Finite Group Databasebeta

← #10947983

Subgroups — #10947983

Order 768.

1252 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86313C211yesmaximal, characteristic
2D3×#410C212yesmaximal, complemented
3C3×#2128C211yesmaximal, characteristic, complemented
4#71861C211yesmaximal, characteristic, complemented
5#70059C212yesmaximal, complemented
6#69151C214yesmaximal, complemented
7#69150C214yesmaximal, complemented
8#33358—31nomaximal, Sylow
9C2×D3×#41C2211yescharacteristic
10C22×#237C411yescharacteristic, complemented
11C3×#535C2211yescharacteristic
12#5825C2211yescharacteristic
13#5805C411yescharacteristic, complemented
14#5716C2212yes—
15#5715C2212yes—
16#5675C412yescomplemented
17#5668C412yescomplemented
18#5608—22no—
19#5606—22no—
20#5589—22no—
21#5586—22no—
22#5526C411yescharacteristic, complemented
23D3×#117—22no—
24D3×#117C411yescharacteristic, complemented
25C3×#410C2212yes—
26#5240C2212yes—
27#5006—22no—
28D3×#99—22no—
29D3×#99C2214yescomplemented
30#4740—22no—
31#4740C2214yescomplemented
32C3×#353C2214yescomplemented
33#4545C2214yescomplemented
34#4473C2218yescomplemented
35#3449—31no—
36C2×#410—32no—
37#2129—32no—
38#2128—31no—
39#2128D311yescharacteristic, complemented
40#2127—34no—
41#2126—34no—
42C2×C3×#41C2311yescharacteristic
43C22×C3×D4C2×C411yescharacteristic, complemented
44C22×#67—21no—
45C23×Dic3—21no—
46D3×#41—42no—
47D3×#41—42no—
48C2×#239C2×C411yescharacteristic
49C2×D3×Q8C2×C411yescharacteristic
50C2×#237—21no—
51C2×#237—21no—
52C2×#237—42no—
53C2×#237—42no—
54C2×#237C2×C412yes—
55C2×#237C2×C412yescomplemented
56C2×#237D412yes—
57C2×#237C2311yescharacteristic
58C2×D3×D4D412yes—
59C2×D3×D4C2311yescharacteristic
60C2×#235C2×C411yescharacteristic
61C2×#235D412yes—
62C22×C4×D3C2×C411yescharacteristic
63C22×C4×D3D412yes—
64C22×Dic6C2×C411yescharacteristic, complemented
65C3×#127C2312yes—
66C3×#124C2×C412yescomplemented
67C3×#121—22no—
68C3×#120—22no—
69C3×#117—22no—
70C3×#117C2×C411yescharacteristic, complemented
71#953—22no—
72#944C2×C412yescomplemented
73#940C2×C412yes—
74#939—24no—
75#938—24no—
76#938—22no—
77#937—24no—
78#936C2×C412yescomplemented
79#936C2312yes—
80#935—22no—
81D4×Dic3—24no—
82#926—22no—
83#926C2×C411yescharacteristic, complemented
84#924—22no—
85#897—22no—
86#892—21no—
87#892—42no—
88D3×C4⋊C4—21no—
89D3×C4⋊C4—42no—
90#886—22no—
91#884—24no—
92#883—24no—
93#67⋊C4—24no—
94D3×C22⋊C4—21no—
95D3×C22⋊C4—24no—
96#878—24no—
97#877—21no—
98#877—24no—
99#875C2×C412yes—
100#872—42no—
101#872C2×C412yes—
102C42×D3—42no—
103C42×D3C2×C412yescomplemented
104#869C2×C412yescomplemented
105C3×#99—22no—
106C3×#99C2314yes—
107#834—22no—
108#834C2314yes—
109#806—24no—
110#806—24no—
111C22×#41—31no—
112C22×#41—31no—
113#537—62no—
114#536—31no—
115#536—32no—
116#536—31no—
117#535—31no—
118#535—32no—
119#535C2×D311yescharacteristic
120#534—62no—
121#532—32no—
122#532—32no—
123#525—62no—
124#525—62no—
125C2×#117—31no—
126C2×#117—62no—
127#410—32no—
128#410—34no—
129#410—62no—
130#410—62no—
131#410C2×D312yes—
132C2×#99—34no—
133C2×#99—62no—
134#354—38no—
135#353—34no—
136#353C2×D314yescomplemented
137C24×C3—21no—
138C23×D3C22⋊C411yescharacteristic
139C23×D3C2×D412yes—
140C3×#41—42no—
141C2×C3×Q8C22×C411yescharacteristic
142C2×C3×D4—24no—
143C2×C3×D4—21no—
144C2×C3×D4—42no—
145C2×C3×D4C22×C412yes—
146C2×C3×D4C2×D412yes—
147C2×C3×D4C2411yescharacteristic
148C22×C3×C4C22×C411yescharacteristic
149C22×C3×C4C2×D412yes—
150C2×#67—22no—
151C2×#67—22no—
152C2×#67—24no—
153C2×#67—44no—
154C2×#67C22⋊C412yes—
155C2×#67C2×D412yes—
156C2×#67C2×D412yes—
157C22×Dic3—21no—
158C22×Dic3—22no—
159C22×Dic3—22no—
160C22×Dic3—42no—
161C22×Dic3—42no—
162C22×Dic3C22⋊C411yescharacteristic
163C22×Dic3C2×D412yes—
164#239—41no—
165#239—41no—
166D3×Q8—42no—
167#237—42no—
168#237—42no—
169#237—41no—
170#237—41no—
171#237—44no—
172#237—44no—
173#237—42no—
174#237—42no—
175#237—42no—
176#237—42no—
177D3×D4—42no—
178D3×D4—42no—
179D3×D4—42no—
180#235—42no—
181#235—42no—
182#235—42no—
183C2×D12C22⋊C412yes—
184C2×D12C22×C411yescharacteristic
185C2×C4×D3—22no—
186C2×C4×D3—22no—
187C2×C4×D3—22no—
188C2×C4×D3—22no—
189C2×C4×D3—22no—
190C2×C4×D3—22no—
191C2×C4×D3—22no—
192C2×C4×D3—44no—
193C2×C4×D3C22⋊C412yes—
194C2×C4×D3C22⋊C412yes—
195C2×C4×D3C22⋊C411yescharacteristic
196C2×C4×D3C22⋊C412yes—
197C2×C4×D3C22⋊C411yescharacteristic
198C2×C4×D3C22×C411yescharacteristic
199C2×C4×D3C22×C411yescharacteristic
200C2×Dic6—21no—
201C2×Dic6—42no—
202C2×Dic6C22⋊C412yes—
203C2×Dic6C22×C412yes—
204C2×Dic6C22×C411yescharacteristic
205C3×C4⋊C4—21no—
206C3×C4⋊C4—42no—
207C3×C22⋊C4—21no—
208C3×C22⋊C4—24no—
209C3×C22⋊C4—24no—
210C3×C42—42no—
211C3×C42C22×C412yes—
212#217—21no—
213#217—24no—
214#217—24no—
215#217—24no—
216(C2×D3)⋊C4—22no—
217(C2×D3)⋊C4—24no—
218(C2×D3)⋊C4—44no—
219C4⋊Dic3—21no—
220C4⋊Dic3—24no—
221C4⋊Dic3—42no—
222Dic3⋊C4—24no—
223Dic3⋊C4—22no—
224Dic3⋊C4—44no—
225C4×Dic3—22no—
226C4×Dic3—44no—
227C4×Dic3—42no—
228C4×Dic3C22×C412yes—
229C2×#41—31no—
230C2×#41—32no—
231C2×#41—31no—
232C2×#41—31no—
233C2×#41—32no—
234C2×#41—32no—
235C2×#41—32no—
236C2×#41—61no—
237C2×#41—61no—
238C2×#41—122no—
239C2×#41—122no—
240C2×#41—122no—
241C2×#41—122no—
242C2×#41C22×D311yescharacteristic
243C22×Q8—31no—
244C22×Q8—31no—
245C22×D4—32no—
246C22×D4—31no—
247C22×D4—61no—
248C22×D4C4×D311yescharacteristic, complemented
249C23×C4—31no—
250C23×C4—32no—
251C23×C4—61no—
252C4⋊Q8—32no—
253C4⋊Q8—32no—
254#127—32no—
255#127C22×D312yes—
256#124—32no—
257#124—32no—
258#124—32no—
259#124C4×D312yescomplemented
260#123—64no—
261#123—62no—
262#123—62no—
263C22⋊Q8—64no—
264C22⋊Q8—62no—
265C22⋊Q8—62no—
266#121—22no—
267#121—64no—
268#121—62no—
269#121—64no—
270#121—64no—
271#120—22no—
272#120—64no—
273#120—62no—
274C4×D4—64no—
275C4×D4—64no—
276#117—22no—
277#117—31no—
278#117—32no—
279#117—61no—
280#117—61no—
281#117—64no—
282#117—62no—
283#117—122no—
284#117—122no—
285#117C4×D311yescharacteristic, complemented
286C2×C4⋊C4—61no—
287C2×C4⋊C4—122no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—64no—
290C2×C42—32no—
291C2×C42—122no—
292#99—22no—
293#99—34no—
294#99—64no—
295#99—64no—
296#99—62no—
297#99C22×D314yes—
298C23×C3—22no—
299C23×C3—42no—
300C23×C3—82no—
301C23×C3C2×C22⋊C411yescharacteristic
302C23×C3C22×D412yes—
303C22×D3—22no—
304C22×D3—22no—
305C22×D3—22no—
306C22×D3—82no—
307C22×D3—81no—
308C22×D3C2×C22⋊C412yes—
309C22×D3C2×C22⋊C411yescharacteristic
310C22×D3C2×C22⋊C411yescharacteristic
311C3×Q8—41no—
312C3×D4—42no—
313C3×D4—41no—
314C3×D4—42no—
315C3×D4—42no—
316C3×D4—84no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—44no—
322C2×C3×C4C2×C22⋊C412yes—
323C2×C3×C4C2×C22⋊C411yescharacteristic
324C2×C3×C4C23×C411yescharacteristic
325#67—42no—
326#67—42no—
327#67—42no—
328#67—44no—
329#67—44no—
330#67—84no—
331C2×Dic3—22no—
332C2×Dic3—22no—
333C2×Dic3—21no—
334C2×Dic3—21no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—44no—
338C2×Dic3—42no—
339C2×Dic3—42no—
340C2×Dic3—42no—
341C2×Dic3—44no—
342C2×Dic3C2×C22⋊C411yescharacteristic
343C2×Dic3C2×C22⋊C412yes—
344C2×Dic3C2×C22⋊C411yescharacteristic
345D12—42no—
346D12—41no—
347C4×D3—22no—
348C4×D3—22no—
349C4×D3—22no—
350C4×D3—22no—
351C4×D3—22no—
352C4×D3—22no—
353C4×D3—22no—
354C4×D3—22no—
355C4×D3—21no—
356C4×D3—21no—
357C4×D3—82no—
358C4×D3—88no—
359C4×D3#9912yes—
360C4×D3#9912yes—
361Dic6—42no—
362Dic6—41no—
363Dic6—42no—
364Dic6—42no—
365C24—21no—
366C24—31no—
367C24—32no—
368#41—42no—
369#41—122no—
370#41—122no—
371#41—122no—
372#41—122no—
373#41—122no—
374#41—121no—
375#41—121no—
376#41—122no—
377#41—122no—
378#41—121no—
379#41—121no—
380#41—124no—
381#41—124no—
382#41—122no—
383#41—122no—
384C2×Q8—32no—
385C2×Q8—31no—
386C2×Q8—32no—
387C2×Q8—61no—
388C2×Q8—122no—
389C2×Q8—122no—
390C2×Q8C2×C4×D311yescharacteristic
391C2×D4—24no—
392C2×D4—21no—
393C2×D4—32no—
394C2×D4—32no—
395C2×D4—32no—
396C2×D4—32no—
397C2×D4—31no—
398C2×D4—42no—
399C2×D4—62no—
400C2×D4—62no—
401C2×D4—64no—
402C2×D4—122no—
403C2×D4—122no—
404C2×D4—122no—
405C2×D4—124no—
406C2×D4C2×C4×D312yes—
407C2×D4D3×D412yes—
408C2×D4C23×D311yescharacteristic
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—32no—
412C22×C4—32no—
413C22×C4—32no—
414C22×C4—31no—
415C22×C4—32no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—124no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4C2×C4×D311yescharacteristic
432C22×C4D3×D412yes—
433C4⋊C4—21no—
434C4⋊C4—42no—
435C4⋊C4—62no—
436C4⋊C4—64no—
437C4⋊C4—64no—
438C4⋊C4—61no—
439C4⋊C4—122no—
440C4⋊C4—124no—
441C22⋊C4—21no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—62no—
445C22⋊C4—64no—
446C22⋊C4—64no—
447C22⋊C4—61no—
448C22⋊C4—64no—
449C22⋊C4—64no—
450C22⋊C4—124no—
451C42—32no—
452C42—42no—
453C42—62no—
454C42—122no—
455C42—124no—
456C42C2×C4×D312yes—
457C22×C3—22no—
458C22×C3—21no—
459C22×C3—42no—
460C22×C3—41no—
461C22×C3—44no—
462C22×C3—42no—
463C22×C3—84no—
464C22×C3C22×C22⋊C411yescharacteristic
465C2×D3—21no—
466C2×D3—22no—
467C2×D3—41no—
468C2×D3—44no—
469C2×D3—41no—
470C2×D3—42no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3C2×#9912yes—
474C3×C4—21no—
475C3×C4—22no—
476C3×C4—42no—
477C3×C4—84no—
478C3×C4C2×#9912yes—
479Dic3—21no—
480Dic3—22no—
481Dic3—42no—
482Dic3—42no—
483Dic3—84no—
484Dic3C2×#9911yescharacteristic
485Dic3C2×#9911yescharacteristic
486C23—22no—
487C23—31no—
488C23—32no—
489C23—31no—
490C23—42no—
491C23—62no—
492C23—62no—
493C23—62no—
494C23—82no—
495C23—242no—
496C23—241no—
497C23D3×C22⋊C411yescharacteristic
498C23C2×D3×D412yes—
499Q8—41no—
500Q8—122no—
501Q8—121no—
502Q8—122no—
503Q8—122no—
504D4—42no—
505D4—41no—
506D4—42no—
507D4—42no—
508D4—84no—
509D4—122no—
510D4—121no—
511D4—122no—
512D4—122no—
513D4—122no—
514D4—124no—
515D4—124no—
516D4—244no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—31no—
522C2×C4—32no—
523C2×C4—32no—
524C2×C4—32no—
525C2×C4—31no—
526C2×C4—44no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—62no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—124no—
546C2×C4—122no—
547C2×C4—124no—
548C2×C4—242no—
549C2×C4—248no—
550C2×C4D3×C22⋊C412yes—
551C2×C4D3×C22⋊C411yescharacteristic
552C2×C4C22×C4×D311yescharacteristic
553C2×C3—21no—
554C2×C3—42no—
555C2×C3—41no—
556C2×C3—82no—
557C2×C3C22×#9911yescharacteristic
558D3—22no—
559D3—42no—
560D3—41no—
561C22—22no—
562C22—21no—
563C22—32no—
564C22—42no—
565C22—41no—
566C22—44no—
567C22—42no—
568C22—61no—
569C22—62no—
570C22—84no—
571C22—121no—
572C22—121no—
573C22—122no—
574C22—124no—
575C22—242no—
576C22—242no—
577C22C2×D3×C22⋊C411yescharacteristic
578C4—22no—
579C4—21no—
580C4—31no—
581C4—31no—
582C4—42no—
583C4—61no—
584C4—62no—
585C4—84no—
586C4—122no—
587C4—122no—
588C4—244no—
589C4D3×#9912yes—
590C3#3335811yescharacteristic, complemented, Sylow
591C2—21no—
592C2—42no—
593C2—41no—
594C2—62no—
595C2—82no—
596C2—122no—
597C2—121no—
598C2C2×D3×#9911yescharacteristic
subgroupG/H
centerC2C2×D3×#99
commutatorC2×C3×C4C23×C4
FrattiniC2×D4C23×D3
FittingC3×#2128C2
socleC2×C3C22×#99
prime radical#86313C2
cosocleC2×C3×D4C24
hypercenter#2128D3
hypocenterC3#33358
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2128

Complements

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

NH ≅ G/Ntype
2. D3×#410C2semidirect
3. C3×#2128C2semidirect
4. #71861C2semidirect
5. #70059C2semidirect
6. #69151C2semidirect
7. #69150C2semidirect
10. C22×#237C4semidirect
13. #5805C4semidirect
16. #5675C4semidirect
17. #5668C4semidirect
22. #5526C4semidirect
24. D3×#117C4semidirect
29. D3×#99C22semidirect
31. #4740C22semidirect
32. C3×#353C22semidirect
33. #4545C22semidirect
34. #4473C22semidirect
39. #2128D3semidirect
43. C22×C3×D4C2×C4semidirect
55. C2×#237C2×C4semidirect
64. C22×Dic6C2×C4semidirect
66. C3×#124C2×C4semidirect
70. C3×#117C2×C4semidirect
72. #944C2×C4semidirect
78. #936C2×C4semidirect
83. #926C2×C4semidirect
103. C42×D3C2×C4semidirect
104. #869C2×C4semidirect
136. #353C2×D3semidirect
248. C22×D4C4×D3semidirect
259. #124C4×D3semidirect
285. #117C4×D3semidirect
590. C3#33358semidirect

Maximal subgroups

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

subgroupindex
#863132
D3×#4102
C3×#21282
#718612
#700592
#691512
#691502
#333583