Finite Group Databasebeta

← #11664497

Subgroups — #11664497

Order 768.

2170 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
1D3×#533C211yesmaximal, characteristic, complemented
2C2×#5725C211yesmaximal, characteristic
3C3×#3139C211yesmaximal, characteristic, complemented
4#83336C211yesmaximal, characteristic, complemented
5#83198C211yesmaximal, characteristic, complemented
6#81740C214yesmaximal, complemented
7#81737C214yesmaximal, complemented
8#81601C214yesmaximal, complemented
9#81598C214yesmaximal, complemented
10#78956C214yesmaximal, complemented
11#78954C214yesmaximal, complemented
12#78526C211yesmaximal, characteristic, complemented
13C4×D3×M16C211yesmaximal, characteristic, complemented
14#60618—31nomaximal, Sylow
15C22×#239C2212yes—
16C3×#533C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×#947C2212yes—
19C2×D3×M16C2212yes—
20#5728C2212yescomplemented
21#5725—22no—
22#5725—22no—
23#5725C2212yes—
24#5725C2212yes—
25#5725C2212yes—
26#5725C2212yes—
27D3×C4⋊Q8C2212yescomplemented
28#5688C2212yescomplemented
29D3×#125C2212yescomplemented
30D3×C22⋊Q8—22no—
31#5622—22no—
32#5578—22no—
33C4×D3×Q8—22no—
34D3×#117C2212yes—
35#5517C2212yes—
36C2×#895—22no—
37C2×#893—22no—
38C2×#874C2211yescharacteristic
39C2×C42×D3C2211yescharacteristic
40#5470C2211yescharacteristic
41C3×#491C2214yescomplemented
42C3×#487C2214yescomplemented
43C3×#419C2214yes—
44C3×C4×M16C2211yescharacteristic
45#5225C2214yes—
46#5131C2214yes—
47Dic3×M16C2212yes—
48#5088C2214yescomplemented
49#5082C2214yescomplemented
50#5079C2214yescomplemented
51#5067C2214yescomplemented
52#5001C2214yes—
53C4×#208C2211yescharacteristic
54#4804C22116yescomplemented
55#4803C22116yescomplemented
56#4800C22116yescomplemented
57#4794C22116yescomplemented
58#4713C2214yescomplemented
59#4707C2212yescomplemented
60D3×#97C2212yescomplemented
61#4694C2214yescomplemented
62C4×ΓL(1,𝔽25)C2212yescomplemented
63C4×D3×C8C2212yescomplemented
64C2×#533—31no—
65C2×#532—31no—
66#3142—34no—
67#3141—34no—
68#3140—34no—
69#3139—31no—
70#3139D311yescharacteristic, complemented
71#3138—31no—
72#3138—31no—
73#3135—34no—
74#3131—34no—
75#2896—34no—
76C2×C4×M16—31no—
77C22×C3×Q8C2312yes—
78C2×#239—22no—
79C2×#239—22no—
80C2×#239—216no—
81C2×#239C2314yes—
82C2×#239C2314yes—
83C2×D3×Q8—22no—
84C22×D12—22no—
85C22×D12—22no—
86C22×D12C2312yes—
87C22×C4×D3—22no—
88C22×C4×D3D412yes—
89C22×C4×D3C2311yescharacteristic
90C2×C3×M16C2312yes—
91C3×C4⋊Q8—22no—
92C3×C4⋊Q8C2312yes—
93C3×C4⋊Q8C2312yes—
94C3×C4⋊Q8C2312yes—
95C3×#125C2312yes—
96C3×C22⋊Q8—22no—
97C3×C4×Q8—22no—
98C3×#117C2312yes—
99C2×C3×C4⋊C4—22no—
100C2×C3×C42C2311yescharacteristic
101#947—24no—
102#947C2314yes—
103#947C2314yes—
104#946—22no—
105Q8×Dic3—22no—
106#944—22no—
107C2×(C2×D3)⋊C4—24no—
108#926C2312yes—
109#924—22no—
110C2×C4×Dic3D412yes—
111C2×C4×Dic3C2311yescharacteristic
112C2×#208C2312yes—
113D3×M16—28no—
114C2×ΓL(1,𝔽25)C2314yes—
115C2×D3×C8C2314yes—
116#897—22no—
117#896—24no—
118#895—24no—
119#895—24no—
120#895—24no—
121#895—24no—
122#893—24no—
123#893—24no—
124#893—24no—
125#893—24no—
126#892C2314yes—
127D3×C4⋊C4—24no—
128D3×C4⋊C4—22no—
129D3×C4⋊C4C2314yes—
130#890C2314yes—
131#889—24no—
132#888—22no—
133#888C2314yes—
134#887—24no—
135#887—22no—
136D3×C22⋊C4—22no—
137#878—24no—
138#877—22no—
139#874—22no—
140#874C2312yes—
141#874C2312yes—
142#872—22no—
143C42×D3—22no—
144C42×D3D412yes—
145C42×D3D412yes—
146C42×D3C2311yescharacteristic
147C42×D3C2311yescharacteristic
148C42×D3C2311yescharacteristic
149C42×D3C2311yescharacteristic
150#870C2312yes—
151#869C2312yes—
152C4×Dic6—22no—
153C4×Dic6—24no—
154C3×#103C23116yes—
155C3×#97C2312yes—
156C3×C4×C8C2312yes—
157#835C23116yes—
158#821C23116yes—
159#815C2314yes—
160C8×Dic3C2314yes—
161#809C23116yes—
162#803C2312yes—
163C4×C3⋊C8C2312yes—
164C22×#41—32no—
165C22×M16—32no—
166#533—32no—
167#533—32no—
168#533—32no—
169#533—31no—
170#533—62no—
171#533—62no—
172#533C2×D311yescharacteristic
173#532—32no—
174#532—32no—
175#532—32no—
176#532—32no—
177#532—62no—
178#532—62no—
179C2×C4⋊Q8—32no—
180C2×C4⋊Q8C2×D311yescharacteristic
181C2×#127—31no—
182C2×#125—32no—
183C2×#124—32no—
184C2×C22⋊Q8—62no—
185C2×#121—62no—
186C2×C4×Q8—62no—
187C2×C4×D4—62no—
188C2×#117—32no—
189C22×C42—31no—
190#491—34no—
191#491C2×D314yescomplemented
192#490—34no—
193#490—34no—
194#489—316no—
195#488—316no—
196#487—34no—
197#487C2×D314yescomplemented
198#486—34no—
199#486—34no—
200#419—34no—
201#419C2×D314yes—
202#418—34no—
203#418—34no—
204#417—316no—
205#416—316no—
206C4×M16—32no—
207C4×M16—31no—
208C4×M16—32no—
209C4×M16—32no—
210C4×M16C2×D311yescharacteristic
211C2×#97—32no—
212C2×C4×C8—32no—
213C23×D3—22no—
214C23×D3C2×D411yescharacteristic
215C2×C3×Q8—22no—
216C2×C3×Q8—22no—
217C2×C3×Q8—28no—
218C2×C3×Q8C2414yes—
219C22×C3×C4—22no—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3C2×D412yes—
223C22×Dic3C2×D411yescharacteristic
224#239—216no—
225#239—216no—
226#239—48no—
227#239—48no—
228D3×Q8—48no—
229C2×D12—28no—
230C2×D12—24no—
231C2×D12—24no—
232C2×D12—22no—
233C2×D12—24no—
234C2×D12—24no—
235C2×D12—48no—
236C2×D12—48no—
237C2×D12C2414yes—
238C2×C4×D3—24no—
239C2×C4×D3—24no—
240C2×C4×D3—24no—
241C2×C4×D3—44no—
242C2×C4×D3—48no—
243C2×C4×D3C2×D412yes—
244C2×C4×D3C2×D412yes—
245C2×C4×D3C2×D412yes—
246C2×C4×D3C2×D412yes—
247C2×C4×D3C2×D411yescharacteristic
248C2×C4×D3C2×D411yescharacteristic
249C2×C4×D3#4118yes—
250C2×C4×D3C2411yescharacteristic
251C2×C4×D3C2411yescharacteristic
252C2×Dic6—22no—
253C2×Dic6—24no—
254C3×M16—24no—
255C2×C3×C8C2414yes—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—24no—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—24no—
260C3×C4⋊C4C2414yes—
261C3×C22⋊C4—22no—
262C3×C42—22no—
263C3×C42C2×D412yes—
264C3×C42C2411yescharacteristic
265C3×C42C2411yescharacteristic
266#217—22no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—28no—
269(C2×D3)⋊C4—28no—
270C4⋊Dic3—22no—
271C4⋊Dic3—24no—
272C4⋊Dic3C2414yes—
273Dic3⋊C4—24no—
274Dic3⋊C4—24no—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277C4×Dic3—22no—
278C4×Dic3—24no—
279C4×Dic3C2×D412yes—
280C4×Dic3C2×D412yes—
281C4×Dic3C2×D412yes—
282C4×Dic3C2×D412yes—
283C4×Dic3C2×D412yes—
284C4×Dic3C2411yescharacteristic
285C4×Dic3C2411yescharacteristic
286#208—24no—
287C2×C3⋊C8C2414yes—
288ΓL(1,𝔽25)—28no—
289D3×C8—28no—
290C2×#41—34no—
291C2×#41—34no—
292C2×#41—62no—
293C2×#41—62no—
294C2×#41—616no—
295C22×Q8—62no—
296C22×Q8C22×D312yes—
297C22×D4—32no—
298C22×D4—62no—
299C22×D4—62no—
300C23×C4—31no—
301C23×C4—32no—
302C23×C4—62no—
303C2×M16—32no—
304C2×M16—34no—
305C2×M16—68no—
306C2×M16C22×D312yes—
307C22×C8—34no—
308C4⋊Q8—22no—
309C4⋊Q8—34no—
310C4⋊Q8—32no—
311C4⋊Q8—62no—
312C4⋊Q8—62no—
313C4⋊Q8C22×D312yes—
314C4⋊Q8C22×D312yes—
315C4⋊Q8C22×D312yes—
316#127—32no—
317#127—32no—
318#127—62no—
319#125—34no—
320#125—32no—
321#125—64no—
322#125C22×D312yes—
323#124—34no—
324#124—34no—
325#124—64no—
326C22⋊Q8—22no—
327C22⋊Q8—64no—
328C22⋊Q8—64no—
329C22⋊Q8—62no—
330C22⋊Q8—62no—
331C22⋊Q8—62no—
332#121—64no—
333#121—64no—
334#121—64no—
335#121—64no—
336C4×Q8—22no—
337C4×Q8—64no—
338C4×Q8—64no—
339C4×Q8—62no—
340C4×Q8—62no—
341C4×Q8—62no—
342C4×D4—64no—
343C4×D4—64no—
344C4×D4—64no—
345C4×D4—64no—
346#117—32no—
347#117—34no—
348#117—62no—
349#117—62no—
350#117C22×D312yes—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—34no—
353C2×C4⋊C4—64no—
354C2×C4⋊C4—62no—
355C2×C22⋊C4—62no—
356C2×C22⋊C4—64no—
357C2×C42—32no—
358C2×C42—31no—
359C2×C42—32no—
360C2×C42—32no—
361C2×C42—31no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—31no—
365C2×C42—62no—
366C2×C42C22×D311yescharacteristic
367#103—316no—
368#103C22×D3116yes—
369#102—316no—
370#102—316no—
371#97—34no—
372#97—32no—
373#97C22×D312yes—
374C4×C8—34no—
375C4×C8—32no—
376C4×C8C22×D312yes—
377C23×C3C22×D411yescharacteristic
378C22×D3—24no—
379C22×D3—42no—
380C22×D3—48no—
381C22×D3—44no—
382C22×D3#12414yes—
383C22×D3C22×D411yescharacteristic
384C22×D3C22×D411yescharacteristic
385C3×Q8—28no—
386C3×Q8—44no—
387C3×Q8—44no—
388C2×C3×C4—24no—
389C2×C3×C4—24no—
390C2×C3×C4—24no—
391C2×C3×C4—44no—
392C2×C3×C4C22×D412yes—
393C2×C3×C4C22×D411yescharacteristic
394C2×C3×C4C22×D412yes—
395C2×C3×C4C2×#4114yes—
396C2×C3×C4C2511yescharacteristic
397C2×Dic3—24no—
398C2×Dic3—42no—
399C2×Dic3#12414yes—
400C2×Dic3C22×D412yes—
401C2×Dic3C22×D412yes—
402C2×Dic3C22×D411yescharacteristic
403C2×Dic3C22×D411yescharacteristic
404D12—28no—
405D12—48no—
406D12—48no—
407D12—48no—
408D12—44no—
409D12—48no—
410C4×D3—24no—
411C4×D3—24no—
412C4×D3—42no—
413C4×D3—42no—
414C4×D3—48no—
415C4×D3—48no—
416C4×D3—48no—
417C4×D3#12414yes—
418C4×D3#12414yes—
419C4×D3C2×#4114yes—
420C4×D3C2×#4114yes—
421Dic6—48no—
422Dic6—44no—
423C3×C8—24no—
424C3⋊C8—24no—
425C24—31no—
426C24—62no—
427#41—616no—
428#41—616no—
429#41—128no—
430#41—128no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—28no—
434C2×Q8—64no—
435C2×Q8—62no—
436C2×Q8—128no—
437C2×Q8C23×D314yes—
438C2×D4—34no—
439C2×D4—64no—
440C2×D4—64no—
441C2×D4—64no—
442C2×D4—64no—
443C2×D4—62no—
444C2×D4—68no—
445C2×D4—128no—
446C2×D4—128no—
447C22×C4—22no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—31no—
451C22×C4—38no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—32no—
455C22×C4—32no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—32no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4—64no—
462C22×C4—124no—
463C22×C4—128no—
464C22×C4D3×D412yes—
465C22×C4C23×D311yescharacteristic
466M16—24no—
467M16—68no—
468M16—64no—
469C2×C8—34no—
470C2×C8—68no—
471C2×C8C23×D314yes—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—34no—
477C4⋊C4—64no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—64no—
481C4⋊C4—64no—
482C4⋊C4—62no—
483C4⋊C4C23×D314yes—
484C22⋊C4—22no—
485C22⋊C4—68no—
486C22⋊C4—68no—
487C22⋊C4—64no—
488C22⋊C4—62no—
489C42—22no—
490C42—32no—
491C42—32no—
492C42—32no—
493C42—32no—
494C42—32no—
495C42—31no—
496C42—31no—
497C42—64no—
498C42—62no—
499C42D3×D412yes—
500C42C23×D311yescharacteristic
501C42C23×D311yescharacteristic
502C22×C3—22no—
503C22×C3C2×#12412yes—
504C22×C3C23×D411yescharacteristic
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—44no—
508C2×D3—44no—
509C2×D3—48no—
510C2×D3—88no—
511C2×D3C2×#12412yes—
512C2×D3C2×#12412yes—
513C3×C4—22no—
514C3×C4—44no—
515C3×C4—44no—
516C3×C4C2×#12412yes—
517C3×C4#57514yes—
518C3×C4C22×#4112yes—
519Dic3—22no—
520Dic3—22no—
521Dic3—44no—
522Dic3C2×#12412yes—
523Dic3C2×#12412yes—
524C23—31no—
525C23—31no—
526C23—34no—
527C23—64no—
528C23—122no—
529C23—128no—
530C23—124no—
531C23C2×D3×D411yescharacteristic
532Q8—28no—
533Q8—44no—
534Q8—44no—
535Q8—128no—
536Q8—124no—
537D4—68no—
538D4—128no—
539D4—128no—
540D4—128no—
541D4—124no—
542D4—128no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—34no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—34no—
550C2×C4—34no—
551C2×C4—34no—
552C2×C4—34no—
553C2×C4—32no—
554C2×C4—32no—
555C2×C4—44no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—122no—
560C2×C4—122no—
561C2×C4—128no—
562C2×C4—128no—
563C2×C4—128no—
564C2×C4—122no—
565C2×C4C2×D3×D412yes—
566C2×C4C2×D3×D411yescharacteristic
567C2×C4C2×D3×D412yes—
568C2×C4D3×#4114yes—
569C2×C4C24×D311yescharacteristic
570C8—24no—
571C8—64no—
572C2×C3—22no—
573C2×C3C22×#12411yescharacteristic
574C2×C3C2×#57512yes—
575D3—22no—
576D3—22no—
577D3—84no—
578C22—22no—
579C22—32no—
580C22—32no—
581C22—62no—
582C22—62no—
583C22—124no—
584C22—124no—
585C22—128no—
586C22—248no—
587C22D3×#12412yes—
588C22C22×D3×D411yescharacteristic
589C4—22no—
590C4—32no—
591C4—32no—
592C4—44no—
593C4—44no—
594C4—62no—
595C4—62no—
596C4—124no—
597C4D3×#12412yes—
598C4#577714yes—
599C4C2×D3×#4112yes—
600C3#6061811yescharacteristic, complemented, Sylow
601C2—22no—
602C2—62no—
603C2—62no—
604C2—244no—
605C2C2×D3×#12411yescharacteristic
606C2C2×#577712yes—
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#3139C2
socleC22×C3C23×D4
prime radicalC2×#5725C2
hypercenter#3139D3
hypocenterC3#60618
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
1. D3×#533C2semidirect
3. C3×#3139C2semidirect
4. #83336C2semidirect
5. #83198C2semidirect
6. #81740C2semidirect
7. #81737C2semidirect
8. #81601C2semidirect
9. #81598C2semidirect
10. #78956C2semidirect
11. #78954C2semidirect
12. #78526C2semidirect
13. C4×D3×M16C2semidirect
20. #5728C22semidirect
27. D3×C4⋊Q8C22semidirect
28. #5688C22semidirect
29. D3×#125C22semidirect
41. C3×#491C22semidirect
42. C3×#487C22semidirect
48. #5088C22semidirect
49. #5082C22semidirect
50. #5079C22semidirect
51. #5067C22semidirect
54. #4804C22semidirect
55. #4803C22semidirect
56. #4800C22semidirect
57. #4794C22semidirect
58. #4713C22semidirect
59. #4707C22semidirect
60. D3×#97C22semidirect
61. #4694C22semidirect
62. C4×ΓL(1,𝔽25)C22semidirect
63. C4×D3×C8C22semidirect
70. #3139D3semidirect
191. #491C2×D3semidirect
197. #487C2×D3semidirect
600. C3#60618semidirect

Maximal subgroups

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

subgroupindex
D3×#5332
C2×#57252
C3×#31392
#833362
#831982
#817402
#817372
#816012
#815982
#789562
#789542
#785262
C4×D3×M162
#606183