Finite Group Databasebeta

← #11659428

Subgroups — #11659428

Order 768.

2658 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×2−1+4C211yesmaximal, characteristic, complemented
2C2×#5838C211yesmaximal, characteristic
3#85887C211yesmaximal, characteristic, complemented
4C3×#2898C211yesmaximal, characteristic, complemented
5#83806C211yesmaximal, characteristic, complemented
6Dic3×#131C211yesmaximal, characteristic, complemented
7#82562C211yesmaximal, characteristic, complemented
8#78948C213yesmaximal, complemented
9#78946C213yesmaximal, complemented
10#78868C213yesmaximal, complemented
11#78867C213yesmaximal, complemented
12#78821C213yesmaximal, complemented
13#78818C213yesmaximal, complemented
14#78774C213yesmaximal, complemented
15C2×#4790C213yesmaximal, complemented
16C4×#578—31nomaximal, Sylow
17C2×C3×2−1+4C2211yescharacteristic
18C2×#1010C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20C3×#520C2211yescharacteristic
21#5838C4116yescomplemented
22#5827—23no—
23#5827C2213yes—
24#5827C2211yescharacteristic
25Dic3×#41—23no—
26Dic3×#41C2213yes—
27Dic3×#41C2211yescharacteristic
28C2×#951C2213yes—
29C2×#950C2213yes—
30C2×#948C2211yescharacteristic
31#5812—21no—
32#5812C2213yes—
33C2×Q8×Dic3—21no—
34C2×Q8×Dic3C2213yes—
35C22×#216C2213yes—
36C2×#942C2213yes—
37#5572—21no—
38C4×D3×Q8—21no—
39#5546—23no—
40C4×#237—23no—
41#5522C2213yes—
42#5516C2213yes—
43#5504C2213yes—
44C2×#887C2213yes—
45C3×#419C2213yes—
46C3×#417C2213yes—
47C3×#416C2213yes—
48C2×C3×#103C2213yes—
49#5133C2213yes—
50#5122C2213yes—
51#5120C2213yes—
52Dic3×M16C2213yes—
53#5113C2213yes—
54C2×#816C2213yes—
55#5101C2213yes—
56C2×C8×Dic3C2213yes—
57#4998C2213yes—
58#4995C2213yes—
59#4964C2213yes—
60C2×#807C2213yes—
61#4817C2216yes—
62#4790C2216yes—
63#4790C2216yes—
64#4790C2213yes—
65#4790C2213yes—
66#4790C2216yes—
67#4789C2216yes—
68#4789C2216yes—
69#4788C2213yes—
70#4788C2213yes—
71#4758C2213yes—
72#4758C2213yes—
73#4757C2216yes—
74#4756C2213yes—
75#4756C2211yescharacteristic
76C2×#578—31no—
77C4×2−1+4—31no—
78C4×2−1+4—31no—
79#2951—33no—
80#2950—33no—
81#2947—33no—
82#2947—33no—
83C4×#137—33no—
84C4×#137—33no—
85C4×#135—33no—
86C2×C4×Q16—33no—
87#2913—31no—
88#2898—31no—
89#2898D311yescharacteristic, complemented
90C4×#131—31no—
91C3×2−1+4—24no—
92C3×2−1+4C2×C418yescomplemented
93C2×C3×#41—23no—
94C2×C3×#41C2313yes—
95C2×C3×#41C2311yescharacteristic
96C22×C3×Q8—21no—
97C22×C3×Q8C2313yes—
98#1010—24no—
99#1010C2×C418yescomplemented
100C2×D3×Q8—21no—
101C2×#237—23no—
102C2×#235C2313yes—
103C22×Dic6C2313yes—
104C3×#131—24no—
105C2×C3×M16C2313yes—
106C22×C3×C8C2313yes—
107C3×C4×Q8—21no—
108C3×C4×D4—23no—
109C3×#117C2313yes—
110C2×C3×C4⋊C4C2313yes—
111#951C2×C4124yescomplemented
112#950C2×C4124yescomplemented
113#948C2×C418yescomplemented
114Q8×Dic3—21no—
115Q8×Dic3—23no—
116Q8×Dic3—23no—
117Q8×Dic3—23no—
118Q8×Dic3—23no—
119Q8×Dic3D411yescharacteristic
120Q8×Dic3D411yescharacteristic
121Q8×Dic3C2313yes—
122Q8×Dic3C2316yes—
123Q8×Dic3C2313yes—
124C2×#216C2×C4124yescomplemented
125C2×#216C2316yes—
126C2×#216C2313yes—
127#942C2×C4124yescomplemented
128C2×#215C2313yes—
129D4×Dic3—23no—
130D4×Dic3—23no—
131D4×Dic3—23no—
132D4×Dic3—26no—
133D4×Dic3D413yes—
134D4×Dic3D413yes—
135D4×Dic3C2313yes—
136D4×Dic3C2311yescharacteristic
137C2×#214C2313yes—
138C2×#213C2311yescharacteristic
139C4×#67—26no—
140#926—26no—
141#926—23no—
142#926—23no—
143#926C2313yes—
144#926C2313yes—
145C2×C4⋊Dic3—23no—
146C2×C4⋊Dic3—23no—
147C2×C4⋊Dic3C2313yes—
148C2×Dic3⋊C4—23no—
149C2×C4×Dic3—23no—
150C2×C4×Dic3—23no—
151C2×C4×Dic3—23no—
152C2×C4×Dic3C2313yes—
153C2×#208C2313yes—
154C22×C3⋊C8C2313yes—
155#893C2313yes—
156#893C2311yescharacteristic
157#892—23no—
158D3×C4⋊C4—23no—
159#887—23no—
160#887—23no—
161#887C2313yes—
162#887C2316yes—
163#887C2313yes—
164#67⋊C4—23no—
165#67⋊C4—23no—
166#877—26no—
167#872—23no—
168C42×D3—23no—
169C4×Dic6—23no—
170C4×Dic6—23no—
171C3×#103C2313yes—
172C3×#103C2316yes—
173C3×#103C2313yes—
174C3×#102C2313yes—
175C3×#102C2311yescharacteristic
176#821C2313yes—
177#821C2311yescharacteristic
178#816C2313yes—
179#816C2316yes—
180#816C2313yes—
181#815C2316yes—
182C8×Dic3C2313yes—
183C8×Dic3C2311yescharacteristic
184C8×Dic3C2313yes—
185C8×Dic3C2313yes—
186#808C2313yes—
187#808C2313yes—
188#807C2313yes—
189#807C2316yes—
190#807C2311yescharacteristic
191C2×2−1+4—31no—
192C2×2−1+4C2×D311yescharacteristic
193#578—316no—
194C2×#137—33no—
195C2×#137—33no—
196C2×#135—33no—
197C22×Q16—33no—
198C2×#131—31no—
199C2×#131C2×D311yescharacteristic
200#520—31no—
201#520—33no—
202#520—33no—
203#520—63no—
204#520—63no—
205#520C2×D311yescharacteristic
206#519—33no—
207#519—33no—
208#519—61no—
209#519—61no—
210C4×#41—31no—
211C4×#41—33no—
212C4×#41—33no—
213C4×#41—63no—
214C4×#41—63no—
215C2×C4×Q8—33no—
216C2×C4×Q8—33no—
217C2×C4×Q8—61no—
218C2×C4×Q8—61no—
219#441—36no—
220#441—36no—
221#440—36no—
222#440—36no—
223C4×Q16—33no—
224C4×Q16—36no—
225C4×Q16—36no—
226C4×Q16—36no—
227C4×Q16—33no—
228C4×SD16—33no—
229C4×SD16—33no—
230C4×SD16—33no—
231C4×SD16—33no—
232C4×D8—31no—
233C4×D8—33no—
234#428—33no—
235#428—33no—
236#427—33no—
237C2×#107—33no—
238#419—33no—
239#419C2×D313yes—
240#417—33no—
241#417C2×D313yes—
242#416—33no—
243#416C2×D313yes—
244C2×#103—33no—
245C2×#103C2×D313yes—
246#405—33no—
247#405—33no—
248C4×M16—33no—
249C2×C4×C8—33no—
250C3×#41—224no—
251C3×#41—26no—
252C3×#41C22×C4112yes—
253C3×#41C22×C414yes—
254C3×#41#4114yes—
255C2×C3×Q8—28no—
256C2×C3×Q8—23no—
257C2×C3×Q8—26no—
258C2×C3×Q8C22×C4112yes—
259C2×C3×Q8C2×D411yescharacteristic
260C2×C3×Q8C2413yes—
261C2×C3×D4—23no—
262C2×C3×D4C2×D413yes—
263C2×C3×D4C2411yescharacteristic
264C22×C3×C4—23no—
265C22×C3×C4—23no—
266C22×C3×C4—23no—
267C22×C3×C4C2413yes—
268C2×#67—23no—
269C22×Dic3—23no—
270C22×Dic3—23no—
271C22×Dic3—21no—
272C22×Dic3—23no—
273C22×Dic3C2×D413yes—
274C22×Dic3C2×D413yes—
275D3×Q8—28no—
276#237—224no—
277#235—26no—
278#235C22×C4112yes—
279C2×D12C2411yescharacteristic
280C2×C4×D3—23no—
281C2×C4×D3—21no—
282C2×C4×D3—23no—
283C2×Dic6—23no—
284C2×Dic6—26no—
285C2×Dic6C22×C4112yes—
286C2×Dic6C2413yes—
287C3×M16—26no—
288C2×C3×C8—26no—
289C2×C3×C8C2411yescharacteristic
290C2×C3×C8C2413yes—
291C3×C4⋊C4—23no—
292C3×C4⋊C4C2413yes—
293C3×C4⋊C4C2411yescharacteristic
294C3×C22⋊C4—23no—
295C3×C42—23no—
296#217—23no—
297#217—21no—
298#217—23no—
299#217C2×D416yes—
300#216C22×C4124yes—
301#216C22×C4112yes—
302#215C22×C4112yes—
303#214C22×C4112yes—
304#213C22×C414yes—
305(C2×D3)⋊C4—21no—
306(C2×D3)⋊C4—23no—
307C4⋊Dic3—23no—
308C4⋊Dic3—26no—
309C4⋊Dic3—23no—
310C4⋊Dic3—23no—
311C4⋊Dic3C2×D413yes—
312C4⋊Dic3C2×D413yes—
313C4⋊Dic3C2413yes—
314C4⋊Dic3C2411yescharacteristic
315Dic3⋊C4—23no—
316Dic3⋊C4—26no—
317Dic3⋊C4—23no—
318C4×Dic3—23no—
319C4×Dic3—23no—
320C4×Dic3—26no—
321C4×Dic3—23no—
322C4×Dic3—23no—
323C4×Dic3—23no—
324C4×Dic3—26no—
325C4×Dic3C2×D413yes—
326C4×Dic3C2×D413yes—
327C4×Dic3C2411yescharacteristic
328C4×Dic3C2413yes—
329#208C22×C4112yes—
330C2×C3⋊C8C22×C4112yes—
331C2×C3⋊C8C2411yescharacteristic
332C2×C3⋊C8C2413yes—
3332−1+4—24no—
3342−1+4—38no—
3352−1+4—64no—
3362−1+4C4×D318yescomplemented
337C2×#41—23no—
338C2×#41—33no—
339C2×#41—63no—
340C2×#41C22×D313yes—
341C2×#41C22×D311yescharacteristic
342C22×Q8—21no—
343C22×Q8—33no—
344C22×Q8—61no—
345C22×Q8C22×D313yes—
346#137—324no—
347#137—324no—
348#135—324no—
349C2×Q16—324no—
350C2×Q16—33no—
351C2×Q16—36no—
352C2×SD16—33no—
353C2×SD16—33no—
354C2×D8—31no—
355#131—24no—
356#131—38no—
357C2×M16—33no—
358C2×M16C22×D313yes—
359C22×C8—33no—
360C22×C8C22×D313yes—
361C4×Q8—21no—
362C4×Q8—31no—
363C4×Q8—31no—
364C4×Q8—33no—
365C4×Q8—36no—
366C4×Q8—33no—
367C4×Q8—33no—
368C4×Q8—36no—
369C4×Q8—33no—
370C4×Q8—63no—
371C4×Q8—63no—
372C4×Q8—63no—
373C4×Q8—63no—
374C4×Q8—63no—
375C4×Q8—63no—
376C4×Q8—63no—
377C4×Q8—63no—
378C4×Q8—61no—
379C4×D4—23no—
380C4×D4—33no—
381C4×D4—33no—
382C4×D4—31no—
383C4×D4—33no—
384C4×D4—31no—
385C4×D4—33no—
386C4×D4—63no—
387C4×D4—63no—
388C4×D4—66no—
389C4×D4—66no—
390C4×D4—63no—
391C4×D4—63no—
392C4×D4—63no—
393#117—33no—
394#117—33no—
395#117—63no—
396#117—63no—
397#117—66no—
398#117—66no—
399#117—63no—
400#117—63no—
401#117C22×D313yes—
402C2×C4⋊C4—33no—
403C2×C4⋊C4—63no—
404C2×C4⋊C4—63no—
405C2×C4⋊C4—63no—
406C2×C4⋊C4—63no—
407C2×C4⋊C4C22×D313yes—
408C2×C42—33no—
409C2×C42—63no—
410C2×C42—63no—
411C2×C42—63no—
412C2×C42—63no—
413#107—33no—
414#107—36no—
415#107—31no—
416#106—33no—
417#106—33no—
418#103—33no—
419#103—36no—
420#103—33no—
421#103C22×D313yes—
422#103C22×D316yes—
423#103C22×D313yes—
424#102—31no—
425#102—33no—
426#102C22×D311yescharacteristic
427#102C22×D313yes—
428#97—36no—
429C4×C8—31no—
430C4×C8—33no—
431C4×C8—33no—
432C4×C8—33no—
433C23×C3—21no—
434C23×C3C22×D413yes—
435C22×D3—21no—
436C3×Q8—212no—
437C3×Q8—23no—
438C3×Q8C4×D414yes—
439C3×Q8C23×C416yes—
440C3×D4—212no—
441C3×D4—21no—
442C3×D4C4×D4112yes—
443C3×D4C23×C412yes—
444C2×C3×C4—212no—
445C2×C3×C4—23no—
446C2×C3×C4—212no—
447C2×C3×C4—21no—
448C2×C3×C4—23no—
449C2×C3×C4—83no—
450C2×C3×C4C23×C416yes—
451C2×C3×C4C22×D413yes—
452C2×C3×C4C2×#4116yes—
453C2×C3×C4C2511yescharacteristic
454#67—212no—
455C2×Dic3—212no—
456C2×Dic3—23no—
457C2×Dic3—26no—
458C2×Dic3—21no—
459C2×Dic3—23no—
460C2×Dic3—43no—
461C2×Dic3—42no—
462C2×Dic3—83no—
463C2×Dic3—83no—
464C2×Dic3C22×D413yes—
465C2×Dic3C22×D413yes—
466C2×Dic3C22×D411yescharacteristic
467C2×Dic3C22×D411yescharacteristic
468D12—21no—
469D12C23×C412yes—
470C4×D3—212no—
471C4×D3—42no—
472C4×D3—83no—
473Dic6—212no—
474Dic6—23no—
475Dic6C23×C416yes—
476C3×C8—21no—
477C3×C8—23no—
478C3⋊C8C23×C412yes—
479C3⋊C8C23×C416yes—
480#41—224no—
481#41—26no—
482#41—312no—
483#41—624no—
484#41—66no—
485#41C2×C4×D3112yes—
486#41C2×C4×D314yes—
487#41#23714yes—
488C2×Q8—28no—
489C2×Q8—23no—
490C2×Q8—26no—
491C2×Q8—312no—
492C2×Q8—33no—
493C2×Q8—68no—
494C2×Q8—63no—
495C2×Q8—66no—
496C2×Q8C2×C4×D3112yes—
497C2×Q8D3×D411yescharacteristic
498C2×Q8C23×D313yes—
499C2×D4—23no—
500C2×D4—31no—
501C2×D4—63no—
502C2×D4D3×D413yes—
503C2×D4C23×D311yescharacteristic
504C22×C4—23no—
505C22×C4—23no—
506C22×C4—23no—
507C22×C4—33no—
508C22×C4—33no—
509C22×C4—61no—
510C22×C4—63no—
511C22×C4—61no—
512C22×C4—63no—
513C22×C4—63no—
514C22×C4—63no—
515C22×C4—63no—
516C22×C4C23×D313yes—
517Q16—312no—
518Q16—324no—
519SD16—312no—
520SD16—312no—
521D8—34no—
522M16—26no—
523M16—312no—
524C2×C8—26no—
525C2×C8—312no—
526C2×C8—31no—
527C2×C8—33no—
528C2×C8C23×D311yescharacteristic
529C2×C8C23×D313yes—
530C4⋊C4—23no—
531C4⋊C4—33no—
532C4⋊C4—33no—
533C4⋊C4—31no—
534C4⋊C4—33no—
535C4⋊C4—63no—
536C4⋊C4—66no—
537C4⋊C4—63no—
538C4⋊C4—66no—
539C4⋊C4—63no—
540C4⋊C4—63no—
541C4⋊C4—63no—
542C4⋊C4C23×D311yescharacteristic
543C4⋊C4C23×D313yes—
544C22⋊C4—23no—
545C22⋊C4—36no—
546C22⋊C4—61no—
547C22⋊C4—63no—
548C22⋊C4—61no—
549C22⋊C4—63no—
550C22⋊C4—63no—
551C42—23no—
552C42—33no—
553C42—33no—
554C42—31no—
555C42—33no—
556C42—63no—
557C42—63no—
558C42—66no—
559C42—63no—
560C42—63no—
561C42—66no—
562C42—63no—
563C22×C3—23no—
564C22×C3—23no—
565C22×C3—22no—
566C22×C3—41no—
567C22×C3—41no—
568C22×C3C2×C4×D416yes—
569C22×C3C23×D411yescharacteristic
570C2×D3—22no—
571C2×D3—41no—
572C2×D3—41no—
573C3×C4—26no—
574C3×C4—43no—
575C3×C4—41no—
576C3×C4C2×C4×D416yes—
577C3×C4C24×C411yescharacteristic
578C3×C4C22×#4111yescharacteristic
579Dic3—23no—
580Dic3—21no—
581Dic3—26no—
582Dic3—23no—
583Dic3—43no—
584Dic3—41no—
585Dic3#57812yescomplemented
586C23—21no—
587C23—61no—
588C23C2×D3×D413yes—
589Q8—23no—
590Q8—212no—
591Q8—36no—
592Q8—63no—
593Q8—612no—
594Q8#67⋊C414yes—
595Q8C22×C4×D316yes—
596D4—212no—
597D4—21no—
598D4—32no—
599D4—612no—
600D4—61no—
601D4#67⋊C4112yes—
602D4C22×C4×D312yes—
603C2×C4—23no—
604C2×C4—212no—
605C2×C4—212no—
606C2×C4—21no—
607C2×C4—23no—
608C2×C4—31no—
609C2×C4—31no—
610C2×C4—33no—
611C2×C4—33no—
612C2×C4—66no—
613C2×C4—63no—
614C2×C4—612no—
615C2×C4—612no—
616C2×C4—61no—
617C2×C4—63no—
618C2×C4—83no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—123no—
622C2×C4—243no—
623C2×C4—243no—
624C2×C4—243no—
625C2×C4C22×C4×D316yes—
626C2×C4C2×D3×D413yes—
627C2×C4C2×#23716yes—
628C2×C4C24×D311yescharacteristic
629C8—21no—
630C8—23no—
631C8—32no—
632C8—36no—
633C2×C3—26no—
634C2×C3—42no—
635C2×C3C22×C4×D411yescharacteristic
636C2×C3C2×#57811yescharacteristic
637C2×C3#358411yescharacteristic
638D3—42no—
639C22—23no—
640C22—23no—
641C22—22no—
642C22—41no—
643C22—41no—
644C22—62no—
645C22—121no—
646C22—121no—
647C22C2×#67⋊C416yes—
648C22C22×D3×D411yescharacteristic
649C4—26no—
650C4—32no—
651C4—43no—
652C4—41no—
653C4—61no—
654C4—63no—
655C4—63no—
656C4—66no—
657C4—123no—
658C4—121no—
659C4C2×#67⋊C416yes—
660C4C23×C4×D311yescharacteristic
661C4C22×#23711yescharacteristic
662C3C4×#57811yescharacteristic, complemented, Sylow
663C2—26no—
664C2—42no—
665C2—122no—
666C2C22×#67⋊C411yescharacteristic
667C2#8720611yescharacteristic
668C2D3×#57811yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutatorC3×C4C24×C4
FrattiniC2×C4C24×D3
FittingC3×#2898C2
socleC22×C3C23×D4
prime radicalC2×#5838C2
cosocleC2×C3×C4C25
hypercenter#2898D3
hypocenterC3C4×#578
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2898

Complements

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

NH ≅ G/Ntype
1. Dic3×2−1+4C2semidirect
3. #85887C2semidirect
4. C3×#2898C2semidirect
5. #83806C2semidirect
6. Dic3×#131C2semidirect
7. #82562C2semidirect
8. #78948C2semidirect
9. #78946C2semidirect
10. #78868C2semidirect
11. #78867C2semidirect
12. #78821C2semidirect
13. #78818C2semidirect
14. #78774C2semidirect
15. C2×#4790C2semidirect
21. #5838C4semidirect
89. #2898D3semidirect
92. C3×2−1+4C2×C4semidirect
99. #1010C2×C4semidirect
111. #951C2×C4semidirect
112. #950C2×C4semidirect
113. #948C2×C4semidirect
124. C2×#216C2×C4semidirect
127. #942C2×C4semidirect
336. 2−1+4C4×D3semidirect
585. Dic3#578semidirect
662. C3C4×#578semidirect

Maximal subgroups

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

subgroupindex
Dic3×2−1+42
C2×#58382
#858872
C3×#28982
#838062
Dic3×#1312
#825622
#789482
#789462
#788682
#788672
#788212
#788182
#787742
C2×#47902
C4×#5783