes—
182C2×C3×D4—216no—
183C2×C3×D4—216no—
184C2×C3×D4—28no—
185C2×C3×D4—24no—
186C2×C3×D4—28no—
187C2×C3×D4—24no—
188C2×C3×D4—42no—
189C2×C3×D4—48no—
190C2×C3×D4—42no—
191C2×C3×D4—464no—
192C2×C3×D4C2×D418yescomplemented
193C2×C3×D4C2×D4116yescomplemented
194C2×C3×D4C2×D418yescomplemented
195C22×C3×C4—28no—
196C22×C3×C4—28no—
197C22×C3×C4—28no—
198C22×C3×C4C2414yes—
199C22×C3×C4C2411yescharacteristic
200C22×C3×C4C2412yes—
201C2×#67—216no—
202C2×#67—216no—
203C2×#67—216no—
204C2×#67—216no—
205C2×#67—28no—
206C2×#67—28no—
207C2×#67—44no—
208C2×#67—48no—
209C2×#67—48no—
210C2×#67—44no—
211C2×#67—464no—
212C2×#67C2×D4116yescomplemented
213C2×#67C2×D4116yescomplemented
214C22×Dic3—216no—
215C22×Dic3—28no—
216C22×Dic3C2414yes—
217C22×Dic3C2414yes—
218C22×Dic3C2411yescharacteristic
219C22×Dic3C2412yes—
220C22×Dic3C2411yescharacteristic
221D3×D4—464no—
222C2×D12—24no—
223C2×D12—24no—
224C2×D12C2×D418yescomplemented
225C2×C4×D3—28no—
226C2×C4×D3—28no—
227C2×C4×D3—28no—
228C3×C4⋊C4—24no—
229C3×C22⋊C4—28no—
230C3×C22⋊C4—28no—
231C3×C42C2×D414yescomplemented
232#217—28no—
233#217—216no—
234#217—28no—
235#217—216no—
236#217—44no—
237#217—44no—
238(C2×D3)⋊C4—216no—
239(C2×D3)⋊C4—28no—
240(C2×D3)⋊C4—28no—
241(C2×D3)⋊C4—44no—
242C4⋊Dic3—24no—
243C4⋊Dic3—28no—
244C4⋊Dic3—42no—
245Dic3⋊C4—28no—
246Dic3⋊C4—44no—
247C4×Dic3—24no—
248C4×Dic3—22no—
249C4×Dic3—41no—
250C4×Dic3C2×D418yescomplemented
251C25—34no—
252C25C22×D314yes—
253C22×D4—216no—
254C22×D4—216no—
255C22×D4—34no—
256C22×D4—34no—
257C22×D4—38no—
258C22×D4—38no—
259C22×D4—32no—
260C22×D4—48no—
261C22×D4—616no—
262C22×D4—632no—
263C22×D4—616no—
264C22×D4—128no—
265C22×D4—128no—
266C22×D4C22×D318yes—
267C22×D4C22×D312yes—
268C22×D4C22×D314yes—
269C22×D4C22×D312yes—
270C23×C4—34no—
271C23×C4—34no—
272C23×C4C22×D314yes—
273#127—316no—
274#127#6718yescomplemented
275#121—216no—
276#121—616no—
277#121—632no—
278#121—616no—
279#121—616no—
280#120—216no—
281#120—632no—
282#120—632no—
283#120—616no—
284C4×D4—28no—
285C4×D4—616no—
286C4×D4—616no—
287C4×D4—68no—
288C4×D4—128no—
289C4×D4—124no—
290C2×C4⋊C4—34no—
291C2×C4⋊C4—34no—
292C2×C4⋊C4—32no—
293C2×C4⋊C4—34no—
294C2×C4⋊C4—32no—
295C2×C4⋊C4C22×D312yes—
296C2×C22⋊C4—34no—
297C2×C22⋊C4—38no—
298C2×C22⋊C4—38no—
299C2×C22⋊C4—34no—
300C2×C22⋊C4—34no—
301C2×C22⋊C4—34no—
302C2×C22⋊C4—38no—
303C2×C22⋊C4—34no—
304C2×C22⋊C4—34no—
305C2×C22⋊C4—34no—
306C2×C22⋊C4C22×D314yes—
307C2×C22⋊C4C22×D314yes—
308C2×C42—32no—
309C2×C42—31no—
310C2×C42—32no—
311C2×C42—31no—
312C2×C42C22×D311yescharacteristic
313#95—38no—
314C23×C3—28no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—28no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—24no—
324C23×C3—416no—
325C23×C3—432no—
326C23×C3—816no—
327C23×C3—88no—
328C23×C3—88no—
329C23×C3C22×D418yes—
330C23×C3C22×D418yes—
331C23×C3C2511yescharacteristic
332C22×D3—24no—
333C22×D3—24no—
334C22×D3—24no—
335C22×D3—24no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3—416no—
339C22×D3—432no—
340C22×D3—816no—
341C22×D3—88no—
342C22×D3—88no—
343C22×D3C22×D418yes—
344C3×D4—232no—
345C3×D4—216no—
346C3×D4—48no—
347C3×D4—432no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—416no—
354C2×C3×C4C22×D414yes—
355C2×C3×C4C22×D412yes—
356C2×C3×C4C22×D414yes—
357C2×C3×C4H(5,𝔽2)14yes—
358#67—232no—
359#67—416no—
360#67—432no—
361#67—432no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—24no—
365C2×Dic3—24no—
366C2×Dic3—28no—
367C2×Dic3—22no—
368C2×Dic3—21no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—416no—
372C2×Dic3C22×D414yes—
373C2×Dic3C22×D414yes—
374C2×Dic3H(5,𝔽2)12yes—
375D12—216no—
376C4×D3—416no—
377C24—216no—
378C24—28no—
379C24—34no—
380C24—48no—
381C24—44no—
382C24—44no—
383C24—616no—
384C24—68no—
385C24—128no—
386C24—124no—
387C24—124no—
388C24C23×D314yes—
389C24C23×D314yes—
390C2×D4—216no—
391C2×D4—216no—
392C2×D4—28no—
393C2×D4—24no—
394C2×D4—28no—
395C2×D4—24no—
396C2×D4—316no—
397C2×D4—316no—
398C2×D4—38no—
399C2×D4—42no—
400C2×D4—48no—
401C2×D4—42no—
402C2×D4—464no—
403C2×D4—68no—
404C2×D4—68no—
405C2×D4—616no—
406C2×D4—616no—
407C2×D4—616no—
408C2×D4—616no—
409C2×D4—64no—
410C2×D4—64no—
411C2×D4—124no—
412C2×D4—128no—
413C2×D4—128no—
414C2×D4—124no—
415C2×D4—1264no—
416C2×D4—1264no—
417C2×D4D3×D418yescomplemented
418C2×D4C2×#67116yescomplemented
419C2×D4C2×#6718yescomplemented
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—31no—
424C22×C4—32no—
425C22×C4—34no—
426C22×C4—34no—
427C22×C4—31no—
428C22×C4—616no—
429C22×C4—68no—
430C22×C4—68no—
431C22×C4—68no—
432C22×C4—68no—
433C22×C4C23×D314yes—
434C22×C4C23×D311yescharacteristic
435C22×C4C23×D312yes—
436C4⋊C4—24no—
437C4⋊C4—68no—
438C4⋊C4—68no—
439C4⋊C4—64no—
440C4⋊C4—124no—
441C4⋊C4—122no—
442C22⋊C4—28no—
443C22⋊C4—28no—
444C22⋊C4—68no—
445C22⋊C4—616no—
446C22⋊C4—68no—
447C22⋊C4—616no—
448C22⋊C4—616no—
449C22⋊C4—68no—
450C22⋊C4—68no—
451C22⋊C4—124no—
452C22⋊C4—124no—
453C22⋊C4—124no—
454C42—38no—
455C42—62no—
456C42—64no—
457C42—121no—
458C42C2×#6714yescomplemented
459C22×C3—28no—
460C22×C3—216no—
461C22×C3—28no—
462C22×C3—28no—
463C22×C3—28no—
464C22×C3—44no—
465C22×C3—44no—
466C22×C3—48no—
467C22×C3—44no—
468C22×C3—44no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—832no—
472C22×C3C23×D412yes—
473C22×C3C23×D411yescharacteristic
474C22×C3C2×H(5,𝔽2)12yes—
475C22×C3C2×H(5,𝔽2)11yescharacteristic
476C22×C3#58511yescharacteristic
477C2×D3—28no—
478C2×D3—28no—
479C2×D3—28no—
480C2×D3—44no—
481C2×D3—44no—
482C2×D3—44no—
483C2×D3—44no—
484C2×D3—832no—
485C3×C4—22no—
486C3×C4—44no—
487C3×C4D4218yescomplemented
488Dic3—24no—
489Dic3—44no—
490Dic3—44no—
491Dic3—41no—
492Dic3D4214yescomplemented
493C23—24no—
494C23—28no—
495C23—24no—
496C23—24no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—38no—
504C23—432no—
505C23—416no—
506C23—64no—
507C23—64no—
508C23—64no—
509C23—64no—
510C23—64no—
511C23—64no—
512C23—816no—
513C23—88no—
514C23—88no—
515C23—1232no—
516C23—1216no—
517C23—248no—
518C23—2416no—
519C23—248no—
520C23C2×D3×D418yes—
521C23C22×#6718yes—
522C23C24×D311yescharacteristic
523D4—216no—
524D4—232no—
525D4—48no—
526D4—432no—
527D4—632no—
528D4—616no—
529D4—1216no—
530D4—1232no—
531D4—1232no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—34no—
538C2×C4—32no—
539C2×C4—34no—
540C2×C4—416no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—68no—
544C2×C4—64no—
545C2×C4—62no—
546C2×C4—64no—
547C2×C4—61no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—1216no—
551C2×C4—1216no—
552C2×C4C2×D3×D414yes—
553C2×C4#100414yes—
554C2×C4C22×#6712yes—
555C2×C4C22×#6714yes—
556C2×C3—48no—
557C2×C3—48no—
558C2×C3C2×D4211yescharacteristic
559C2×C3C2×D4212yes—
560C2×C3#349612yes—
561C2×C3#349611yescharacteristic
562C2×C3#354911yescharacteristic
563D3—48no—
564C22—28no—
565C22—28no—
566C22—28no—
567C22—216no—
568C22—28no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—48no—
574C22—44no—
575C22—44no—
576C22—68no—
577C22—68no—
578C22—68no—
579C22—832no—
580C22—124no—
581C22—124no—
582C22—124no—
583C22—124no—
584C22—2432no—
585C22C22×D3×D412yes—
586C22C2×#100412yes—
587C22D3×H(5,𝔽2)11yescharacteristic
588C22#596611yescharacteristic
589C22C23×#6711yescharacteristic
590C4—22no—
591C4—34no—
592C4—44no—
593C4—64no—
594C4—121no—
595C4—124no—
596C4—124no—
597C4D4×#6718yescomplemented
598C3#5981611yescharacteristic, complemented, Sylow
599C2—48no—
600C2—48no—
601C2—128no—
602C2#8661711yescharacteristic
603C2D3×D4211yescharacteristic
604C2#8662112yes—
605C2C2×D4×#6712yes—
606C2#8728111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×D42C2
hypercenterC2×D42D3
hypocenterC3#59816
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×D42

Complements

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

NH ≅ G/Ntype
1. C2×C3×D42C2semidirect
2. C2×D4×#67C2semidirect
3. C2×#5717C2semidirect
4. #84053C2semidirect
5. Dic3×#127C2semidirect
6. #82667C2semidirect
7. #82640C2semidirect
8. #82597C2semidirect
9. #82582C2semidirect
10. #82417C2semidirect
11. C4×#940C2semidirect
13. C23×C3×D4C22semidirect
14. C23×#67C22semidirect
15. C22×D3×D4C22semidirect
18. C2×C3×#121C22semidirect
19. C2×C3×#120C22semidirect
20. C2×C3×C4×D4C22semidirect
21. C2×#953C22semidirect
23. C2×#939C22semidirect
25. C2×#938C22semidirect
26. C2×#938C22semidirect
27. C2×#937C22semidirect
28. C2×#937C22semidirect
29. C2×D4×Dic3C22semidirect
30. C2×D4×Dic3C22semidirect
31. C2×#930C22semidirect
32. C2×C4×#67C22semidirect
33. C2×C4×#67C22semidirect
35. C2×C4×D12C22semidirect
36. #5247C22semidirect
37. #5223C22semidirect
38. #5222C22semidirect
39. #5221C22semidirect
40. #5221C22semidirect
41. #5221C22semidirect
42. #5212C22semidirect
43. #4988C22semidirect
44. #4987C22semidirect
45. #4975C22semidirect
46. #4961C22semidirect
47. #4959C22semidirect
48. #4957C22semidirect
49. #4957C22semidirect
50. #4955C22semidirect
51. #4942C22semidirect
52. #4935C22semidirect
53. C42×Dic3C22semidirect
56. C2×D42D3semidirect
87. C3×#127D4semidirect
102. #940D4semidirect
130. C23×D4C2×D3semidirect
140. C2×#121C2×D3semidirect
144. C2×#120C2×D3semidirect
150. C2×C4×D4C2×D3semidirect
192. C2×C3×D4C2×D4semidirect
193. C2×C3×D4C2×D4semidirect
194. C2×C3×D4C2×D4semidirect
212. C2×#67C2×D4semidirect
213. C2×#67C2×D4semidirect
224. C2×D12C2×D4semidirect
231. C3×C42C2×D4semidirect
250. C4×Dic3C2×D4semidirect
274. #127#67semidirect
417. C2×D4D3×D4semidirect
418. C2×D4C2×#67semidirect
419. C2×D4C2×#67semidirect
458. C42C2×#67semidirect
487. C3×C4D42semidirect
492. Dic3D42semidirect
597. C4D4×#67semidirect
598. C3#59816semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D422
C2×D4×#672
C2×#57172
#840532
Dic3×#1272
#826672
#826402
#825972
#825822
#824172
C4×#9402
#598163