Finite Group Databasebeta

← #11659383

Subgroups — #11659383

Order 768.

2666 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic, complemented
2C22×#942C211yesmaximal, characteristic
3#85731C211yesmaximal, characteristic
4C2×#5225C211yesmaximal, characteristic, complemented
5#82549C214yesmaximal, complemented
6#82548C214yesmaximal, complemented
7#82547C218yesmaximal, complemented
8C4×#942C218yesmaximal, complemented
9#82213C211yesmaximal, characteristic
10#82207C211yesmaximal, characteristic, complemented
11#82202C211yesmaximal, characteristic, complemented
12#60169—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#235C2211yescharacteristic
15C3×#519—22no—
16C3×#519—22no—
17C3×#519C2214yes—
18C3×#519C2214yescomplemented
19C2×C3×C4×Q8—22no—
20C2×C3×C4×Q8C2214yescomplemented
21C2×C3×#117—22no—
22C2×C3×#117C2211yescharacteristic
23C2×C3×#117C2211yescharacteristic
24C22×#216C2212yes—
25C2×#942C4116yescomplemented
26C2×#942C2214yes—
27C2×#942C2214yes—
28C22×#215C2212yes—
29C2×#926C2211yescharacteristic
30C22×#208C2211yescharacteristic
31C22×#208C2211yescharacteristic
32#5523C2214yes—
33#5522—22no—
34#5521C2212yes—
35D3×#117—22no—
36#5516C2212yes—
37#5504—22no—
38#5486—22no—
39C4×#235C2214yes—
40#5225C2214yes—
41#5225C2214yescomplemented
42C2×#835C2214yescomplemented
43#5030C22116yescomplemented
44C4×#216C22116yescomplemented
45#5026C22116yescomplemented
46C4×#215C22116yescomplemented
47#5004C2212yes—
48#5002C2214yes—
49#5001C2212yes—
50#4998C2214yes—
51#4997C2214yes—
52#4997C2214yescomplemented
53#4995C2214yescomplemented
54#4966C2214yes—
55#4964C2214yescomplemented
56C4×#208C2214yescomplemented
57C22×#137—31no—
58#3428—31no—
59C2×#519D311yescharacteristic, complemented
60#2950—34no—
61#2949—34no—
62#2947—38no—
63C4×#137—38no—
64#2911—31no—
65#2896—31no—
66C2×#419—31no—
67#2873—31no—
68C22×C3×Q8—21no—
69C22×C3×Q8—21no—
70C22×C3×Q8—21no—
71C22×C3×Q8—24no—
72C22×C3×Q8C2×C418yescomplemented
73C22×C3×Q8C2312yes—
74C22×C3×Q8C2312yes—
75C22×C3×Q8C2312yes—
76C23×C3×C4—21no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C2×#235—21no—
80C2×#235—21no—
81C2×#235—24no—
82C2×#235C2×C418yes—
83C2×#235C2312yes—
84C2×#235C2312yes—
85C22×D12C2311yescharacteristic
86C22×C4×D3—21no—
87C22×Dic6C2311yescharacteristic
88C3×C4×Q8—28no—
89C3×C4×Q8—28no—
90C3×C4×Q8—28no—
91C3×C4×Q8C23116yes—
92C3×#117—21no—
93C3×#117—21no—
94C3×#117—28no—
95C3×#117—28no—
96C3×#117D412yes—
97C3×#117D412yes—
98C3×#117C2312yes—
99C3×#117C2312yes—
100C3×#117C2312yes—
101C3×#117C2312yes—
102C2×C3×C4⋊C4—24no—
103C2×C3×C4⋊C4D412yes—
104C2×C3×C4⋊C4C2312yes—
105C2×C3×C22⋊C4—21no—
106C2×C3×C22⋊C4D412yes—
107C2×C3×C42—24no—
108C2×C3×C42—21no—
109C2×C3×C42C2312yes—
110C2×#217—21no—
111C2×#216C2×C4116yescomplemented
112C2×#216C2318yes—
113#942C2×C4132yescomplemented
114#942C2×C4132yescomplemented
115C2×#215C2×C4116yescomplemented
116C2×#215C2318yes—
117C4×#67—28no—
118#926—21no—
119#926—21no—
120#926C2312yes—
121#926C2312yes—
122C2×C4⋊Dic3C2312yes—
123C2×C4×Dic3—21no—
124C2×#208—24no—
125C2×#208C2×C418yescomplemented
126C2×#208C2312yes—
127C2×#208C2312yes—
128C2×#208C2312yes—
129C2×#208C2312yes—
130C22×C3⋊C8C2312yes—
131C22×C3⋊C8C2312yes—
132#893—24no—
133#892—24no—
134D3×C4⋊C4—24no—
135#887—24no—
136#67⋊C4—28no—
137D3×C22⋊C4—24no—
138#877—24no—
139C4×D12C2318yes—
140#872—24no—
141C42×D3—24no—
142C4×Dic6C2318yes—
143#835C23116yes—
144#810C2318yes—
145#809C2318yes—
146#808C2318yes—
147#807C2318yes—
148#803C2318yes—
149C4×C3⋊C8C2318yes—
150C22×#41—31no—
151C23×Q8C2×D311yescharacteristic
152C2×#137—316no—
153C2×#137—34no—
154C2×#137—34no—
155C22×Q16—32no—
156C22×SD16—32no—
157C22×M16—31no—
158C22×M16—31no—
159#520—34no—
160#520—62no—
161#519—22no—
162#519—22no—
163#519—32no—
164#519C2×D314yes—
165#519C2×D314yescomplemented
166#518—32no—
167#518—62no—
168C4×#41—34no—
169C4×#41—62no—
170C2×C4×Q8—22no—
171C2×C4×Q8C2×D314yescomplemented
172C2×#117—22no—
173C2×#117—31no—
174C2×#117—62no—
175C2×#117C2×D311yescharacteristic
176C2×#117C2×D311yescharacteristic
177#441—316no—
178#440—316no—
179C4×Q16—316no—
180C4×SD16—316no—
181#428—34no—
182#428—34no—
183#427—34no—
184#419—34no—
185#419—34no—
186#419—32no—
187#418—32no—
188#417—34no—
189#416—34no—
190C2×#103—34no—
191#405—34no—
192#405—34no—
193C4×M16—34no—
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C2×C3×Q8—28no—
197C2×C3×Q8—28no—
198C2×C3×Q8—28no—
199C2×C3×Q8—22no—
200C2×C3×Q8—22no—
201C2×C3×Q8—22no—
202C2×C3×Q8—24no—
203C2×C3×Q8—24no—
204C2×C3×Q8—24no—
205C2×C3×Q8C22×C418yes—
206C2×C3×Q8C22×C418yes—
207C2×C3×Q8C22×C418yes—
208C2×C3×Q8C2414yes—
209C22×C3×C4—28no—
210C22×C3×C4—22no—
211C22×C3×C4—22no—
212C22×C3×C4—22no—
213C22×C3×C4—22no—
214C22×C3×C4C22×C414yes—
215C22×C3×C4C2×D414yes—
216C22×C3×C4C2×D411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4C2×D411yescharacteristic
219C22×C3×C4#4114yes—
220C22×C3×C4C2411yescharacteristic
221C22×C3×C4C2411yescharacteristic
222C22×C3×C4C2411yescharacteristic
223C2×#67—28no—
224C2×#67—22no—
225C2×#67—22no—
226C22×Dic3—21no—
227C22×Dic3—22no—
228#235—28no—
229#235—28no—
230#235—24no—
231#235—24no—
232#235C22×C418yes—
233#235C22×C418yes—
234C2×D12—21no—
235C2×D12—22no—
236C2×D12C22×C414yes—
237C2×D12C2412yes—
238C2×C4×D3—28no—
239C2×C4×D3—22no—
240C2×C4×D3—22no—
241C2×C4×D3—28no—
242C2×Dic6—21no—
243C2×Dic6—22no—
244C2×Dic6C22×C414yes—
245C2×Dic6C2412yes—
246C3×C4⋊C4—216no—
247C3×C4⋊C4—22no—
248C3×C4⋊C4C2×D414yes—
249C3×C4⋊C4C2×D414yes—
250C3×C4⋊C4C2414yes—
251C3×C22⋊C4—22no—
252C3×C22⋊C4—22no—
253C3×C22⋊C4C2×D414yes—
254C3×C22⋊C4C2×D414yes—
255C3×C42—216no—
256C3×C42—22no—
257C3×C42—22no—
258C3×C42C2×D414yes—
259C3×C42C2414yes—
260#217—22no—
261#217—22no—
262#216C22×C4132yes—
263#215C22×C4132yes—
264(C2×D3)⋊C4—28no—
265C4⋊Dic3—22no—
266C4⋊Dic3C2414yes—
267Dic3⋊C4—28no—
268C4×Dic3—28no—
269C4×Dic3—22no—
270C4×Dic3—22no—
271#208—24no—
272#208—24no—
273#208C22×C418yes—
274#208C22×C418yes—
275C2×C3⋊C8—24no—
276C2×C3⋊C8C22×C418yes—
277C2×C3⋊C8C2414yes—
278C2×C3⋊C8C2414yes—
279C2×#41—38no—
280C2×#41—32no—
281C2×#41—32no—
282C2×#41—64no—
283C2×#41—61no—
284C2×#41—61no—
285C22×Q8—24no—
286C22×Q8—21no—
287C22×Q8—21no—
288C22×Q8—21no—
289C22×Q8—31no—
290C22×Q8C4×D318yescomplemented
291C22×Q8C22×D312yes—
292C22×Q8C22×D312yes—
293C22×Q8C22×D312yes—
294C22×D4—31no—
295C22×D4—61no—
296C23×C4—21no—
297C23×C4—61no—
298C23×C4C22×D311yescharacteristic
299#137—332no—
300#137—332no—
301C2×Q16—316no—
302C2×Q16—38no—
303C2×SD16—316no—
304C2×SD16—38no—
305C2×M16—38no—
306C2×M16—32no—
307C2×M16—32no—
308C2×M16—32no—
309C2×M16—32no—
310C2×M16—64no—
311C22×C8—32no—
312C22×C8—32no—
313C4×Q8—28no—
314C4×Q8—28no—
315C4×Q8—28no—
316C4×Q8—38no—
317C4×Q8—64no—
318C4×Q8C22×D3116yes—
319C4×D4—38no—
320C4×D4—68no—
321C4×D4—64no—
322C4×D4—68no—
323#117—21no—
324#117—21no—
325#117—28no—
326#117—28no—
327#117—32no—
328#117—32no—
329#117—61no—
330#117—61no—
331#117—64no—
332#117—64no—
333#117—64no—
334#117#6712yes—
335#117#6712yes—
336#117C22×D312yes—
337#117C22×D312yes—
338#117C22×D312yes—
339#117C22×D312yes—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—64no—
343C2×C4⋊C4#6712yes—
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—64no—
348C2×C22⋊C4#6712yes—
349C2×C42—24no—
350C2×C42—21no—
351C2×C42—61no—
352C2×C42—64no—
353C2×C42C22×D312yes—
354#107—38no—
355#106—38no—
356#103—316no—
357#103—38no—
358#102—38no—
359#97—38no—
360C4×C8—38no—
361C23×C3—42no—
362C23×C3C4×D414yes—
363C23×C3C22×D411yescharacteristic
364C23×C3C22×D411yescharacteristic
365C23×C3C22×D411yescharacteristic
366C22×D3—22no—
367C22×D3—24no—
368C22×D3—82no—
369C3×Q8—28no—
370C3×Q8—28no—
371C3×Q8—28no—
372C3×Q8—24no—
373C3×Q8C23×C418yes—
374C2×C3×C4—28no—
375C2×C3×C4—28no—
376C2×C3×C4—28no—
377C2×C3×C4—24no—
378C2×C3×C4—24no—
379C2×C3×C4—44no—
380C2×C3×C4—44no—
381C2×C3×C4C4×D414yes—
382C2×C3×C4C4×D414yes—
383C2×C3×C4C4×D414yes—
384C2×C3×C4C23×C412yes—
385C2×C3×C4C23×C412yes—
386C2×C3×C4C23×C412yes—
387C2×C3×C4C22×D418yes—
388C2×C3×C4C22×D411yescharacteristic
389C2×C3×C4C22×D411yescharacteristic
390C2×C3×C4C22×D411yescharacteristic
391C2×C3×C4C2×#4112yes—
392C2×C3×C4C2×#4112yes—
393C2×C3×C4C2×#4112yes—
394C2×C3×C4C2511yescharacteristic
395#67—28no—
396#67—28no—
397C2×Dic3—24no—
398C2×Dic3—22no—
399C2×Dic3—24no—
400C2×Dic3—82no—
401D12—24no—
402D12—22no—
403D12C23×C414yes—
404C4×D3—28no—
405C4×D3—28no—
406C4×D3—88no—
407Dic6—24no—
408Dic6—22no—
409Dic6C23×C414yes—
410C3⋊C8—24no—
411C3⋊C8C23×C418yes—
412C24—61no—
413C24C2×#6711yescharacteristic
414#41—38no—
415#41—38no—
416#41—68no—
417#41—68no—
418#41—64no—
419#41—64no—
420C2×Q8—28no—
421C2×Q8—28no—
422C2×Q8—28no—
423C2×Q8—24no—
424C2×Q8—24no—
425C2×Q8—24no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—34no—
430C2×Q8—32no—
431C2×Q8—62no—
432C2×Q8—61no—
433C2×Q8C2×C4×D318yes—
434C2×Q8C2×C4×D318yes—
435C2×Q8C2×C4×D318yes—
436C2×Q8C23×D314yes—
437C2×D4—34no—
438C2×D4—32no—
439C2×D4—68no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—62no—
443C2×D4—61no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—61no—
450C22×C4—68no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—68no—
455C22×C4C2×C4×D314yes—
456C22×C4#23514yes—
457C22×C4C2×#6714yes—
458C22×C4C2×#6711yescharacteristic
459C22×C4C2×#6711yescharacteristic
460C22×C4C2×#6711yescharacteristic
461C22×C4C23×D311yescharacteristic
462C22×C4C23×D311yescharacteristic
463C22×C4C23×D311yescharacteristic
464Q16—332no—
465SD16—332no—
466M16—38no—
467M16—38no—
468M16—64no—
469M16—64no—
470C2×C8—38no—
471C2×C8—34no—
472C2×C8—34no—
473C2×C8—64no—
474C4⋊C4—216no—
475C4⋊C4—22no—
476C4⋊C4—34no—
477C4⋊C4—62no—
478C4⋊C4—68no—
479C4⋊C4C2×#6714yes—
480C4⋊C4C2×#6714yes—
481C4⋊C4C23×D314yes—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—68no—
487C22⋊C4C2×#6714yes—
488C22⋊C4C2×#6714yes—
489C42—216no—
490C42—22no—
491C42—22no—
492C42—62no—
493C42—62no—
494C42—68no—
495C42C2×#6714yes—
496C42C23×D314yes—
497C22×C3—22no—
498C22×C3—22no—
499C22×C3—22no—
500C22×C3—44no—
501C22×C3C2×C4×D412yes—
502C22×C3C2×C4×D412yes—
503C22×C3C2×C4×D412yes—
504C22×C3C23×D411yescharacteristic
505C2×D3—24no—
506C2×D3—44no—
507C2×D3—84no—
508C3×C4—28no—
509C3×C4—28no—
510C3×C4—44no—
511C3×C4C2×C4×D412yes—
512C3×C4C2×C4×D412yes—
513C3×C4C2×C4×D412yes—
514C3×C4C24×C411yescharacteristic
515C3×C4C22×#4111yescharacteristic
516Dic3—24no—
517Dic3—44no—
518C23—42no—
519C23—62no—
520C23—64no—
521C23—242no—
522C23C4×#6714yes—
523C23C22×#6711yescharacteristic
524C23C22×#6711yescharacteristic
525C23C22×#6711yescharacteristic
526Q8—24no—
527Q8—28no—
528Q8—28no—
529Q8—28no—
530Q8—34no—
531Q8—62no—
532Q8—64no—
533Q8C22×C4×D318yes—
534D4—34no—
535D4—68no—
536D4—68no—
537D4—62no—
538D4—64no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—62no—
547C2×C4—64no—
548C2×C4—68no—
549C2×C4—68no—
550C2×C4—64no—
551C2×C4—248no—
552C2×C4—242no—
553C2×C4C4×#6714yes—
554C2×C4C4×#6714yes—
555C2×C4C4×#6714yes—
556C2×C4C22×C4×D312yes—
557C2×C4C22×C4×D312yes—
558C2×C4C22×C4×D312yes—
559C2×C4C2×#23512yes—
560C2×C4C2×#23512yes—
561C2×C4C2×#23512yes—
562C2×C4C22×#6718yes—
563C2×C4C22×#6711yescharacteristic
564C2×C4C22×#6711yescharacteristic
565C2×C4C22×#6711yescharacteristic
566C2×C4C24×D311yescharacteristic
567C8—38no—
568C8—64no—
569C2×C3—22no—
570C2×C3—22no—
571C2×C3—22no—
572C2×C3C22×C4×D411yescharacteristic
573C2×C3#358612yes—
574D3—44no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—44no—
579C22—64no—
580C22—124no—
581C22—244no—
582C22C2×C4×#6712yes—
583C22C2×C4×#6712yes—
584C22C2×C4×#6712yes—
585C22C23×#6711yescharacteristic
586C4—28no—
587C4—28no—
588C4—44no—
589C4—64no—
590C4—124no—
591C4C2×C4×#6712yes—
592C4C2×C4×#6712yes—
593C4C2×C4×#6712yes—
594C4C23×C4×D311yescharacteristic
595C4C22×#23511yescharacteristic
596C3#6016911yescharacteristic, complemented, Sylow
597C2—22no—
598C2—22no—
599C2—22no—
600C2—124no—
601C2C22×C4×#6711yescharacteristic
602C2#8728712yes—
subgroupG/H
centerC22C23×#67
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC2×C3×#519C2
socleC22×C3C23×D4
prime radicalC22×#235C22
cosocleC2×C3×C4C25
hypercenterC2×#519D3
hypocenterC3#60169
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#117 ◁ C2×#519

Complements

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

NH ≅ G/Ntype
1. C2×C3×#519C2semidirect
4. C2×#5225C2semidirect
5. #82549C2semidirect
6. #82548C2semidirect
7. #82547C2semidirect
8. C4×#942C2semidirect
10. #82207C2semidirect
11. #82202C2semidirect
18. C3×#519C22semidirect
20. C2×C3×C4×Q8C22semidirect
25. C2×#942C4semidirect
41. #5225C22semidirect
42. C2×#835C22semidirect
43. #5030C22semidirect
44. C4×#216C22semidirect
45. #5026C22semidirect
46. C4×#215C22semidirect
52. #4997C22semidirect
53. #4995C22semidirect
55. #4964C22semidirect
56. C4×#208C22semidirect
59. C2×#519D3semidirect
72. C22×C3×Q8C2×C4semidirect
111. C2×#216C2×C4semidirect
113. #942C2×C4semidirect
114. #942C2×C4semidirect
115. C2×#215C2×C4semidirect
125. C2×#208C2×C4semidirect
165. #519C2×D3semidirect
171. C2×C4×Q8C2×D3semidirect
290. C22×Q8C4×D3semidirect
596. C3#60169semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C22×#9422
#857312
C2×#52252
#825492
#825482
#825472
C4×#9422
#822132
#822072
#822022
#601693