Finite Group Databasebeta

← #11659385

Subgroups — #11659385

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
1C22×#909C211yesmaximal, characteristic
2#85731C211yesmaximal, characteristic
3C2×#5516C211yesmaximal, characteristic, complemented
4C3×#2873C211yesmaximal, characteristic, complemented
5C2×#5120C211yesmaximal, characteristic, complemented
6#83768C211yesmaximal, characteristic
7#83752C211yesmaximal, characteristic, complemented
8#78573C214yesmaximal, complemented
9#78572C214yesmaximal, complemented
10#78571C218yesmaximal, complemented
11C4×#909C218yesmaximal, complemented
12#60169—31nomaximal, Sylow
13C22×#235C2211yescharacteristic
14C23×Dic6C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C22×C3×M16C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#909C4116yescomplemented
21C2×#909C2214yes—
22C2×#909C2214yes—
23C22×Dic12C2212yes—
24C22×#204C2212yes—
25#5523C2214yes—
26#5522—22no—
27#5521C2212yes—
28D3×#117—22no—
29#5516—22no—
30#5516—22no—
31#5516C2214yes—
32#5516C2212yes—
33#5516C2214yescomplemented
34#5504—22no—
35C2×#887—22no—
36#5486—22no—
37C2×#877—22no—
38C4×#235C2214yes—
39C2×C4×Dic6C2214yescomplemented
40C3×#405C2214yes—
41C3×#405C2214yescomplemented
42C3×C4×M16C2214yescomplemented
43#5133C2214yes—
44#5131C2212yes—
45#5121C2214yescomplemented
46#5121C2214yescomplemented
47#5120C2212yes—
48#5120C2214yes—
49#5120C2214yescomplemented
50#5113C2214yes—
51#5106C2214yes—
52C2×#816C2214yescomplemented
53#4716C22116yescomplemented
54#4710C22116yescomplemented
55C4×Dic12C22116yescomplemented
56C4×#204C22116yescomplemented
57C22×#137—31no—
58#3428—31no—
59C2×#519—31no—
60#2950—34no—
61#2949—34no—
62#2947—38no—
63C4×#137—38no—
64#2911—31no—
65#2896—31no—
66C2×#419—31no—
67#2873D311yescharacteristic, complemented
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C23×Dic3—21no—
71C2×#235—21no—
72C2×#235—21no—
73C2×#235—24no—
74C2×#235C2×C418yes—
75C2×#235C2312yes—
76C2×#235C2312yes—
77C22×D12C2311yescharacteristic
78C22×C4×D3—21no—
79C22×Dic6—21no—
80C22×Dic6—21no—
81C22×Dic6—21no—
82C22×Dic6—24no—
83C22×Dic6C2×C418yescomplemented
84C22×Dic6C2312yes—
85C22×Dic6C2311yescharacteristic
86C22×Dic6C2312yes—
87C22×Dic6C2312yes—
88C2×C3×M16—24no—
89C2×C3×M16C2×C418yescomplemented
90C2×C3×M16C2312yes—
91C2×C3×M16C2312yes—
92C2×C3×M16C2312yes—
93C2×C3×M16C2312yes—
94C22×C3×C8C2312yes—
95C22×C3×C8C2312yes—
96C3×#117D412yes—
97C3×#117D412yes—
98C3×#117C2312yes—
99C3×#117C2312yes—
100C2×C3×C4⋊C4D412yes—
101C2×C3×C22⋊C4D412yes—
102C2×C3×C42C2312yes—
103C2×#217—21no—
104C2×#217—21no—
105C4×#67—28no—
106#926—21no—
107#926—21no—
108#926—21no—
109#926—21no—
110#926C2312yes—
111#926C2312yes—
112#926C2312yes—
113#926C2312yes—
114C2×C4⋊Dic3C2312yes—
115C2×C4⋊Dic3C2312yes—
116C2×Dic3⋊C4—24no—
117C2×C4×Dic3—24no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120#909C2×C4132yescomplemented
121#909C2×C4132yescomplemented
122C2×Dic12C2×C4116yescomplemented
123C2×Dic12C2318yes—
124C2×#204C2×C4116yescomplemented
125C2×#204C2318yes—
126#893—24no—
127#892—24no—
128D3×C4⋊C4—24no—
129#887—28no—
130#887—28no—
131#887—24no—
132#67⋊C4—28no—
133D3×C22⋊C4—24no—
134#877—28no—
135#877—28no—
136#877—24no—
137C4×D12C2318yes—
138#872—24no—
139C42×D3—24no—
140C4×Dic6—28no—
141C4×Dic6C23116yes—
142C4×Dic6C2318yes—
143C3×#97C2318yes—
144C3×C4×C8C2318yes—
145#821C2318yes—
146#818C2318yes—
147#817C2318yes—
148#816C23116yes—
149#816C2318yes—
150C22×#41—31no—
151C23×Q8—31no—
152C2×#137—316no—
153C2×#137—34no—
154C2×#137—34no—
155C22×Q16—32no—
156C22×SD16—32no—
157C22×M16C2×D311yescharacteristic
158C22×M16C2×D311yescharacteristic
159#520—34no—
160#520—62no—
161#519—34no—
162#519—34no—
163#519—32no—
164#519—62no—
165#519—62no—
166#518—32no—
167#518—62no—
168C4×#41—34no—
169C4×#41—62no—
170C2×C4×Q8—34no—
171C2×C4×Q8—62no—
172C2×#117—31no—
173C2×#117—31no—
174C2×#117—62no—
175C2×#117—62no—
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—32no—
186#419—34no—
187#418—32no—
188#417—34no—
189#416—34no—
190C2×#103—34no—
191#405C2×D314yes—
192#405C2×D314yescomplemented
193C4×M16C2×D314yescomplemented
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C22×C3×C4C22×C414yes—
197C22×C3×C4C2×D414yes—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4#4114yes—
202C22×C3×C4C2411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×C3×C4C2411yescharacteristic
205C2×#67—28no—
206C2×#67—22no—
207C2×#67—22no—
208C22×Dic3—28no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—21no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215#235—28no—
216#235—28no—
217#235—24no—
218#235—24no—
219#235C22×C418yes—
220#235C22×C418yes—
221C2×D12—21no—
222C2×D12—22no—
223C2×D12C22×C414yes—
224C2×D12C2412yes—
225C2×C4×D3—28no—
226C2×C4×D3—22no—
227C2×C4×D3—22no—
228C2×C4×D3—28no—
229C2×Dic6—28no—
230C2×Dic6—28no—
231C2×Dic6—28no—
232C2×Dic6—22no—
233C2×Dic6—22no—
234C2×Dic6—22no—
235C2×Dic6—21no—
236C2×Dic6—24no—
237C2×Dic6—24no—
238C2×Dic6—24no—
239C2×Dic6—22no—
240C2×Dic6C22×C418yes—
241C2×Dic6C22×C414yes—
242C2×Dic6C22×C418yes—
243C2×Dic6C22×C418yes—
244C2×Dic6C2414yes—
245C2×Dic6C2412yes—
246C3×M16—24no—
247C3×M16—24no—
248C3×M16C22×C418yes—
249C3×M16C22×C418yes—
250C2×C3×C8—24no—
251C2×C3×C8C22×C418yes—
252C2×C3×C8C2414yes—
253C2×C3×C8C2414yes—
254C3×C4⋊C4C2×D414yes—
255C3×C4⋊C4C2×D414yes—
256C3×C22⋊C4C2×D414yes—
257C3×C22⋊C4C2×D414yes—
258C3×C42C2×D414yes—
259C3×C42C2414yes—
260#217—22no—
261#217—22no—
262#217—22no—
263#217—22no—
264(C2×D3)⋊C4—28no—
265C4⋊Dic3—22no—
266C4⋊Dic3—22no—
267C4⋊Dic3C2414yes—
268C4⋊Dic3C2414yes—
269Dic3⋊C4—216no—
270Dic3⋊C4—28no—
271C4×Dic3—216no—
272C4×Dic3—28no—
273C4×Dic3—22no—
274C4×Dic3—22no—
275C4×Dic3—22no—
276C4×Dic3—22no—
277Dic12C22×C4132yes—
278#204C22×C4132yes—
279C2×#41—38no—
280C2×#41—32no—
281C2×#41—32no—
282C2×#41—61no—
283C2×#41—61no—
284C2×#41—64no—
285C22×Q8—38no—
286C22×Q8—32no—
287C22×Q8—32no—
288C22×Q8—31no—
289C22×Q8—32no—
290C22×Q8—61no—
291C22×Q8—61no—
292C22×Q8—61no—
293C22×Q8—64no—
294C22×D4—31no—
295C22×D4—61no—
296C23×C4—61no—
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—24no—
306C2×M16C4×D318yescomplemented
307C2×M16C22×D312yes—
308C2×M16C22×D312yes—
309C2×M16C22×D312yes—
310C2×M16C22×D312yes—
311C22×C8C22×D312yes—
312C22×C8C22×D312yes—
313C4×Q8—316no—
314C4×Q8—38no—
315C4×Q8—68no—
316C4×Q8—68no—
317C4×Q8—64no—
318C4×Q8—68no—
319C4×D4—38no—
320C4×D4—68no—
321C4×D4—64no—
322C4×D4—68no—
323#117—32no—
324#117—32no—
325#117—32no—
326#117—32no—
327#117—61no—
328#117—61no—
329#117—61no—
330#117—61no—
331#117—64no—
332#117—64no—
333#117—68no—
334#117—68no—
335#117—64no—
336#117D1212yes—
337#117D1212yes—
338#117C22×D312yes—
339#117C22×D312yes—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—64no—
343C2×C4⋊C4—64no—
344C2×C4⋊C4D1212yes—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—64no—
348C2×C22⋊C4D1212yes—
349C2×C42—61no—
350C2×C42—61no—
351C2×C42—64no—
352C2×C42—64no—
353C2×C42C22×D312yes—
354#107—38no—
355#106—38no—
356#103—316no—
357#103—38no—
358#102—38no—
359#97C22×D318yes—
360C4×C8C22×D318yes—
361C23×C3—42no—
362C23×C3C4×D414yes—
363C23×C3C22×D411yescharacteristic
364C23×C3C22×D411yescharacteristic
365C23×C3C22×D411yescharacteristic
366C22×D3—22no—
367C22×D3—24no—
368C22×D3—82no—
369C2×C3×C4—44no—
370C2×C3×C4C4×D414yes—
371C2×C3×C4C4×D414yes—
372C2×C3×C4C4×D414yes—
373C2×C3×C4C23×C412yes—
374C2×C3×C4C23×C412yes—
375C2×C3×C4C23×C412yes—
376C2×C3×C4C22×D418yes—
377C2×C3×C4C22×D411yescharacteristic
378C2×C3×C4C22×D411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4112yes—
381C2×C3×C4C2×#4112yes—
382C2×C3×C4C2×#4112yes—
383C2×C3×C4C2511yescharacteristic
384#67—28no—
385#67—28no—
386C2×Dic3—28no—
387C2×Dic3—28no—
388C2×Dic3—28no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—22no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—44no—
395C2×Dic3—82no—
396D12—24no—
397D12—22no—
398D12C23×C414yes—
399C4×D3—28no—
400C4×D3—28no—
401C4×D3—88no—
402Dic6—28no—
403Dic6—28no—
404Dic6—28no—
405Dic6—24no—
406Dic6—24no—
407Dic6—22no—
408Dic6C23×C418yes—
409Dic6C23×C414yes—
410C3×C8—24no—
411C3×C8C23×C418yes—
412C24—61no—
413C24C2×D1211yescharacteristic
414#41—38no—
415#41—38no—
416#41—68no—
417#41—68no—
418#41—64no—
419#41—64no—
420C2×Q8—38no—
421C2×Q8—34no—
422C2×Q8—38no—
423C2×Q8—38no—
424C2×Q8—34no—
425C2×Q8—32no—
426C2×Q8—68no—
427C2×Q8—68no—
428C2×Q8—68no—
429C2×Q8—62no—
430C2×Q8—62no—
431C2×Q8—62no—
432C2×Q8—61no—
433C2×Q8—64no—
434C2×Q8—64no—
435C2×Q8—64no—
436C2×Q8—62no—
437C2×D4—34no—
438C2×D4—32no—
439C2×D4—68no—
440C2×D4—62no—
441C2×D4—62no—
442C2×D4—61no—
443C2×D4—62no—
444C22×C4—68no—
445C22×C4—62no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—68no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—68no—
455C22×C4C2×C4×D314yes—
456C22×C4C2×D1214yes—
457C22×C4C2×D1211yescharacteristic
458C22×C4C2×D1211yescharacteristic
459C22×C4C2×D1211yescharacteristic
460C22×C4#23514yes—
461C22×C4C23×D311yescharacteristic
462C22×C4C23×D311yescharacteristic
463C22×C4C23×D311yescharacteristic
464Q16—332no—
465SD16—332no—
466M16—24no—
467M16—24no—
468M16C2×C4×D318yes—
469M16C2×C4×D318yes—
470C2×C8—24no—
471C2×C8C2×C4×D318yes—
472C2×C8C23×D314yes—
473C2×C8C23×D314yes—
474C4⋊C4—34no—
475C4⋊C4—34no—
476C4⋊C4—62no—
477C4⋊C4—62no—
478C4⋊C4—68no—
479C4⋊C4—616no—
480C4⋊C4C2×D1214yes—
481C4⋊C4C2×D1214yes—
482C22⋊C4—62no—
483C22⋊C4—62no—
484C22⋊C4—62no—
485C22⋊C4—62no—
486C22⋊C4—68no—
487C22⋊C4C2×D1214yes—
488C22⋊C4C2×D1214yes—
489C42—62no—
490C42—62no—
491C42—62no—
492C42—62no—
493C42—68no—
494C42—616no—
495C42C2×D1214yes—
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×C4C2×C4×D412yes—
510C3×C4C2×C4×D412yes—
511C3×C4C2×C4×D412yes—
512C3×C4C24×C411yescharacteristic
513C3×C4C22×#4111yescharacteristic
514Dic3—28no—
515Dic3—24no—
516Dic3—44no—
517Dic3—44no—
518C23—42no—
519C23—62no—
520C23—64no—
521C23—242no—
522C23C4×D1214yes—
523C23C22×D1211yescharacteristic
524C23C22×D1211yescharacteristic
525C23C22×D1211yescharacteristic
526Q8—38no—
527Q8—34no—
528Q8—64no—
529Q8—62no—
530Q8—68no—
531Q8—68no—
532Q8—68no—
533Q8—64no—
534D4—34no—
535D4—68no—
536D4—68no—
537D4—62no—
538D4—64no—
539C2×C4—44no—
540C2×C4—62no—
541C2×C4—64no—
542C2×C4—64no—
543C2×C4—68no—
544C2×C4—68no—
545C2×C4—68no—
546C2×C4—68no—
547C2×C4—68no—
548C2×C4—64no—
549C2×C4—64no—
550C2×C4—124no—
551C2×C4—248no—
552C2×C4—242no—
553C2×C4C4×D1214yes—
554C2×C4C4×D1214yes—
555C2×C4C4×D1214yes—
556C2×C4C22×C4×D312yes—
557C2×C4C22×C4×D312yes—
558C2×C4C22×C4×D312yes—
559C2×C4C22×D1218yes—
560C2×C4C22×D1211yescharacteristic
561C2×C4C22×D1211yescharacteristic
562C2×C4C22×D1211yescharacteristic
563C2×C4C2×#23512yes—
564C2×C4C2×#23512yes—
565C2×C4C2×#23512yes—
566C2×C4C24×D311yescharacteristic
567C8—24no—
568C8C22×C4×D318yes—
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×D1212yes—
583C22C2×C4×D1212yes—
584C22C2×C4×D1212yes—
585C22C23×D1211yescharacteristic
586C4—28no—
587C4—64no—
588C4—68no—
589C4—124no—
590C4—124no—
591C4C2×C4×D1212yes—
592C4C2×C4×D1212yes—
593C4C2×C4×D1212yes—
594C4C23×C4×D311yescharacteristic
595C4C22×#23511yescharacteristic
596C3#6016911yescharacteristic, complemented, Sylow
597C2—22no—
598C2—22no—
599C2—22no—
600C2—124no—
601C2C22×C4×D1211yescharacteristic
602C2#8716812yes—
subgroupG/H
centerC22C23×D12
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC3×#2873C2
socleC22×C3C23×D4
prime radicalC22×#235C22
cosocleC2×C3×C4C25
hypercenter#2873D3
hypocenterC3#60169
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#117 ◁ #2873

Complements

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

NH ≅ G/Ntype
3. C2×#5516C2semidirect
4. C3×#2873C2semidirect
5. C2×#5120C2semidirect
7. #83752C2semidirect
8. #78573C2semidirect
9. #78572C2semidirect
10. #78571C2semidirect
11. C4×#909C2semidirect
20. C2×#909C4semidirect
33. #5516C22semidirect
39. C2×C4×Dic6C22semidirect
41. C3×#405C22semidirect
42. C3×C4×M16C22semidirect
45. #5121C22semidirect
46. #5121C22semidirect
49. #5120C22semidirect
52. C2×#816C22semidirect
53. #4716C22semidirect
54. #4710C22semidirect
55. C4×Dic12C22semidirect
56. C4×#204C22semidirect
67. #2873D3semidirect
83. C22×Dic6C2×C4semidirect
89. C2×C3×M16C2×C4semidirect
120. #909C2×C4semidirect
121. #909C2×C4semidirect
122. C2×Dic12C2×C4semidirect
124. C2×#204C2×C4semidirect
192. #405C2×D3semidirect
193. C4×M16C2×D3semidirect
306. C2×M16C4×D3semidirect
596. C3#60169semidirect

Maximal subgroups

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

subgroupindex
C22×#9092
#857312
C2×#55162
C3×#28732
C2×#51202
#837682
#837522
#785732
#785722
#785712
C4×#9092
#601693