Finite Group Databasebeta

← #11664443

Subgroups — #11664443

Order 768.

2050 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87268C211yesmaximal, characteristic
2#85725C211yesmaximal, characteristic, complemented
3C3×#2896C211yesmaximal, characteristic, complemented
4C2×#5233C211yesmaximal, characteristic, complemented
5#83768C211yesmaximal, characteristic, complemented
6C2×#5122C211yesmaximal, characteristic, complemented
7C2×#5002C211yesmaximal, characteristic, complemented
8#78952C214yesmaximal, complemented
9#78920C214yesmaximal, complemented
10#78877C214yesmaximal, complemented
11#78876C214yesmaximal, complemented
12#78826C214yesmaximal, complemented
13#78825C214yesmaximal, complemented
14#60612—31nomaximal, Sylow
15C22×C3×#41C2211yescharacteristic
16C22×#235C2211yescharacteristic
17C22×C3×M16C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19#5833—22no—
20#5832C2214yes—
21#5830C2212yes—
22#5828—22no—
23#5825C2212yes—
24#5816C2212yes—
25#5805C2212yes—
26#5804—22no—
27#5791—22no—
28C2×#926C2211yescharacteristic
29C2×#926C2211yescharacteristic
30C23×C3⋊C8C2211yescharacteristic
31#5519—22no—
32#5519C2214yescomplemented
33#5517—22no—
34#5517C2214yescomplemented
35#5500C2214yescomplemented
36#5483—22no—
37#5470—22no—
38C3×#419C2212yescomplemented
39C3×#418C2212yescomplemented
40C3×#417C2214yes—
41C3×#416C2214yescomplemented
42#5233C2214yes—
43#5233C2212yescomplemented
44#5233C2212yescomplemented
45C2×#835C2212yescomplemented
46C2×#832C2212yescomplemented
47#5133C2214yes—
48#5131C2212yescomplemented
49#5122C2214yescomplemented
50#5122C2214yescomplemented
51#5120C2212yescomplemented
52#5113C2214yescomplemented
53C2×#815C2212yescomplemented
54C2×C8×Dic3C2212yescomplemented
55#5002C2214yes—
56#5002C2212yescomplemented
57#5002C2212yescomplemented
58C2×#810C2212yescomplemented
59C2×#809C2212yescomplemented
60#4800C2218yescomplemented
61#4797C2218yescomplemented
62#4794C2218yescomplemented
63#4792C2218yescomplemented
64#4768C2218yescomplemented
65#4765C2218yescomplemented
66#4764C2218yescomplemented
67#4760C2218yescomplemented
68#3458—31no—
69#3449—31no—
70#3137—34no—
71#3137—34no—
72#3136—34no—
73#3136—34no—
74#3135—34no—
75#3131—34no—
76#2896—31no—
77#2896D311yescharacteristic, complemented
78C2×#416—31no—
79C2×#416—31no—
80C2×#405—31no—
81C2×C3×#41—21no—
82C2×C3×#41—21no—
83C2×C3×#41—24no—
84C2×C3×#41—24no—
85C2×C3×#41C2312yes—
86C2×C3×#41C2312yes—
87C22×C3×Q8C2311yescharacteristic
88C22×C3×D4—21no—
89C22×C3×D4C2311yescharacteristic
90C23×C3×C4—21no—
91C23×C3×C4C2311yescharacteristic
92C22×#67—21no—
93C2×#235—21no—
94C2×#235—21no—
95C2×#235—24no—
96C2×#235—24no—
97C2×#235C2312yes—
98C2×#235C2312yes—
99C22×D12C2311yescharacteristic
100C22×C4×D3—21no—
101C22×Dic6C2311yescharacteristic
102C2×C3×M16—24no—
103C2×C3×M16C2312yes—
104C2×C3×M16C2312yes—
105C22×C3×C8C2312yes—
106C3×#117—21no—
107C3×#117—21no—
108C3×#117C2312yes—
109C3×#117C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C22⋊C4—21no—
112C2×C3×C42—21no—
113#953—24no—
114C2×#217—21no—
115C2×#217D412yes—
116#947C2314yes—
117#946—24no—
118#944C2314yes—
119#940C2314yes—
120#939—28no—
121#938—24no—
122#937—24no—
123#936C2314yes—
124#935—24no—
125#930—24no—
126#929—24no—
127C4×#67—28no—
128#926—21no—
129#926—21no—
130#926D412yes—
131#926D412yes—
132#926C2312yes—
133#926C2312yes—
134#926C2312yes—
135#926C2312yes—
136C2×C4⋊Dic3D412yes—
137C2×C4⋊Dic3C2312yes—
138#924—28no—
139#924—24no—
140C2×C4×Dic3—21no—
141C2×C4×Dic3C2312yes—
142C22×C3⋊C8—24no—
143C22×C3⋊C8C2312yes—
144C22×C3⋊C8C2312yes—
145C22×C3⋊C8C2311yescharacteristic
146C22×C3⋊C8C2311yescharacteristic
147#890—24no—
148#890C2318yes—
149#888—24no—
150#888C2318yes—
151#878—28no—
152#878—28no—
153#870—24no—
154#869—24no—
155C4×Dic6—28no—
156C3×#103C2318yes—
157C3×#102C2318yes—
158#835C2314yes—
159#835C2314yes—
160#832C2314yes—
161#832C2314yes—
162#821C2318yes—
163#816C2318yes—
164#815C2318yes—
165C8×Dic3C2318yes—
166#810C2314yes—
167#810C2314yes—
168#809C2314yes—
169#809C2314yes—
170C22×#41—31no—
171C22×#41C2×D311yescharacteristic
172C22×M16C2×D311yescharacteristic
173C23×C8—31no—
174#544—34no—
175#544—34no—
176#544—62no—
177#543—62no—
178#537—62no—
179#536—32no—
180#536—32no—
181#535—32no—
182#535—32no—
183#534—62no—
184#533—34no—
185#533—62no—
186#533—62no—
187#532—34no—
188#525—62no—
189#525—62no—
190C2×#117—31no—
191C2×#117—31no—
192C2×#117C2×D311yescharacteristic
193#491—38no—
194#490—38no—
195#489—38no—
196#489—38no—
197#488—38no—
198#488—38no—
199#487—38no—
200#486—38no—
201#419—32no—
202#419C2×D312yescomplemented
203#418—32no—
204#418C2×D312yescomplemented
205#417—34no—
206#417C2×D314yes—
207#416—32no—
208#416—32no—
209#416—34no—
210#416—34no—
211#416—32no—
212#416—32no—
213#416—34no—
214#416C2×D314yescomplemented
215C2×#103—32no—
216C2×#103—32no—
217C2×#102—32no—
218C2×#102—32no—
219#405—34no—
220#405—34no—
221C2×#97—32no—
222C2×C4×C8—32no—
223C24×C3—21no—
224C24×C3C2×D411yescharacteristic
225C23×D3—21no—
226C3×#41—24no—
227C3×#41—24no—
228C3×#41—24no—
229C3×#41—24no—
230C3×#41—44no—
231C3×#41—44no—
232C2×C3×Q8—21no—
233C2×C3×Q8—22no—
234C2×C3×Q8—22no—
235C2×C3×Q8C2412yes—
236C2×C3×D4—22no—
237C2×C3×D4—22no—
238C2×C3×D4—21no—
239C2×C3×D4—22no—
240C2×C3×D4—22no—
241C2×C3×D4—44no—
242C2×C3×D4C2412yes—
243C22×C3×C4—21no—
244C22×C3×C4—22no—
245C22×C3×C4—22no—
246C22×C3×C4—22no—
247C22×C3×C4—44no—
248C22×C3×C4C2×D411yescharacteristic
249C22×C3×C4C2×D411yescharacteristic
250C22×C3×C4C2×D411yescharacteristic
251C22×C3×C4#4114yes—
252C22×C3×C4#4114yes—
253C22×C3×C4C2411yescharacteristic
254C22×C3×C4C2411yescharacteristic
255C22×C3×C4C2411yescharacteristic
256C2×#67—22no—
257C2×#67—22no—
258C2×#67—28no—
259C2×#67—44no—
260C22×Dic3—21no—
261C22×Dic3—22no—
262C22×Dic3C2×D414yes—
263#235—24no—
264#235—24no—
265#235—24no—
266#235—24no—
267#235—44no—
268#235—44no—
269C2×D12—22no—
270C2×D12—22no—
271C2×D12—21no—
272C2×D12C2412yes—
273C2×C4×D3—22no—
274C2×C4×D3—22no—
275C2×C4×D3—44no—
276C2×Dic6—22no—
277C2×Dic6—22no—
278C2×Dic6—21no—
279C2×Dic6—28no—
280C2×Dic6C2412yes—
281C3×M16—24no—
282C3×M16—24no—
283C2×C3×C8—24no—
284C2×C3×C8C2414yes—
285C3×C4⋊C4—22no—
286C3×C4⋊C4C2414yes—
287C3×C22⋊C4—22no—
288C3×C22⋊C4—22no—
289C3×C42—22no—
290C3×C42—22no—
291#217—22no—
292#217—22no—
293#217—28no—
294#217C2×D414yes—
295#217C2×D414yes—
296(C2×D3)⋊C4—28no—
297C4⋊Dic3—22no—
298C4⋊Dic3—28no—
299C4⋊Dic3C2×D414yes—
300C4⋊Dic3C2×D414yes—
301C4⋊Dic3C2414yes—
302Dic3⋊C4—28no—
303Dic3⋊C4—28no—
304Dic3⋊C4—28no—
305C4×Dic3—22no—
306C4×Dic3—22no—
307C4×Dic3C2×D414yes—
308C4×Dic3C2414yes—
309C2×C3⋊C8—24no—
310C2×C3⋊C8—24no—
311C2×C3⋊C8—22no—
312C2×C3⋊C8—22no—
313C2×C3⋊C8C2412yes—
314C2×C3⋊C8C2412yes—
315C2×#41—21no—
316C2×#41—21no—
317C2×#41—24no—
318C2×#41—24no—
319C2×#41—32no—
320C2×#41—32no—
321C2×#41—61no—
322C2×#41—61no—
323C2×#41—64no—
324C2×#41—64no—
325C2×#41C22×D312yes—
326C2×#41C22×D312yes—
327C22×Q8—31no—
328C22×Q8C22×D311yescharacteristic
329C22×D4—21no—
330C22×D4—31no—
331C22×D4—61no—
332C22×D4C22×D311yescharacteristic
333C23×C4—21no—
334C23×C4—61no—
335C23×C4C22×D311yescharacteristic
336C2×M16—24no—
337C2×M16C22×D312yes—
338C2×M16C22×D312yes—
339C22×C8—31no—
340C22×C8—31no—
341C22×C8—32no—
342C22×C8—32no—
343C22×C8—64no—
344C22×C8C22×D312yes—
345C4⋊Q8—38no—
346C4⋊Q8—34no—
347C4⋊Q8—64no—
348C4⋊Q8—64no—
349#127—34no—
350#125—38no—
351#125—64no—
352#125—64no—
353#124—34no—
354#124—34no—
355#123—64no—
356#123—64no—
357C22⋊Q8—68no—
358C22⋊Q8—68no—
359C22⋊Q8—68no—
360C22⋊Q8—64no—
361C22⋊Q8—64no—
362#121—64no—
363#121—64no—
364#121—68no—
365#120—64no—
366#120—64no—
367C4×Q8—68no—
368C4×D4—68no—
369#117—21no—
370#117—21no—
371#117—32no—
372#117—32no—
373#117—32no—
374#117—32no—
375#117—32no—
376#117—32no—
377#117—61no—
378#117—61no—
379#117C22×D312yes—
380#117C22×D312yes—
381C2×C4⋊C4—32no—
382C2×C4⋊C4—32no—
383C2×C4⋊C4C22×D312yes—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—32no—
386C2×C22⋊C4—61no—
387C2×C42—21no—
388C2×C42—32no—
389C2×C42—61no—
390#103—34no—
391#103—34no—
392#103—38no—
393#103—34no—
394#103—34no—
395#103C22×D318yes—
396#102—34no—
397#102—34no—
398#102—38no—
399#102—34no—
400#102—34no—
401#102C22×D318yes—
402#97—38no—
403C4×C8—38no—
404C23×C3—22no—
405C23×C3—42no—
406C23×C3—42no—
407C23×C3—82no—
408C23×C3#12414yes—
409C23×C3C22×D411yescharacteristic
410C23×C3C22×D411yescharacteristic
411C23×C3C22×D411yescharacteristic
412C22×D3—22no—
413C22×D3—42no—
414C22×D3—82no—
415C3×Q8—22no—
416C3×Q8—22no—
417C3×Q8—42no—
418C3×D4—22no—
419C3×D4—22no—
420C3×D4—44no—
421C3×D4—44no—
422C3×D4—42no—
423C2×C3×C4—24no—
424C2×C3×C4—22no—
425C2×C3×C4—42no—
426C2×C3×C4—44no—
427C2×C3×C4—44no—
428C2×C3×C4—82no—
429C2×C3×C4#12414yes—
430C2×C3×C4#12414yes—
431C2×C3×C4#12414yes—
432C2×C3×C4C22×D411yescharacteristic
433C2×C3×C4C22×D411yescharacteristic
434C2×C3×C4C22×D411yescharacteristic
435C2×C3×C4C2×#4112yes—
436C2×C3×C4C2×#4112yes—
437C2×C3×C4C2×#4112yes—
438C2×C3×C4C2×#4112yes—
439C2×C3×C4C2×#4112yes—
440C2×C3×C4C2×#4112yes—
441C2×C3×C4C2511yescharacteristic
442#67—44no—
443#67—44no—
444#67—88no—
445C2×Dic3—24no—
446C2×Dic3—22no—
447C2×Dic3—44no—
448C2×Dic3—42no—
449C2×Dic3—82no—
450C2×Dic3C22×D418yes—
451D12—22no—
452D12—22no—
453D12—42no—
454C4×D3—44no—
455C4×D3—44no—
456Dic6—22no—
457Dic6—22no—
458Dic6—42no—
459Dic6—88no—
460C3×C8—24no—
461C3⋊C8—22no—
462C3⋊C8—22no—
463C24—21no—
464C24—61no—
465C24D3×D411yescharacteristic
466#41—24no—
467#41—24no—
468#41—24no—
469#41—24no—
470#41—44no—
471#41—44no—
472#41—64no—
473#41—64no—
474#41—64no—
475#41—64no—
476#41—124no—
477#41—124no—
478C2×Q8—21no—
479C2×Q8—22no—
480C2×Q8—22no—
481C2×Q8—32no—
482C2×Q8—68no—
483C2×Q8—61no—
484C2×Q8—62no—
485C2×Q8—62no—
486C2×Q8C23×D312yes—
487C2×D4—22no—
488C2×D4—22no—
489C2×D4—22no—
490C2×D4—21no—
491C2×D4—22no—
492C2×D4—32no—
493C2×D4—44no—
494C2×D4—68no—
495C2×D4—62no—
496C2×D4—62no—
497C2×D4—62no—
498C2×D4—61no—
499C2×D4—62no—
500C2×D4—124no—
501C2×D4C23×D312yes—
502C22×C4—21no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—34no—
507C22×C4—44no—
508C22×C4—61no—
509C22×C4—62no—
510C22×C4—62no—
511C22×C4—62no—
512C22×C4—124no—
513C22×C4#23514yes—
514C22×C4D3×D411yescharacteristic
515C22×C4D3×D411yescharacteristic
516C22×C4D3×D411yescharacteristic
517C22×C4#23714yes—
518C22×C4C23×D311yescharacteristic
519C22×C4C23×D311yescharacteristic
520C22×C4C23×D311yescharacteristic
521M16—24no—
522M16—24no—
523C2×C8—24no—
524C2×C8—32no—
525C2×C8—32no—
526C2×C8—64no—
527C2×C8—64no—
528C2×C8—62no—
529C2×C8—62no—
530C2×C8C23×D314yes—
531C4⋊C4—22no—
532C4⋊C4—34no—
533C4⋊C4—34no—
534C4⋊C4—34no—
535C4⋊C4—68no—
536C4⋊C4—68no—
537C4⋊C4—68no—
538C4⋊C4—68no—
539C4⋊C4—62no—
540C4⋊C4C23×D314yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—34no—
544C22⋊C4—34no—
545C22⋊C4—68no—
546C22⋊C4—68no—
547C22⋊C4—62no—
548C22⋊C4—62no—
549C42—22no—
550C42—22no—
551C42—34no—
552C42—34no—
553C42—62no—
554C42—62no—
555C22×C3—22no—
556C22×C3—22no—
557C22×C3—22no—
558C22×C3—44no—
559C22×C3—44no—
560C22×C3—42no—
561C22×C3—84no—
562C22×C3C2×#12412yes—
563C22×C3C2×#12412yes—
564C22×C3C2×#12412yes—
565C22×C3C23×D411yescharacteristic
566C2×D3—44no—
567C2×D3—42no—
568C2×D3—84no—
569C3×C4—44no—
570C3×C4—42no—
571C3×C4C2×#12412yes—
572C3×C4C2×#12412yes—
573C3×C4C2×#12412yes—
574C3×C4C22×#4111yescharacteristic
575C3×C4C22×#4111yescharacteristic
576Dic3—28no—
577Dic3—44no—
578Dic3—42no—
579C23—22no—
580C23—42no—
581C23—42no—
582C23—62no—
583C23—82no—
584C23—122no—
585C23—242no—
586C23#88514yes—
587C23C2×D3×D411yescharacteristic
588C23C2×D3×D411yescharacteristic
589C23C2×D3×D411yescharacteristic
590Q8—22no—
591Q8—22no—
592Q8—42no—
593Q8—62no—
594Q8—62no—
595Q8—122no—
596Q8—248no—
597D4—22no—
598D4—22no—
599D4—44no—
600D4—44no—
601D4—42no—
602D4—62no—
603D4—62no—
604D4—124no—
605D4—124no—
606D4—122no—
607D4—248no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—38no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—64no—
615C2×C4—62no—
616C2×C4—82no—
617C2×C4—124no—
618C2×C4—122no—
619C2×C4—124no—
620C2×C4—124no—
621C2×C4—242no—
622C2×C4#88514yes—
623C2×C4#88514yes—
624C2×C4#88514yes—
625C2×C4C2×#23512yes—
626C2×C4C2×#23512yes—
627C2×C4C2×#23512yes—
628C2×C4C2×D3×D411yescharacteristic
629C2×C4C2×D3×D411yescharacteristic
630C2×C4C2×D3×D411yescharacteristic
631C2×C4C2×#23712yes—
632C2×C4C2×#23712yes—
633C2×C4C2×#23712yes—
634C2×C4C24×D311yescharacteristic
635C8—24no—
636C8—62no—
637C8—62no—
638C2×C3—22no—
639C2×C3—22no—
640C2×C3—22no—
641C2×C3—82no—
642C2×C3C22×#12411yescharacteristic
643C2×C3#358412yes—
644D3—82no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—84no—
652C22—124no—
653C22—122no—
654C22—244no—
655C22C2×#88512yes—
656C22C2×#88512yes—
657C22C2×#88512yes—
658C22C22×D3×D411yescharacteristic
659C4—44no—
660C4—42no—
661C4—68no—
662C4—124no—
663C4—122no—
664C4C2×#88512yes—
665C4C2×#88512yes—
666C4C2×#88512yes—
667C4C22×#23511yescharacteristic
668C4C22×#23711yescharacteristic
669C3#6061211yescharacteristic, complemented, Sylow
670C2—22no—
671C2—22no—
672C2—22no—
673C2—82no—
674C2—242no—
675C2C22×#88511yescharacteristic
676C2#8719912yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2896C2
socleC22×C3C23×D4
prime radical#87268C2
hypercenter#2896D3
hypocenterC3#60612
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2896

Complements

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

NH ≅ G/Ntype
2. #85725C2semidirect
3. C3×#2896C2semidirect
4. C2×#5233C2semidirect
5. #83768C2semidirect
6. C2×#5122C2semidirect
7. C2×#5002C2semidirect
8. #78952C2semidirect
9. #78920C2semidirect
10. #78877C2semidirect
11. #78876C2semidirect
12. #78826C2semidirect
13. #78825C2semidirect
32. #5519C22semidirect
34. #5517C22semidirect
35. #5500C22semidirect
38. C3×#419C22semidirect
39. C3×#418C22semidirect
41. C3×#416C22semidirect
43. #5233C22semidirect
44. #5233C22semidirect
45. C2×#835C22semidirect
46. C2×#832C22semidirect
48. #5131C22semidirect
49. #5122C22semidirect
50. #5122C22semidirect
51. #5120C22semidirect
52. #5113C22semidirect
53. C2×#815C22semidirect
54. C2×C8×Dic3C22semidirect
56. #5002C22semidirect
57. #5002C22semidirect
58. C2×#810C22semidirect
59. C2×#809C22semidirect
60. #4800C22semidirect
61. #4797C22semidirect
62. #4794C22semidirect
63. #4792C22semidirect
64. #4768C22semidirect
65. #4765C22semidirect
66. #4764C22semidirect
67. #4760C22semidirect
77. #2896D3semidirect
202. #419C2×D3semidirect
204. #418C2×D3semidirect
214. #416C2×D3semidirect
669. C3#60612semidirect

Maximal subgroups

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

subgroupindex
#872682
#857252
C3×#28962
C2×#52332
#837682
C2×#51222
C2×#50022
#789522
#789202
#788772
#788762
#788262
#788252
#606123