Finite Group Databasebeta

← #10824903

Subgroups — #10824903

Order 768.

808 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
2#78866C211yesmaximal, characteristic, complemented
3C2×#4785C211yesmaximal, characteristic, complemented
4#78649C211yesmaximal, characteristic, complemented
5C3×#2051C211yesmaximal, characteristic, complemented
6#72095C211yesmaximal, characteristic
7#70715C211yesmaximal, characteristic, complemented
8#70692C211yesmaximal, characteristic, complemented
9#70302C211yesmaximal, characteristic, complemented
10#70197C211yesmaximal, characteristic, complemented
11#70083C211yesmaximal, characteristic, complemented
12Dic3×#100C211yesmaximal, characteristic, complemented
13#70003C211yesmaximal, characteristic, complemented
14#68981C211yesmaximal, characteristic, complemented
15#68972C211yesmaximal, characteristic, complemented
16#23719—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18Dic3×#41C2211yescharacteristic
19C2×#949C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#213C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×C3×#108C2211yescharacteristic
24C3×#417C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#100C2211yescharacteristic
27C3×#408C2211yescharacteristic
28#5234C2211yescharacteristic
29#5232C2211yescharacteristic
30C2×#833C2211yescharacteristic
31#5215C2211yescharacteristic
32#5164—21no—
33Dic3×SD16—21no—
34#5152—21no—
35Dic3×D8—21no—
36#5131C2211yescharacteristic
37C2×#823C2211yescharacteristic
38C2×#823C2211yescharacteristic
39#5121C2211yescharacteristic
40#5119C2211yescharacteristic
41Dic3×M16C2211yescharacteristic
42Dic3×M16C2211yescharacteristic
43Dic3×M16C2211yescharacteristic
44C2×#821C2211yescharacteristic
45C2×#819C2211yescharacteristic
46#5102C2211yescharacteristic
47C2×#815C2211yescharacteristic
48#4964C2211yescharacteristic
49C2×#808C2211yescharacteristic
50#4817—21no—
51#4788—21no—
52#4785—21no—
53#4785—22no—
54#4785—22no—
55#4756—21no—
56#4720—22no—
57#4719—22no—
58C3×#343C2211yescharacteristic
59#4558C2211yescharacteristic
60#4558C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4530C2211yescharacteristic
64C4×AGL(1,ℤ/8)—31no—
65C4×AGL(1,ℤ/8)—31no—
66C2×#442—31no—
67#2923—31no—
68#2090—31no—
69#2079—31no—
70#2063—31no—
71#2063—31no—
72#2051—31no—
73#2051D311yescharacteristic, complemented
74#2045—31no—
75#2018—31no—
76#1778—31no—
77C4×#100—31no—
78C4×#100—31no—
79C2×C3×#41C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×D12C2311yescharacteristic
83C22×C4×D3—21no—
84C3×AGL(1,ℤ/8)—22no—
85C3×AGL(1,ℤ/8)—22no—
86C2×C3×SD16—21no—
87C2×C3×D8—21no—
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C2×C3×M16C2311yescharacteristic
91C22×C3×C8C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×#217—21no—
94#949—22no—
95#949—22no—
96Q8×Dic3D411yescharacteristic
97Q8×Dic3D411yescharacteristic
98C2×#215—21no—
99D4×Dic3—21no—
100D4×Dic3—21no—
101D4×Dic3—21no—
102D4×Dic3—21no—
103D4×Dic3—21no—
104D4×Dic3—21no—
105D4×Dic3D411yescharacteristic
106D4×Dic3D411yescharacteristic
107C2×#213—21no—
108C2×#213—22no—
109C2×#213—22no—
110C2×#213—22no—
111C2×(C2×D3)⋊C4—21no—
112#926—21no—
113#926C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#893—21no—
122#893—21no—
123#893—21no—
124#893—21no—
125C3×#108—22no—
126C3×#103—21no—
127C3×#102—21no—
128C3×#102—22no—
129C3×#100—22no—
130C3×C22⋊C8—22no—
131#835—21no—
132#833—22no—
133#832—21no—
134#832—21no—
135#832—21no—
136#830—22no—
137#823—22no—
138#823—22no—
139#821—22no—
140#821—21no—
141#821—21no—
142#819—22no—
143#818—21no—
144#817—21no—
145#815—21no—
146#815—21no—
147#815—21no—
148#815D412yes—
149#815D412yes—
150Dic3⋊C8—22no—
151Dic3⋊C8—22no—
152C8×Dic3—21no—
153C8×Dic3—21no—
154C8×Dic3—21no—
155#808—21no—
156#808—22no—
157#807—21no—
158C2×AGL(1,ℤ/8)—31no—
159C2×AGL(1,ℤ/8)C2×D311yescharacteristic
160C22×D8—31no—
161C4×#41—31no—
162C2×C4×D4—31no—
163C2×C4×D4—31no—
164#442—62no—
165#442—62no—
166#442—61no—
167#442—61no—
168#440—61no—
169#440—61no—
170C4×SD16—61no—
171C4×SD16—61no—
172C4×D8—61no—
173C4×D8—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108C2×D311yescharacteristic
178#428—31no—
179#428—31no—
180C2×#106—31no—
181#424—31no—
182#424—31no—
183#418—31no—
184#418—31no—
185#417—31no—
186#417C2×D311yescharacteristic
187C2×#102—31no—
188C2×#102C2×D311yescharacteristic
189C2×#100—31no—
190C2×#100—31no—
191C2×#100—31no—
192C2×#100C2×D311yescharacteristic
193#408—31no—
194#408C2×D311yescharacteristic
195C4×M16—31no—
196C4×M16—31no—
197C4×M16—31no—
198C2×#97—31no—
199#343—31no—
200#343—31no—
201#343—31no—
202#343C2×D311yescharacteristic
203#341—31no—
204#341—31no—
205C24×C3—21no—
206C23×D3—21no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—21no—
210C3×#41—21no—
211C2×C3×Q8C2×D411yescharacteristic
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4C2×D411yescharacteristic
219C2×C3×D4#4112yes—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4C2411yescharacteristic
223C22×Dic3—21no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3—41no—
227C22×Dic3—41no—
228C22×Dic3—82no—
229C22×Dic3#4111yescharacteristic
230C22×Dic3#4111yescharacteristic
231C2×D12—21no—
232C2×D12—21no—
233C2×D12—21no—
234C2×D12—21no—
235C2×D12—21no—
236C2×D12#4112yes—
237C2×C4×D3—41no—
238C2×C4×D3—41no—
239C2×C4×D3—82no—
240C3×SD16—24no—
241C3×D8—24no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16—21no—
247C3×M16#4112yes—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8C2×D412yes—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256#217—21no—
257#217—21no—
258#217—21no—
259#217—41no—
260#217—41no—
261#215—24no—
262#213—28no—
263#213—24no—
264(C2×D3)⋊C4—41no—
265(C2×D3)⋊C4—41no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3C2×D411yescharacteristic
271C4⋊Dic3C2×D411yescharacteristic
272C4×Dic3—21no—
273C4×Dic3—21no—
274C4×Dic3—21no—
275C4×Dic3C2×D411yescharacteristic
276C4×Dic3C2×D411yescharacteristic
277C4×Dic3C2×D411yescharacteristic
278C4×Dic3C2×D411yescharacteristic
279#208—21no—
280#208—21no—
281#208—21no—
282#208—21no—
283#208—21no—
284#208#4112yes—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8C2×D412yes—
291C2×#41C22×D311yescharacteristic
292C22×D4—31no—
293C22×D4C22×D311yescharacteristic
294C23×C4—61no—
295C23×C4—61no—
296AGL(1,ℤ/8)—22no—
297AGL(1,ℤ/8)—22no—
298AGL(1,ℤ/8)—62no—
299AGL(1,ℤ/8)—62no—
300C2×SD16—21no—
301C2×SD16—61no—
302C2×D8—21no—
303C2×D8—62no—
304C2×D8—62no—
305C2×D8—61no—
306C2×D8—62no—
307C2×M16—31no—
308C2×M16—31no—
309C2×M16—31no—
310C2×M16C22×D311yescharacteristic
311C2×M16C22×D311yescharacteristic
312C2×M16C22×D311yescharacteristic
313C22×C8—31no—
314C22×C8C22×D311yescharacteristic
315C4×Q8—31no—
316C4×Q8—31no—
317C4×D4—31no—
318C4×D4—31no—
319C4×D4—61no—
320C4×D4—61no—
321C4×D4—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327C4×D4—61no—
328C4×D4—61no—
329#117—31no—
330#117—61no—
331C2×C4⋊C4—31no—
332C2×C4⋊C4C22×D311yescharacteristic
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C42—31no—
336C2×C42—61no—
337#108—22no—
338#108—62no—
339#107—61no—
340#107—61no—
341#106—61no—
342#106—62no—
343#106—61no—
344C4⋊C8—62no—
345C4⋊C8—62no—
346#103—21no—
347#103—61no—
348#102—21no—
349#102—22no—
350#102—61no—
351#102—61no—
352#102—61no—
353#102—61no—
354#102—61no—
355#102—62no—
356#100—22no—
357#100—62no—
358#100—62no—
359#100—62no—
360C22⋊C8—22no—
361C22⋊C8—62no—
362#97—32no—
363#97—32no—
364#97—61no—
365#97—61no—
366#97—61no—
367C4×C8—61no—
368C4×C8—61no—
369C4×C8—61no—
370C23×C3—21no—
371C23×C3—22no—
372C23×C3—41no—
373C23×C3—41no—
374C23×C3—81no—
375C23×C3—81no—
376C23×C3C2×#4111yescharacteristic
377C22×D3—22no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—81no—
381C22×D3—81no—
382C3×Q8—21no—
383C3×Q8—21no—
384C3×D4—22no—
385C3×D4—21no—
386C3×D4—21no—
387C3×D4—22no—
388C3×D4—41no—
389C3×D4—41no—
390C3×D4—41no—
391C3×D4—41no—
392C2×C3×C4—21no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—81no—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×Dic3—21no—
404C2×Dic3—21no—
405C2×Dic3—41no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—81no—
409C2×Dic3—81no—
410C2×Dic3—82no—
411C2×Dic3—82no—
412C2×Dic3—81no—
413C2×Dic3#12111yescharacteristic
414C2×Dic3#12111yescharacteristic
415C2×Dic3#12111yescharacteristic
416C2×Dic3#12111yescharacteristic
417D12—22no—
418D12—22no—
419D12—41no—
420D12—41no—
421C4×D3—82no—
422C4×D3—82no—
423C3×C8—22no—
424C3×C8—22no—
425C3×C8—41no—
426C3×C8—41no—
427C3⋊C8—22no—
428C3⋊C8—22no—
429C3⋊C8—41no—
430C3⋊C8—41no—
431C24—21no—
432C24—61no—
433#41—21no—
434#41—21no—
435#41—21no—
436#41—21no—
437C2×Q8D3×D411yescharacteristic
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—32no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4D3×D411yescharacteristic
451C2×D4#23712yes—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—31no—
455C22×C4—31no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—121no—
460C22×C4—121no—
461C22×C4—121no—
462C22×C4—121no—
463C22×C4—242no—
464C22×C4—242no—
465C22×C4C23×D311yescharacteristic
466SD16—24no—
467SD16—64no—
468D8—24no—
469D8—68no—
470D8—64no—
471M16—21no—
472M16—21no—
473M16—21no—
474M16—21no—
475M16—21no—
476M16—32no—
477M16—61no—
478M16—61no—
479M16—61no—
480M16—61no—
481M16—61no—
482M16#23712yes—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—32no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8D3×D412yes—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—31no—
498C4⋊C4—31no—
499C4⋊C4—61no—
500C4⋊C4—61no—
501C4⋊C4—61no—
502C4⋊C4—61no—
503C22⋊C4—61no—
504C22⋊C4—61no—
505C22⋊C4—61no—
506C22⋊C4—121no—
507C22⋊C4—121no—
508C22⋊C4—121no—
509C22⋊C4—121no—
510C42—31no—
511C42—31no—
512C42—31no—
513C42—31no—
514C42—61no—
515C42—61no—
516C42—61no—
517C22×C3—21no—
518C22×C3—21no—
519C22×C3—41no—
520C22×C3—42no—
521C22×C3—41no—
522C22×C3—41no—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3—81no—
526C22×C3—81no—
527C22×C3C2×#12111yescharacteristic
528C22×C3#52911yescharacteristic
529C22×C3#52911yescharacteristic
530C2×D3—42no—
531C2×D3—82no—
532C2×D3—81no—
533C2×D3—81no—
534C2×D3—82no—
535C3×C4—41no—
536C3×C4—81no—
537C3×C4#52511yescharacteristic
538C3×C4#52511yescharacteristic
539C3×C4#52511yescharacteristic
540C3×C4#53211yescharacteristic
541Dic3—21no—
542Dic3—21no—
543Dic3—21no—
544Dic3—21no—
545Dic3—41no—
546Dic3—41no—
547Dic3—81no—
548C23—21no—
549C23—22no—
550C23—41no—
551C23—41no—
552C23—62no—
553C23—81no—
554C23—81no—
555C23—121no—
556C23—121no—
557C23—241no—
558C23—241no—
559C23D3×#4111yescharacteristic
560Q8—21no—
561Q8—21no—
562D4—21no—
563D4—21no—
564D4—22no—
565D4—22no—
566D4—41no—
567D4—41no—
568D4—41no—
569D4—41no—
570D4—62no—
571D4—62no—
572D4—121no—
573D4—121no—
574C2×C4—21no—
575C2×C4—31no—
576C2×C4—31no—
577C2×C4—31no—
578C2×C4—31no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—81no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—242no—
589C2×C4—242no—
590C2×C4—241no—
591C2×C4—241no—
592C2×C4—242no—
593C2×C4—242no—
594C2×C4—241no—
595C2×C4C2×D3×D411yescharacteristic
596C2×C4C2×D3×D411yescharacteristic
597C2×C4C2×#23711yescharacteristic
598C2×C4C2×#23711yescharacteristic
599C2×C4D3×#4111yescharacteristic
600C2×C4D3×#4111yescharacteristic
601C8—22no—
602C8—22no—
603C8—41no—
604C8—41no—
605C8—62no—
606C8—62no—
607C8—121no—
608C8—121no—
609C2×C3—21no—
610C2×C3—21no—
611C2×C3—81no—
612C2×C3—82no—
613C2×C3#239211yescharacteristic
614C2×C3#306811yescharacteristic
615C2×C3#306811yescharacteristic
616D3—82no—
617C22—21no—
618C22—21no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—42no—
623C22—82no—
624C22—82no—
625C22—81no—
626C22—81no—
627C22—122no—
628C22—242no—
629C22—242no—
630C22—241no—
631C22—241no—
632C22D3×#12111yescharacteristic
633C22#564911yescharacteristic
634C22#568011yescharacteristic
635C4—41no—
636C4—61no—
637C4—61no—
638C4—61no—
639C4—61no—
640C4—81no—
641C4—121no—
642C4—121no—
643C4—241no—
644C4#558611yescharacteristic
645C4#559611yescharacteristic
646C4#560611yescharacteristic
647C4#565611yescharacteristic
648C3#2371911yescharacteristic, complemented, Sylow
649C2—21no—
650C2—21no—
651C2—81no—
652C2—82no—
653C2—242no—
654C2#7689011yescharacteristic
655C2#8093011yescharacteristic
656C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2051C2
socleC22×C3C2×#121
prime radical#72095C2
hypercenter#2051D3
hypocenterC3#23719
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2051

Complements

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

NH ≅ G/Ntype
1. Dic3×AGL(1,ℤ/8)C2semidirect
2. #78866C2semidirect
3. C2×#4785C2semidirect
4. #78649C2semidirect
5. C3×#2051C2semidirect
7. #70715C2semidirect
8. #70692C2semidirect
9. #70302C2semidirect
10. #70197C2semidirect
11. #70083C2semidirect
12. Dic3×#100C2semidirect
13. #70003C2semidirect
14. #68981C2semidirect
15. #68972C2semidirect
73. #2051D3semidirect
648. C3#23719semidirect

Maximal subgroups

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

subgroupindex
Dic3×AGL(1,ℤ/8)2
#788662
C2×#47852
#786492
C3×#20512
#720952
#707152
#706922
#703022
#701972
#700832
Dic3×#1002
#700032
#689812
#689722
#237193