Finite Group Databasebeta

← #11664526

Subgroups — #11664526

Order 768.

2178 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#87282C211yesmaximal, characteristic
2#85889C211yesmaximal, characteristic, complemented
3C3×#2897C211yesmaximal, characteristic, complemented
4#84099C211yesmaximal, characteristic, complemented
5#83797C211yesmaximal, characteristic, complemented
6Dic3×#131C211yesmaximal, characteristic, complemented
7#82563C211yesmaximal, characteristic, complemented
8#78921C213yesmaximal, complemented
9#78920C213yesmaximal, complemented
10#78878C213yesmaximal, complemented
11#78876C213yesmaximal, complemented
12#78827C213yesmaximal, complemented
13#78825C213yesmaximal, complemented
14#78737C213yesmaximal, complemented
15C2×#4760C213yesmaximal, complemented
16#60622—31nomaximal, Sylow
17C2×C3×H(5,𝔽2)C2211yescharacteristic
18C2×#1009C2211yescharacteristic
19C2×C3×#131C2211yescharacteristic
20C3×#520C2211yescharacteristic
21#5834—21no—
22#5834—21no—
23#5832C2213yes—
24#5830C2213yes—
25#5827C2211yescharacteristic
26Dic3×#41C2211yescharacteristic
27C2×#948C2211yescharacteristic
28#5805C2213yes—
29C2×#940C2213yes—
30D4×#67—23no—
31D4×#67—23no—
32#5569—21no—
33Q8×Dic6—21no—
34#5539—23no—
35D4×Dic6—23no—
36#5519C2213yescomplemented
37#5517C2213yescomplemented
38#5500C2213yescomplemented
39C2×#888C2213yescomplemented
40C3×#418C2213yescomplemented
41C3×#417C2213yescomplemented
42C3×#416C2213yescomplemented
43C2×C3×#102C2213yescomplemented
44#5233C2213yescomplemented
45#5232C2213yescomplemented
46#5215C2213yescomplemented
47C2×#832C2213yescomplemented
48#5133C2213yescomplemented
49#5131C2213yescomplemented
50#5122C2213yescomplemented
51Dic3×M16C2213yescomplemented
52#5113C2213yescomplemented
53C2×#821C2213yescomplemented
54#5101C2213yescomplemented
55C2×C8×Dic3C2213yescomplemented
56#5002C2213yescomplemented
57#5001C2213yescomplemented
58#4966C2213yescomplemented
59C2×#809C2213yescomplemented
60#4800C2213yescomplemented
61#4800C2213yescomplemented
62#4794C2216yescomplemented
63#4792C2213yescomplemented
64#4792C2211yescharacteristic, complemented
65#4768C2213yescomplemented
66#4768C2213yescomplemented
67#4765C2216yescomplemented
68#4764C2216yescomplemented
69#4764C2216yescomplemented
70#4760C2216yescomplemented
71#4760C2216yescomplemented
72#4760C2213yescomplemented
73#4760C2213yescomplemented
74#4760C2216yescomplemented
75#3503—31no—
76#3498—31no—
77#3141—33no—
78#3138—33no—
79#3136—33no—
80#3136—33no—
81#3136—33no—
82#3136—33no—
83#3131—33no—
84C2×#486—33no—
85#2897—31no—
86#2897—31no—
87#2897—31no—
88#2897D311yescharacteristic, complemented
89C4×#131—31no—
90C3×H(5,𝔽2)—24no—
91C3×H(5,𝔽2)—24no—
92C2×C3×#41—21no—
93C2×C3×#41C2313yes—
94C2×C3×#41C2311yescharacteristic
95C22×C3×D4—23no—
96C22×C3×D4C2313yes—
97C22×#67—23no—
98C22×#67—23no—
99#1009—24no—
100#1009—24no—
101C2×#239—21no—
102C2×D3×D4—23no—
103C2×#235C2313yes—
104C22×D12C2313yes—
105C22×Dic6—23no—
106C3×#131—24no—
107C2×C3×M16C2313yes—
108C22×C3×C8C2313yes—
109C3×C4×Q8—21no—
110C3×C4×D4—23no—
111C3×#117C2313yes—
112C2×C3×C4⋊C4C2313yes—
113#953—26no—
114#948—24no—
115#947C2313yes—
116Q8×Dic3—21no—
117Q8×Dic3D411yescharacteristic
118Q8×Dic3D411yescharacteristic
119#944C2311yescharacteristic
120#940—23no—
121#940—23no—
122#940C2313yes—
123#940C2316yes—
124#939—23no—
125#939—23no—
126#938—23no—
127#938—23no—
128#937—26no—
129#936C2313yes—
130D4×Dic3—23no—
131D4×Dic3D413yes—
132D4×Dic3D413yes—
133#930—23no—
134#930—23no—
135C4×#67—23no—
136C4×#67—23no—
137#926C2313yes—
138#926C2313yes—
139C2×C4⋊Dic3C2313yes—
140#924—26no—
141C2×Dic3⋊C4—23no—
142C2×C4×Dic3C2313yes—
143C2×#208C2313yes—
144C22×C3⋊C8C2313yes—
145#890—23no—
146#890C2313yes—
147#890C2313yes—
148#888—23no—
149#888—23no—
150#888—23no—
151#888C2313yes—
152#888C2316yes—
153#888C2311yescharacteristic
154#878—23no—
155#878—23no—
156#878—26no—
157#870—23no—
158#869—23no—
159C4×Dic6—23no—
160C4×Dic6—23no—
161C3×#103C2311yescharacteristic
162C3×#103C2313yes—
163C3×#102C2313yes—
164C3×#102C2316yes—
165C3×#102C2313yes—
166#835C2313yes—
167#835C2311yescharacteristic
168#832C2313yes—
169#832C2316yes—
170#832C2313yes—
171#821C2313yes—
172#821C2316yes—
173#821C2313yes—
174#816C2313yes—
175#816C2311yescharacteristic
176#815C2316yes—
177C8×Dic3C2313yes—
178C8×Dic3C2311yescharacteristic
179C8×Dic3C2313yes—
180C8×Dic3C2313yes—
181#810C2313yes—
182#810C2311yescharacteristic
183#809C2313yes—
184#809C2316yes—
185#809C2313yes—
186C2×H(5,𝔽2)—31no—
187C2×H(5,𝔽2)C2×D311yescharacteristic
188C2×#131—31no—
189C2×#131C2×D311yescharacteristic
190Q82—61no—
191#557—61no—
192#554—63no—
193#550—61no—
194#550—61no—
195D4×Q8—63no—
196D42—63no—
197D42—63no—
198#544—33no—
199#544—33no—
200#535—33no—
201#535—33no—
202#533—33no—
203#532—33no—
204C2×C4⋊Q8—33no—
205C2×#127—33no—
206#520—31no—
207#520C2×D311yescharacteristic
208C4×#41—31no—
209#490—36no—
210#490—36no—
211#489—36no—
212#489—36no—
213#488—33no—
214#488—33no—
215#488—33no—
216#488—33no—
217#487—31no—
218#487—33no—
219#486—33no—
220#486—36no—
221#486—36no—
222#486—36no—
223#486—33no—
224#418—33no—
225#418—33no—
226#418—33no—
227#418C2×D313yescomplemented
228#417—33no—
229#417—33no—
230#417—33no—
231#417C2×D313yescomplemented
232#416—33no—
233#416—33no—
234#416—33no—
235#416C2×D313yescomplemented
236C2×#102—33no—
237C2×#102—33no—
238C2×#102—33no—
239C2×#102C2×D313yescomplemented
240#405—33no—
241#405—33no—
242C4×M16—33no—
243C2×C4×C8—33no—
244C24×C3—23no—
245C23×D3—23no—
246C3×#41—26no—
247C3×#41—26no—
248C3×#41—44no—
249C3×#41#4114yes—
250C3×#41#4114yes—
251C2×C3×Q8C2×D411yescharacteristic
252C2×C3×Q8C2411yescharacteristic
253C2×C3×D4—23no—
254C2×C3×D4—23no—
255C2×C3×D4—26no—
256C2×C3×D4—26no—
257C2×C3×D4—412no—
258C2×C3×D4C2×D413yes—
259C2×C3×D4C2413yes—
260C22×C3×C4—23no—
261C22×C3×C4—23no—
262C22×C3×C4C2413yes—
263C2×#67—23no—
264C2×#67—23no—
265C2×#67—23no—
266C2×#67—26no—
267C2×#67—26no—
268C2×#67—86no—
269C2×#67—86no—
270C22×Dic3—23no—
271C22×Dic3C2×D413yes—
272C22×Dic3C2×D413yes—
273#239—44no—
274D3×D4—412no—
275#235—26no—
276#235—26no—
277C2×D12—26no—
278C2×D12—26no—
279C2×D12—23no—
280C2×D12C2413yes—
281C2×C4×D3—23no—
282C2×Dic6—21no—
283C2×Dic6—23no—
284C2×Dic6—26no—
285C2×Dic6—86no—
286C2×Dic6C2411yescharacteristic
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—23no—
298#217—23no—
299#217C2×D416yes—
300(C2×D3)⋊C4—23no—
301(C2×D3)⋊C4—23no—
302C4⋊Dic3—23no—
303C4⋊Dic3—23no—
304C4⋊Dic3—23no—
305C4⋊Dic3C2×D413yes—
306C4⋊Dic3C2×D413yes—
307C4⋊Dic3C2413yes—
308C4⋊Dic3C2411yescharacteristic
309Dic3⋊C4—21no—
310Dic3⋊C4—23no—
311Dic3⋊C4—23no—
312Dic3⋊C4—23no—
313Dic3⋊C4—26no—
314Dic3⋊C4—21no—
315Dic3⋊C4—21no—
316C4×Dic3—23no—
317C4×Dic3C2×D413yes—
318C4×Dic3C2×D413yes—
319C4×Dic3C2411yescharacteristic
320C4×Dic3C2413yes—
321#208—26no—
322C2×C3⋊C8—26no—
323C2×C3⋊C8C2411yescharacteristic
324C2×C3⋊C8C2413yes—
325H(5,𝔽2)—24no—
326H(5,𝔽2)—24no—
327H(5,𝔽2)—64no—
328H(5,𝔽2)—64no—
329C2×#41—21no—
330C2×#41—33no—
331C2×#41—61no—
332C2×#41C22×D313yes—
333C2×#41C22×D311yescharacteristic
334C22×Q8—63no—
335C22×D4—23no—
336C22×D4—33no—
337C22×D4—63no—
338C22×D4—63no—
339C22×D4—63no—
340C22×D4C22×D313yes—
341#131—24no—
342#131—64no—
343C2×M16—33no—
344C2×M16C22×D313yes—
345C22×C8—33no—
346C22×C8C22×D313yes—
347C4⋊Q8—33no—
348C4⋊Q8—36no—
349C4⋊Q8—31no—
350C4⋊Q8—31no—
351C4⋊Q8—63no—
352C4⋊Q8—63no—
353C4⋊Q8—63no—
354C4⋊Q8—63no—
355#127—33no—
356#127—36no—
357#127—63no—
358#127—63no—
359#125—33no—
360#125—33no—
361#125—63no—
362#125—63no—
363#124—33no—
364#124—33no—
365C22⋊Q8—66no—
366C22⋊Q8—63no—
367C22⋊Q8—63no—
368C22⋊Q8—66no—
369#121—63no—
370#121—63no—
371#121—63no—
372#121—63no—
373#121—63no—
374#121—63no—
375#120—66no—
376#120—66no—
377C4×Q8—21no—
378C4×Q8—31no—
379C4×Q8—31no—
380C4×Q8—63no—
381C4×Q8—63no—
382C4×Q8—61no—
383C4×D4—23no—
384C4×D4—33no—
385C4×D4—33no—
386C4×D4—63no—
387C4×D4—63no—
388C4×D4—63no—
389#117—33no—
390#117—33no—
391#117C22×D313yes—
392C2×C4⋊C4—33no—
393C2×C4⋊C4—63no—
394C2×C4⋊C4C22×D313yes—
395C2×C42—33no—
396#103—31no—
397#103—33no—
398#103—31no—
399#103—33no—
400#103—31no—
401#103—33no—
402#103C22×D311yescharacteristic
403#103C22×D313yes—
404#102—33no—
405#102—36no—
406#102—33no—
407#102—33no—
408#102—36no—
409#102—33no—
410#102—33no—
411#102—36no—
412#102—33no—
413#102C22×D313yes—
414#102C22×D316yes—
415#102C22×D313yes—
416#97—36no—
417C4×C8—31no—
418C4×C8—33no—
419C4×C8—33no—
420C4×C8—33no—
421C23×C3—23no—
422C23×C3—46no—
423C23×C3—83no—
424C23×C3—83no—
425C23×C3C22×D413yes—
426C22×D3—23no—
427C22×D3—46no—
428C22×D3—83no—
429C22×D3—83no—
430C3×Q8—21no—
431C3×Q8—21no—
432C3×Q8#12414yes—
433C3×D4—23no—
434C3×D4—23no—
435C3×D4—46no—
436C3×D4—46no—
437C3×D4#124112yes—
438C2×C3×C4—21no—
439C2×C3×C4—23no—
440C2×C3×C4—21no—
441C2×C3×C4—46no—
442C2×C3×C4—83no—
443C2×C3×C4C22×D413yes—
444C2×C3×C4C2×#4116yes—
445C2×C3×C4C2×#4116yes—
446C2×C3×C4C2511yescharacteristic
447#67—46no—
448#67—46no—
449#67—46no—
450#67—86no—
451#67—86no—
452C2×Dic3—26no—
453C2×Dic3—26no—
454C2×Dic3—21no—
455C2×Dic3—23no—
456C2×Dic3—21no—
457C2×Dic3—83no—
458C2×Dic3C22×D413yes—
459C2×Dic3C22×D413yes—
460C2×Dic3C22×D411yescharacteristic
461C2×Dic3C22×D411yescharacteristic
462D12—23no—
463D12—23no—
464D12—46no—
465C4×D3—46no—
466Dic6—21no—
467Dic6—21no—
468Dic6—46no—
469Dic6—42no—
470Dic6—86no—
471C3×C8—21no—
472C3×C8—23no—
473C3⋊C8—21no—
474C3⋊C8—23no—
475C24—23no—
476C24—63no—
477#41—26no—
478#41—26no—
479#41—44no—
480#41—66no—
481#41—66no—
482#41—124no—
483#41#23514yes—
484#41#23714yes—
485C2×Q8—31no—
486C2×Q8—61no—
487C2×Q8—63no—
488C2×Q8—66no—
489C2×Q8—246no—
490C2×Q8D3×D411yescharacteristic
491C2×Q8C23×D311yescharacteristic
492C2×D4—23no—
493C2×D4—26no—
494C2×D4—23no—
495C2×D4—26no—
496C2×D4—33no—
497C2×D4—412no—
498C2×D4—63no—
499C2×D4—63no—
500C2×D4—66no—
501C2×D4—66no—
502C2×D4—63no—
503C2×D4—66no—
504C2×D4—63no—
505C2×D4—66no—
506C2×D4—1212no—
507C2×D4—246no—
508C2×D4—246no—
509C2×D4D3×D413yes—
510C2×D4C23×D313yes—
511C22×C4—23no—
512C22×C4—23no—
513C22×C4—33no—
514C22×C4—33no—
515C22×C4—63no—
516C22×C4—63no—
517C22×C4C23×D313yes—
518M16—26no—
519M16—66no—
520C2×C8—26no—
521C2×C8—31no—
522C2×C8—33no—
523C2×C8—66no—
524C2×C8C23×D311yescharacteristic
525C2×C8C23×D313yes—
526C4⋊C4—23no—
527C4⋊C4—33no—
528C4⋊C4—33no—
529C4⋊C4—33no—
530C4⋊C4—31no—
531C4⋊C4—63no—
532C4⋊C4—66no—
533C4⋊C4—61no—
534C4⋊C4—61no—
535C4⋊C4—61no—
536C4⋊C4—63no—
537C4⋊C4—63no—
538C4⋊C4—63no—
539C4⋊C4—63no—
540C4⋊C4—63no—
541C4⋊C4C23×D313yes—
542C4⋊C4C23×D311yescharacteristic
543C22⋊C4—23no—
544C22⋊C4—36no—
545C22⋊C4—63no—
546C22⋊C4—63no—
547C22⋊C4—63no—
548C22⋊C4—63no—
549C22⋊C4—63no—
550C42—23no—
551C42—33no—
552C42—33no—
553C42—31no—
554C42—33no—
555C42—63no—
556C22×C3—23no—
557C22×C3—23no—
558C22×C3—43no—
559C22×C3—43no—
560C22×C3—43no—
561C22×C3—166no—
562C22×C3C2×#12416yes—
563C22×C3C23×D411yescharacteristic
564C2×D3—43no—
565C2×D3—43no—
566C2×D3—43no—
567C2×D3—166no—
568C3×C4—41no—
569C3×C4—43no—
570C3×C4—41no—
571C3×C4C2×#12416yes—
572C3×C4C22×#4111yescharacteristic
573C3×C4C22×#4111yescharacteristic
574Dic3—23no—
575Dic3—23no—
576Dic3—41no—
577Dic3—43no—
578Dic3—41no—
579Dic3#57612yes—
580Dic3#57712yes—
581C23—23no—
582C23—46no—
583C23—63no—
584C23—83no—
585C23—83no—
586C23—126no—
587C23—243no—
588C23—243no—
589C23C2×D3×D413yes—
590Q8—21no—
591Q8—21no—
592Q8—61no—
593Q8—61no—
594Q8—126no—
595Q8—122no—
596Q8—246no—
597Q8#88514yes—
598D4—23no—
599D4—23no—
600D4—46no—
601D4—46no—
602D4—63no—
603D4—63no—
604D4—126no—
605D4—126no—
606D4—126no—
607D4—126no—
608D4—246no—
609D4—246no—
610D4#885112yes—
611C2×C4—21no—
612C2×C4—23no—
613C2×C4—21no—
614C2×C4—31no—
615C2×C4—31no—
616C2×C4—33no—
617C2×C4—33no—
618C2×C4—46no—
619C2×C4—66no—
620C2×C4—66no—
621C2×C4—61no—
622C2×C4—63no—
623C2×C4—61no—
624C2×C4—83no—
625C2×C4—126no—
626C2×C4—243no—
627C2×C4C2×#23516yes—
628C2×C4C2×D3×D413yes—
629C2×C4C2×#23716yes—
630C2×C4C24×D311yescharacteristic
631C8—21no—
632C8—23no—
633C8—61no—
634C8—63no—
635C2×C3—26no—
636C2×C3—83no—
637C2×C3C22×#12411yescharacteristic
638C2×C3C2×#57611yescharacteristic
639C2×C3C2×#57711yescharacteristic
640D3—83no—
641C22—23no—
642C22—23no—
643C22—43no—
644C22—43no—
645C22—43no—
646C22—123no—
647C22—123no—
648C22—123no—
649C22—166no—
650C22—486no—
651C22C2×#88516yes—
652C22C22×D3×D411yescharacteristic
653C4—32no—
654C4—32no—
655C4—43no—
656C4—41no—
657C4—41no—
658C4—63no—
659C4—63no—
660C4—123no—
661C4—121no—
662C4—121no—
663C4C2×#88516yes—
664C4C22×#23511yescharacteristic
665C4C22×#23711yescharacteristic
666C3#6062211yescharacteristic, complemented, Sylow
667C2—26no—
668C2—83no—
669C2—243no—
670C2C22×#88511yescharacteristic
671C2D3×#57611yescharacteristic
672C2D3×#57711yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×#2897C2
socleC22×C3C23×D4
prime radical#87282C2
hypercenter#2897D3
hypocenterC3#60622
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2897

Complements

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

NH ≅ G/Ntype
2. #85889C2semidirect
3. C3×#2897C2semidirect
4. #84099C2semidirect
5. #83797C2semidirect
6. Dic3×#131C2semidirect
7. #82563C2semidirect
8. #78921C2semidirect
9. #78920C2semidirect
10. #78878C2semidirect
11. #78876C2semidirect
12. #78827C2semidirect
13. #78825C2semidirect
14. #78737C2semidirect
15. C2×#4760C2semidirect
36. #5519C22semidirect
37. #5517C22semidirect
38. #5500C22semidirect
39. C2×#888C22semidirect
40. C3×#418C22semidirect
41. C3×#417C22semidirect
42. C3×#416C22semidirect
43. C2×C3×#102C22semidirect
44. #5233C22semidirect
45. #5232C22semidirect
46. #5215C22semidirect
47. C2×#832C22semidirect
48. #5133C22semidirect
49. #5131C22semidirect
50. #5122C22semidirect
51. Dic3×M16C22semidirect
52. #5113C22semidirect
53. C2×#821C22semidirect
54. #5101C22semidirect
55. C2×C8×Dic3C22semidirect
56. #5002C22semidirect
57. #5001C22semidirect
58. #4966C22semidirect
59. C2×#809C22semidirect
60. #4800C22semidirect
61. #4800C22semidirect
62. #4794C22semidirect
63. #4792C22semidirect
64. #4792C22semidirect
65. #4768C22semidirect
66. #4768C22semidirect
67. #4765C22semidirect
68. #4764C22semidirect
69. #4764C22semidirect
70. #4760C22semidirect
71. #4760C22semidirect
72. #4760C22semidirect
73. #4760C22semidirect
74. #4760C22semidirect
88. #2897D3semidirect
227. #418C2×D3semidirect
231. #417C2×D3semidirect
235. #416C2×D3semidirect
239. C2×#102C2×D3semidirect
666. C3#60622semidirect

Maximal subgroups

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

subgroupindex
#872822
#858892
C3×#28972
#840992
#837972
Dic3×#1312
#825632
#789212
#789202
#788782
#788762
#788272
#788252
#787372
C2×#47602
#606223