Finite Group Databasebeta

← #10824917

Subgroups — #10824917

Order 768.

776 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Dic3×#137C211yesmaximal, characteristic, complemented
2Dic3×AGL(1,ℤ/8)C211yesmaximal, characteristic
3C2×#5164C211yesmaximal, characteristic, complemented
4#78649C211yesmaximal, characteristic, complemented
5C3×#2019C211yesmaximal, characteristic
6#72105C211yesmaximal, characteristic, complemented
7#72093C211yesmaximal, characteristic
8#72053C211yesmaximal, characteristic, complemented
9#70109C211yesmaximal, characteristic, complemented
10Dic3×#101C211yesmaximal, characteristic, complemented
11#70075C211yesmaximal, characteristic, complemented
12Dic3×#100C211yesmaximal, characteristic, complemented
13#69994C211yesmaximal, characteristic, complemented
14#68923C211yesmaximal, characteristic, complemented
15#68917C211yesmaximal, characteristic, complemented
16#23721—31nomaximal, Sylow
17C2×C3×#137C2211yescharacteristic
18C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
19C22×C3×SD16C2211yescharacteristic
20Dic3×#41C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×D4×Dic3C2211yescharacteristic
23C2×C3×#101C2211yescharacteristic
24C2×C3×#100C2211yescharacteristic
25C3×#408C2211yescharacteristic
26#5234C2211yescharacteristic
27#5232C2211yescharacteristic
28#5232C2211yescharacteristic
29C2×#836C2211yescharacteristic
30#5225C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#833C2211yescharacteristic
33#5215C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#5184—21no—
36Dic3×Q16—21no—
37#5164—21no—
38#5164—21no—
39#5164—22no—
40#5164—22no—
41Dic3×SD16—21no—
42Dic3×SD16—21no—
43#5152—21no—
44Dic3×D8—21no—
45C2×#822C2211yescharacteristic
46C2×#822C2211yescharacteristic
47#5121C2211yescharacteristic
48#5121C2211yescharacteristic
49#5119C2211yescharacteristic
50Dic3×M16C2211yescharacteristic
51Dic3×M16C2211yescharacteristic
52Dic3×M16C2211yescharacteristic
53C2×#818C2211yescharacteristic
54#5102C2211yescharacteristic
55C2×#815C2211yescharacteristic
56#4720—22no—
57#4719—22no—
58C3×#341C2211yescharacteristic
59#4554C2211yescharacteristic
60#4554C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4552C2211yescharacteristic
64C4×#137—31no—
65C4×AGL(1,ℤ/8)—31no—
66C2×#440—31no—
67#2923—31no—
68#2089—31no—
69#2078—31no—
70#2064—31no—
71#2063—31no—
72#2052—31no—
73#2051—31no—
74#2046—31no—
75#2019D311yescharacteristic
76#1778—31no—
77C4×#101—31no—
78C4×#100—31no—
79C2×C3×#41C2311yescharacteristic
80C22×C3×Q8C2311yescharacteristic
81C22×C3×D4C2311yescharacteristic
82C23×Dic3—21no—
83C3×#137—22no—
84C3×#137—22no—
85C3×AGL(1,ℤ/8)—22no—
86C3×AGL(1,ℤ/8)—22no—
87C2×C3×Q16—21no—
88C2×C3×SD16—21no—
89C2×C3×SD16—22no—
90C2×C3×SD16—21no—
91C2×C3×SD16—22no—
92C2×C3×SD16—22no—
93C2×C3×D8—21no—
94C2×C3×M16C2311yescharacteristic
95C2×C3×M16C2311yescharacteristic
96C2×C3×M16C2311yescharacteristic
97C22×C3×C8C2311yescharacteristic
98C2×#217—21no—
99Q8×Dic3—21no—
100Q8×Dic3—21no—
101Q8×Dic3—21no—
102Q8×Dic3—21no—
103Q8×Dic3D411yescharacteristic
104Q8×Dic3D411yescharacteristic
105D4×Dic3—21no—
106D4×Dic3—21no—
107D4×Dic3—21no—
108D4×Dic3—21no—
109D4×Dic3—21no—
110D4×Dic3—21no—
111D4×Dic3D411yescharacteristic
112D4×Dic3D411yescharacteristic
113#926—21no—
114#926C2311yescharacteristic
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3C2311yescharacteristic
117C2×C4⋊Dic3C2311yescharacteristic
118C2×C4×Dic3—21no—
119C2×C4×Dic3—21no—
120C2×C4×Dic3C2311yescharacteristic
121C2×#208C2311yescharacteristic
122C2×#208C2311yescharacteristic
123C2×#208C2311yescharacteristic
124C22×C3⋊C8C2311yescharacteristic
125C3×#101—22no—
126C3×#100—22no—
127C3×C22⋊C8—22no—
128#836—22no—
129#835—21no—
130#835—21no—
131#835—21no—
132#835—21no—
133#835—22no—
134#833—22no—
135#832—21no—
136#832—21no—
137#832—21no—
138#832—21no—
139#832—22no—
140#830—22no—
141#822—22no—
142#822—22no—
143#818—22no—
144#818—21no—
145#818—21no—
146#817—21no—
147#817—21no—
148#815—21no—
149#815—21no—
150#815—21no—
151#815D412yes—
152#815D412yes—
153Dic3⋊C8—22no—
154Dic3⋊C8—22no—
155C8×Dic3—21no—
156C8×Dic3—21no—
157C8×Dic3—21no—
158C2×#137C2×D311yescharacteristic
159C2×AGL(1,ℤ/8)C2×D311yescharacteristic
160C22×SD16C2×D311yescharacteristic
161C4×#41—31no—
162C2×C4×Q8—31no—
163C2×C4×D4—31no—
164#442—61no—
165#441—61no—
166#440—62no—
167#440—62no—
168#440—61no—
169#440—61no—
170C4×Q16—61no—
171C4×SD16—61no—
172C4×SD16—61no—
173C4×D8—61no—
174#433—62no—
175#432—62no—
176C2×#108—31no—
177C2×#108—31no—
178#428—31no—
179#428—31no—
180C2×#107—31no—
181#424—31no—
182#424—31no—
183#419—31no—
184#418—31no—
185#417—31no—
186#417—31no—
187C2×#103—31no—
188C2×#102—31no—
189C2×#101—31no—
190C2×#101C2×D311yescharacteristic
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#343—31no—
203#341—31no—
204#341C2×D311yescharacteristic
205C24×C3—21no—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—21no—
215C2×C3×Q8C2×D411yescharacteristic
216C2×C3×Q8#4112yes—
217C2×C3×D4—21no—
218C2×C3×D4—21no—
219C2×C3×D4—21no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—21no—
223C2×C3×D4C2×D411yescharacteristic
224C2×C3×D4#4112yes—
225C22×C3×C4—21no—
226C22×C3×C4—21no—
227C22×C3×C4C2411yescharacteristic
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—41no—
233C22×Dic3—41no—
234C22×Dic3—82no—
235C22×Dic3#4111yescharacteristic
236C22×Dic3#4111yescharacteristic
237C3×Q16—24no—
238C3×SD16—28no—
239C3×SD16—24no—
240C3×SD16—24no—
241C3×D8—24no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—21no—
245C3×M16—21no—
246C3×M16#4112yes—
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—
254#217—21no—
255#217—21no—
256#217—21no—
257#217—41no—
258#217—41no—
259C4⋊Dic3—21no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—41no—
266C4⋊Dic3—41no—
267C4⋊Dic3C2×D411yescharacteristic
268C4⋊Dic3C2×D411yescharacteristic
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—21no—
272C4×Dic3—41no—
273C4×Dic3—41no—
274C4×Dic3C2×D411yescharacteristic
275C4×Dic3C2×D411yescharacteristic
276C4×Dic3C2×D411yescharacteristic
277C4×Dic3C2×D411yescharacteristic
278#208—21no—
279#208—21no—
280#208—21no—
281#208—21no—
282#208—21no—
283#208—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8C2×D412yes—
290C2×#41C22×D311yescharacteristic
291C22×Q8C22×D311yescharacteristic
292C22×D4C22×D311yescharacteristic
293C23×C4—61no—
294#137—22no—
295#137—22no—
296AGL(1,ℤ/8)—22no—
297AGL(1,ℤ/8)—22no—
298C2×Q16—21no—
299C2×SD16—21no—
300C2×SD16—22no—
301C2×SD16—21no—
302C2×SD16—22no—
303C2×SD16—22no—
304C2×D8—21no—
305C2×M16—31no—
306C2×M16—31no—
307C2×M16—31no—
308C2×M16C22×D311yescharacteristic
309C2×M16C22×D311yescharacteristic
310C2×M16C22×D311yescharacteristic
311C22×C8—31no—
312C22×C8C22×D311yescharacteristic
313C4×Q8—31no—
314C4×Q8—31no—
315C4×Q8—61no—
316C4×Q8—61no—
317C4×Q8—61no—
318C4×Q8—61no—
319C4×D4—31no—
320C4×D4—31no—
321C4×D4—61no—
322C4×D4—61no—
323C4×D4—61no—
324C4×D4—61no—
325C4×D4—61no—
326C4×D4—61no—
327#117—31no—
328#117—61no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C42—31no—
334C2×C42—61no—
335C2×C42—61no—
336#108—62no—
337#108—62no—
338#107—61no—
339#107—61no—
340#107—62no—
341#106—61no—
342#106—61no—
343C4⋊C8—62no—
344C4⋊C8—62no—
345#103—61no—
346#103—61no—
347#103—62no—
348#103—61no—
349#103—61no—
350#102—61no—
351#102—61no—
352#102—62no—
353#102—61no—
354#102—61no—
355#101—22no—
356#101—62no—
357#100—22no—
358#100—62no—
359C22⋊C8—22no—
360C22⋊C8—62no—
361#97—32no—
362#97—32no—
363#97—61no—
364#97—61no—
365#97—61no—
366C4×C8—61no—
367C4×C8—61no—
368C4×C8—61no—
369C23×C3—21no—
370C23×C3—22no—
371C23×C3—41no—
372C23×C3—41no—
373C23×C3—81no—
374C23×C3—81no—
375C23×C3C2×#4111yescharacteristic
376C3×Q8—22no—
377C3×Q8—21no—
378C3×Q8—22no—
379C3×Q8—21no—
380C3×Q8—41no—
381C3×Q8—41no—
382C3×D4—21no—
383C3×D4—22no—
384C3×D4—22no—
385C3×D4—21no—
386C3×D4—41no—
387C3×D4—41no—
388C3×D4—41no—
389C3×D4—41no—
390C2×C3×C4—21no—
391C2×C3×C4—22no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4C22×D411yescharacteristic
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×Dic3—21no—
403C2×Dic3—21no—
404C2×Dic3—41no—
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—81no—
414C2×Dic3#12111yescharacteristic
415C2×Dic3#12111yescharacteristic
416C2×Dic3#12111yescharacteristic
417C2×Dic3#12111yescharacteristic
418C3×C8—22no—
419C3×C8—22no—
420C3×C8—22no—
421C3×C8—41no—
422C3⋊C8—22no—
423C3⋊C8—41no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C24—21no—
427#41—21no—
428#41—21no—
429#41—21no—
430#41—21no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8D3×D411yescharacteristic
437C2×Q8#23712yes—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4D3×D411yescharacteristic
445C2×D4#23712yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—121no—
455C22×C4—121no—
456C22×C4—242no—
457C22×C4C23×D311yescharacteristic
458Q16—24no—
459SD16—28no—
460SD16—24no—
461SD16—24no—
462D8—24no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—61no—
468M16—61no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16#23712yes—
474M16#23712yes—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—32no—
481C2×C8—61no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8D3×D412yes—
487C4⋊C4—31no—
488C4⋊C4—31no—
489C4⋊C4—61no—
490C4⋊C4—61no—
491C4⋊C4—61no—
492C4⋊C4—61no—
493C4⋊C4—61no—
494C4⋊C4—61no—
495C4⋊C4—121no—
496C4⋊C4—121no—
497C22⋊C4—61no—
498C22⋊C4—61no—
499C22⋊C4—61no—
500C22⋊C4—121no—
501C22⋊C4—121no—
502C42—31no—
503C42—31no—
504C42—31no—
505C42—31no—
506C42—61no—
507C42—61no—
508C42—61no—
509C42—121no—
510C42—121no—
511C22×C3—21no—
512C22×C3—21no—
513C22×C3—41no—
514C22×C3—42no—
515C22×C3—41no—
516C22×C3—41no—
517C22×C3—82no—
518C22×C3—82no—
519C22×C3—81no—
520C22×C3—81no—
521C22×C3C2×#12111yescharacteristic
522C22×C3#52911yescharacteristic
523C22×C3#52911yescharacteristic
524C3×C4—41no—
525C3×C4—42no—
526C3×C4#52511yescharacteristic
527C3×C4#52511yescharacteristic
528C3×C4#52511yescharacteristic
529C3×C4#53211yescharacteristic
530Dic3—21no—
531Dic3—21no—
532Dic3—21no—
533Dic3—21no—
534Dic3—41no—
535Dic3—41no—
536Dic3—81no—
537Dic3—81no—
538C23—21no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—81no—
543C23—81no—
544C23D3×#4111yescharacteristic
545Q8—22no—
546Q8—21no—
547Q8—22no—
548Q8—21no—
549Q8—41no—
550Q8—41no—
551D4—21no—
552D4—22no—
553D4—22no—
554D4—21no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—31no—
562C2×C4—31no—
563C2×C4—31no—
564C2×C4—31no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—241no—
576C2×C4—241no—
577C2×C4—242no—
578C2×C4—242no—
579C2×C4—241no—
580C2×C4—241no—
581C2×C4C2×D3×D411yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×#23711yescharacteristic
584C2×C4C2×#23711yescharacteristic
585C2×C4C2×#23711yescharacteristic
586C2×C4C2×#23711yescharacteristic
587C8—22no—
588C8—22no—
589C8—22no—
590C8—41no—
591C8—62no—
592C8—121no—
593C8—121no—
594C8—121no—
595C2×C3—21no—
596C2×C3—21no—
597C2×C3—81no—
598C2×C3—82no—
599C2×C3#239211yescharacteristic
600C2×C3#306811yescharacteristic
601C2×C3#306911yescharacteristic
602C22—21no—
603C22—21no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—82no—
609C22—82no—
610C22—81no—
611C22—81no—
612C22#560011yescharacteristic
613C22#560011yescharacteristic
614C22D3×#12111yescharacteristic
615C4—41no—
616C4—42no—
617C4—61no—
618C4—61no—
619C4—61no—
620C4—61no—
621C4—121no—
622C4—121no—
623C4—241no—
624C4—241no—
625C4#558611yescharacteristic
626C4#558611yescharacteristic
627C4#558611yescharacteristic
628C4#571611yescharacteristic
629C3#2372111yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—81no—
633C2—82no—
634C2#7666611yescharacteristic
635C2#8093011yescharacteristic
636C2#8093111yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2019C2
socleC22×C3C2×#121
prime radicalC2×C3×AGL(1,ℤ/8)C22
hypercenter#2019D3
hypocenterC3#23721
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2019

Complements

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

NH ≅ G/Ntype
1. Dic3×#137C2semidirect
3. C2×#5164C2semidirect
4. #78649C2semidirect
6. #72105C2semidirect
8. #72053C2semidirect
9. #70109C2semidirect
10. Dic3×#101C2semidirect
11. #70075C2semidirect
12. Dic3×#100C2semidirect
13. #69994C2semidirect
14. #68923C2semidirect
15. #68917C2semidirect
629. C3#23721semidirect

Maximal subgroups

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

subgroupindex
Dic3×#1372
Dic3×AGL(1,ℤ/8)2
C2×#51642
#786492
C3×#20192
#721052
#720932
#720532
#701092
Dic3×#1012
#700752
Dic3×#1002
#699942
#689232
#689172
#237213