Finite Group Databasebeta

← #11017266

Subgroups — #11017266

Order 768.

3114 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×#5607C212yesmaximal, complemented
2C3×#2619C211yesmaximal, characteristic, complemented
3#83175C212yesmaximal, complemented
4#82700C212yesmaximal, complemented
5#82286C211yesmaximal, characteristic, complemented
6#82228C211yesmaximal, characteristic, complemented
7#77138C214yesmaximal, complemented
8#77047C214yesmaximal, complemented
9#76764C218yesmaximal, complemented
10#76334C212yesmaximal, complemented
11#76314C212yesmaximal, complemented
12#74166C212yesmaximal, complemented
13#37297—31nomaximal, Sylow
14C22×D3×D4C2212yescomplemented
15C22×#235C2211yescharacteristic
16C2×C3×#127C2211yescharacteristic
17C2×C3×#121C2212yescomplemented
18C2×C3×#117C2211yescharacteristic, complemented
19C2×#940C2214yes—
20C2×#940C2212yes—
21C2×#939C2218yescomplemented
22C2×#938C2212yes—
23C2×#937C2214yescomplemented
24C2×#937C2218yescomplemented
25C2×#930C2212yescomplemented
26C22×C4×Dic3C2211yescharacteristic, complemented
27#5607—216no—
28C2×#895C2212yes—
29C2×#892C2212yescomplemented
30C2×#885C2214yescomplemented
31C2×#882C2214yes—
32C2×#882C2218yescomplemented
33C2×#874C2211yescharacteristic
34C3×#394C2214yescomplemented
35C3×#392C2214yescomplemented
36C3×#390C2212yescomplemented
37C3×#382C2211yescharacteristic, complemented
38#5223C2218yescomplemented
39#5221C2212yescomplemented
40#4991C2214yescomplemented
41#4988C2212yes—
42#4988C2212yescomplemented
43#4975C2212yescomplemented
44#4973C2212yescomplemented
45#4960C2218yescomplemented
46#4960C2214yescomplemented
47#4959C2218yescomplemented
48#4958C2214yescomplemented
49#4957C2218yescomplemented
50#4945C2218yescomplemented
51#4939C2212yescomplemented
52C4×(C2×D3)⋊C4C2212yescomplemented
53C4×(C2×D3)⋊C4C2212yescomplemented
54#4936C2211yescharacteristic, complemented
55#4672C2218yescomplemented
56#4669C2218yescomplemented
57C2×#535—32no—
58#2856—32no—
59#2733—34no—
60#2679—34no—
61#2669—38no—
62#2619—31no—
63#2619D311yescharacteristic, complemented
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×C4C2311yescharacteristic
74C22×#67C2314yes—
75C22×#67C2318yes—
76C22×#67C2314yes—
77C22×#67C2314yes—
78C22×#67C2312yes—
79C23×Dic3C2312yes—
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×#117—42no—
95C2×C3×C4⋊C4C2312yes—
96C2×C3×C4⋊C4C2312yes—
97C2×C3×C22⋊C4C2314yes—
98C2×C3×C22⋊C4C2318yes—
99C2×C3×C22⋊C4C2312yes—
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#217C2314yes—
103C2×#217C2318yes—
104#940—216no—
105#940—44no—
106#939—416no—
107#938—28no—
108#937—216no—
109#937—416no—
110#930—28no—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2318yes—
113C2×(C2×D3)⋊C4C2314yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117C2×(C2×D3)⋊C4C2318yes—
118C2×C4⋊Dic3C2312yes—
119C2×Dic3⋊C4C2318yes—
120C2×C4×Dic3—42no—
121C2×C4×Dic3C2314yes—
122C2×C4×Dic3C2312yes—
123C2×C4×Dic3C2312yes—
124#895—28no—
125#892—28no—
126#885—216no—
127#882—216no—
128#882—416no—
129#874—42no—
130C3×#95C2314yes—
131C3⋊#95C2314yes—
132C3⋊#95C2314yes—
133C3⋊#95C2314yes—
134C22×#41—31no—
135C23×D4—32no—
136#535—616no—
137C2×#127—34no—
138C2×#127—32no—
139C2×#127—31no—
140C2×#127C2×D311yescharacteristic
141C2×#124—34no—
142C2×#121—32no—
143C2×#121—32no—
144C2×#121—38no—
145C2×#121—32no—
146C2×#121C2×D312yescomplemented
147C2×#120—34no—
148C2×#120—38no—
149C2×#120—34no—
150C2×#120—38no—
151C2×#117—32no—
152C2×#117C2×D311yescharacteristic, complemented
153C22×C42—31no—
154#394—34no—
155#394—38no—
156#394—38no—
157#394C2×D314yescomplemented
158#392—38no—
159#392—34no—
160#392—38no—
161#392C2×D314yescomplemented
162#391—32no—
163#390—32no—
164#390—32no—
165#390—32no—
166#390—38no—
167#390—32no—
168#390C2×D312yescomplemented
169#388—38no—
170#388—38no—
171#386—34no—
172#384—32no—
173#382—31no—
174#382C2×D311yescharacteristic, complemented
175C4×C22⋊C4—32no—
176C4×C22⋊C4—32no—
177C24×C3C2414yes—
178C24×C3C2411yescharacteristic
179C23×D3—28no—
180C23×D3—44no—
181C23×D3—44no—
182C23×D3—48no—
183C23×D3—48no—
184C23×D3C2414yes—
185C23×D3C2412yes—
186C2×C3×D4—28no—
187C2×C3×D4—24no—
188C2×C3×D4—24no—
189C2×C3×D4—24no—
190C2×C3×D4—48no—
191C2×C3×D4—48no—
192C2×C3×D4—44no—
193C22×C3×C4—22no—
194C22×C3×C4—22no—
195C22×C3×C4C2×D414yes—
196C22×C3×C4C2412yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2414yes—
199C22×C3×C4C2411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C2×#67—28no—
202C2×#67—28no—
203C2×#67—28no—
204C2×#67—216no—
205C2×#67—42no—
206C2×#67—42no—
207C2×#67—44no—
208C2×#67—44no—
209C2×#67—48no—
210C2×#67—48no—
211C2×#67—48no—
212C2×#67—48no—
213C2×#67C2×D418yes—
214C22×Dic3—28no—
215C22×Dic3—42no—
216C22×Dic3C2414yes—
217C22×Dic3C2412yes—
218C22×Dic3C2414yes—
219D3×D4—432no—
220#235—416no—
221C2×D12—24no—
222C2×D12—24no—
223C2×D12—24no—
224C2×D12—28no—
225C2×D12—41no—
226C2×D12—41no—
227C2×D12—44no—
228C2×D12—48no—
229C2×D12—48no—
230C2×D12C2×D414yes—
231C2×C4×D3—24no—
232C2×C4×D3—24no—
233C2×C4×D3C2×D418yes—
234C2×Dic6—41no—
235C2×Dic6—41no—
236C2×Dic6C2×D414yes—
237C3×C4⋊C4—24no—
238C3×C4⋊C4—42no—
239C3×C22⋊C4—28no—
240C3×C22⋊C4—48no—
241C3×C22⋊C4—42no—
242C3×C42—22no—
243C3×C42—22no—
244#217—28no—
245#217—216no—
246(C2×D3)⋊C4—28no—
247(C2×D3)⋊C4—28no—
248(C2×D3)⋊C4—28no—
249(C2×D3)⋊C4—28no—
250(C2×D3)⋊C4—216no—
251(C2×D3)⋊C4—28no—
252(C2×D3)⋊C4—48no—
253C4⋊Dic3—24no—
254Dic3⋊C4—48no—
255C4×Dic3—24no—
256C4×Dic3—28no—
257C4×Dic3—42no—
258C25—34no—
259C2×#41—38no—
260C2×#41—122no—
261C2×#41—122no—
262C22×Q8—31no—
263C22×D4—32no—
264C22×D4—34no—
265C22×D4—38no—
266C22×D4—34no—
267C22×D4—34no—
268C22×D4—31no—
269C22×D4—32no—
270C22×D4—34no—
271C22×D4—34no—
272C22×D4—68no—
273C22×D4—124no—
274C22×D4—124no—
275C22×D4C22×D312yes—
276C22×D4C22×D314yes—
277C22×D4C22×D314yes—
278C23×C4—32no—
279C23×C4—32no—
280C23×C4C22×D311yescharacteristic
281#127—42no—
282#127—616no—
283#127—122no—
284#127—124no—
285#124—616no—
286#121—28no—
287#121—68no—
288#121—68no—
289#121—68no—
290#121—1216no—
291#120—616no—
292#120—616no—
293#120—1216no—
294#120—1216no—
295#117—42no—
296#117—68no—
297C2×C4⋊C4—32no—
298C2×C4⋊C4—38no—
299C2×C4⋊C4C22×D312yes—
300C2×C4⋊C4C22×D312yes—
301C2×C22⋊C4—34no—
302C2×C22⋊C4—38no—
303C2×C22⋊C4—34no—
304C2×C22⋊C4—34no—
305C2×C22⋊C4—34no—
306C2×C22⋊C4—34no—
307C2×C22⋊C4—34no—
308C2×C22⋊C4—38no—
309C2×C22⋊C4—38no—
310C2×C22⋊C4C22×D314yes—
311C2×C22⋊C4C22×D312yes—
312C2×C22⋊C4C22×D318yes—
313C2×C42—34no—
314C2×C42—32no—
315C2×C42—32no—
316C2×C42—122no—
317C2×C42C22×D311yescharacteristic
318C2×C42C22×D311yescharacteristic
319#95—34no—
320#95—34no—
321#95—34no—
322#95C22×D314yes—
323C23×C3—21no—
324C23×C3—24no—
325C23×C3—21no—
326C23×C3—24no—
327C23×C3—28no—
328C23×C3—28no—
329C23×C3—24no—
330C23×C3—24no—
331C23×C3C22×D412yes—
332C23×C3C2511yescharacteristic
333C22×D3—24no—
334C22×D3—24no—
335C22×D3—22no—
336C22×D3—22no—
337C22×D3—24no—
338C22×D3—28no—
339C22×D3—28no—
340C22×D3—24no—
341C22×D3—24no—
342C22×D3—416no—
343C22×D3—88no—
344C22×D3—88no—
345C22×D3—816no—
346C22×D3—816no—
347C22×D3C22×D414yes—
348C3×D4—48no—
349C3×D4—88no—
350C3×D4—88no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—28no—
354C2×C3×C4—24no—
355C2×C3×C4—28no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4C22×D412yes—
361C2×C3×C4C22×D412yes—
362C2×C3×C4H(5,𝔽2)12yes—
363C2×C3×C4H(5,𝔽2)12yes—
364#67—48no—
365#67—416no—
366#67—816no—
367#67—88no—
368#67—88no—
369C2×Dic3—24no—
370C2×Dic3—24no—
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—24no—
374C2×Dic3—28no—
375C2×Dic3—48no—
376C2×Dic3C22×D414yes—
377C2×Dic3H(5,𝔽2)18yes—
378D12—44no—
379D12—48no—
380D12—88no—
381D12—88no—
382C4×D3—216no—
383Dic6—44no—
384C24—32no—
385C24—34no—
386C24—68no—
387C24—124no—
388C24—124no—
389C24—128no—
390C24—128no—
391C24C23×D314yes—
392C24C23×D311yescharacteristic
393#41—1216no—
394C2×Q8—34no—
395C2×Q8—121no—
396C2×Q8—121no—
397C2×D4—28no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—38no—
402C2×D4—34no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—44no—
406C2×D4—68no—
407C2×D4—68no—
408C2×D4—616no—
409C2×D4—68no—
410C2×D4—68no—
411C2×D4—64no—
412C2×D4—64no—
413C2×D4—64no—
414C2×D4—121no—
415C2×D4—121no—
416C2×D4—128no—
417C2×D4—128no—
418C2×D4—124no—
419C2×D4—1232no—
420C2×D4—124no—
421C2×D4—124no—
422C2×D4—122no—
423C2×D4—122no—
424C2×D4—128no—
425C2×D4—128no—
426C2×D4—128no—
427C2×D4—128no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—32no—
431C22×C4—34no—
432C22×C4—38no—
433C22×C4—34no—
434C22×C4—68no—
435C22×C4—64no—
436C22×C4—64no—
437C22×C4—122no—
438C22×C4D3×D414yes—
439C22×C4C23×D312yes—
440C22×C4C23×D312yes—
441C22×C4C23×D314yes—
442C22×C4C23×D311yescharacteristic
443C22×C4C23×D311yescharacteristic
444C4⋊C4—24no—
445C4⋊C4—42no—
446C4⋊C4—64no—
447C4⋊C4—128no—
448C22⋊C4—28no—
449C22⋊C4—42no—
450C22⋊C4—48no—
451C22⋊C4—68no—
452C22⋊C4—68no—
453C22⋊C4—68no—
454C22⋊C4—616no—
455C22⋊C4—68no—
456C22⋊C4—68no—
457C22⋊C4—616no—
458C22⋊C4—68no—
459C22⋊C4—128no—
460C42—22no—
461C42—22no—
462C42—68no—
463C42—64no—
464C42—122no—
465C22×C3—22no—
466C22×C3—24no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3—44no—
471C22×C3—44no—
472C22×C3—48no—
473C22×C3—48no—
474C22×C3C23×D411yescharacteristic
475C22×C3C2×H(5,𝔽2)11yescharacteristic
476C22×C3C2×H(5,𝔽2)11yescharacteristic
477C22×C3C2×H(5,𝔽2)14yes—
478C2×D3—24no—
479C2×D3—24no—
480C2×D3—24no—
481C2×D3—44no—
482C2×D3—44no—
483C2×D3—44no—
484C2×D3—44no—
485C2×D3—44no—
486C2×D3—48no—
487C2×D3—48no—
488C2×D3—832no—
489C3×C4—22no—
490C3×C4—44no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4#53414yes—
494Dic3—24no—
495Dic3—28no—
496Dic3—44no—
497C23—21no—
498C23—21no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—28no—
504C23—28no—
505C23—34no—
506C23—62no—
507C23—62no—
508C23—64no—
509C23—64no—
510C23—64no—
511C23—64no—
512C23—64no—
513C23—68no—
514C23—68no—
515C23—1216no—
516C23—248no—
517C23—248no—
518C23—2416no—
519C23—2416no—
520C23C2×D3×D412yes—
521C23C24×D311yescharacteristic
522Q8—124no—
523D4—48no—
524D4—88no—
525D4—88no—
526D4—124no—
527D4—128no—
528D4—1216no—
529D4—128no—
530D4—248no—
531D4—248no—
532D4—2416no—
533D4—248no—
534D4—248no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—34no—
545C2×C4—38no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—62no—
549C2×C4—64no—
550C2×C4—616no—
551C2×C4—64no—
552C2×C4—68no—
553C2×C4—128no—
554C2×C4C2×D3×D412yes—
555C2×C4C2×D3×D412yes—
556C2×C4#100412yes—
557C2×C4#100912yes—
558C2×C3—42no—
559C2×C3—84no—
560C2×C3C2×#53411yescharacteristic
561C2×C3#349812yes—
562C2×C3#353712yes—
563C2×C3#354912yes—
564D3—44no—
565D3—84no—
566C22—24no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—44no—
571C22—44no—
572C22—48no—
573C22—44no—
574C22—48no—
575C22—64no—
576C22—64no—
577C22—64no—
578C22—124no—
579C22—124no—
580C22—124no—
581C22—124no—
582C22—128no—
583C22—124no—
584C22—128no—
585C22—2432no—
586C22C22×D3×D411yescharacteristic
587C22C2×#100411yescharacteristic
588C22C2×#100911yescharacteristic
589C22D3×H(5,𝔽2)14yes—
590C4—22no—
591C4—44no—
592C4—42no—
593C4—42no—
594C4—64no—
595C4—68no—
596C4—124no—
597C4#560514yes—
598C3#3729711yescharacteristic, complemented, Sylow
599C2—42no—
600C2—84no—
601C2—124no—
602C2—244no—
603C2C2×#560511yescharacteristic
604C2#8631612yes—
605C2#8632612yes—
606C2#8649112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2619C2
hypercenter#2619D3
hypocenterC3#37297
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. C2×#5607C2semidirect
2. C3×#2619C2semidirect
3. #83175C2semidirect
4. #82700C2semidirect
5. #82286C2semidirect
6. #82228C2semidirect
7. #77138C2semidirect
8. #77047C2semidirect
9. #76764C2semidirect
10. #76334C2semidirect
11. #76314C2semidirect
12. #74166C2semidirect
14. C22×D3×D4C22semidirect
17. C2×C3×#121C22semidirect
18. C2×C3×#117C22semidirect
21. C2×#939C22semidirect
23. C2×#937C22semidirect
24. C2×#937C22semidirect
25. C2×#930C22semidirect
26. C22×C4×Dic3C22semidirect
29. C2×#892C22semidirect
30. C2×#885C22semidirect
32. C2×#882C22semidirect
34. C3×#394C22semidirect
35. C3×#392C22semidirect
36. C3×#390C22semidirect
37. C3×#382C22semidirect
38. #5223C22semidirect
39. #5221C22semidirect
40. #4991C22semidirect
42. #4988C22semidirect
43. #4975C22semidirect
44. #4973C22semidirect
45. #4960C22semidirect
46. #4960C22semidirect
47. #4959C22semidirect
48. #4958C22semidirect
49. #4957C22semidirect
50. #4945C22semidirect
51. #4939C22semidirect
52. C4×(C2×D3)⋊C4C22semidirect
53. C4×(C2×D3)⋊C4C22semidirect
54. #4936C22semidirect
55. #4672C22semidirect
56. #4669C22semidirect
63. #2619D3semidirect
85. C2×#235D4semidirect
146. C2×#121C2×D3semidirect
152. C2×#117C2×D3semidirect
157. #394C2×D3semidirect
161. #392C2×D3semidirect
168. #390C2×D3semidirect
174. #382C2×D3semidirect
598. 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×#26192
#831752
#827002
#822862
#822282
#771382
#770472
#767642
#763342
#763142
#741662
#372973