Finite Group Databasebeta

← #10824925

Subgroups — #10824925

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×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#78867C211yesmaximal, characteristic, complemented
3C2×#4757C211yesmaximal, characteristic, complemented
4#78649C211yesmaximal, characteristic, complemented
5C3×#2051C211yesmaximal, characteristic
6#72097C211yesmaximal, characteristic
7#70717C211yesmaximal, characteristic, complemented
8#70692C211yesmaximal, characteristic, complemented
9#70301C211yesmaximal, characteristic, complemented
10#70197C211yesmaximal, characteristic, complemented
11#70085C211yesmaximal, characteristic, complemented
12Dic3×#100C211yesmaximal, characteristic, complemented
13#70004C211yesmaximal, characteristic, complemented
14#68980C211yesmaximal, characteristic, complemented
15#68973C211yesmaximal, characteristic, complemented
16#23721—31nomaximal, Sylow
17C2×C3×AGL(1,ℤ/8)C2211yescharacteristic
18Dic3×#41C2211yescharacteristic
19C2×#951C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C2×#887C2211yescharacteristic
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—
36C2×#824C2211yescharacteristic
37C2×#824C2211yescharacteristic
38#5121C2211yescharacteristic
39#5120C2211yescharacteristic
40#5119C2211yescharacteristic
41Dic3×M16C2211yescharacteristic
42Dic3×M16C2211yescharacteristic
43Dic3×M16C2211yescharacteristic
44C2×#819C2211yescharacteristic
45C2×#816C2211yescharacteristic
46#5102C2211yescharacteristic
47C2×#815C2211yescharacteristic
48#4964C2211yescharacteristic
49C2×#807C2211yescharacteristic
50#4790—21no—
51#4789—21no—
52#4758—21no—
53#4757—21no—
54#4757—22no—
55#4757—22no—
56#4720—22no—
57#4719—22no—
58C3×#343C2211yescharacteristic
59#4558C2211yescharacteristic
60#4558C2211yescharacteristic
61#4554C2211yescharacteristic
62#4554C2211yescharacteristic
63#4530C2211yescharacteristic
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#2051D311yescharacteristic
74#2046—31no—
75#2019—31no—
76#1778—31no—
77C4×#101—31no—
78C4×#100—31no—
79C2×C3×#41C2311yescharacteristic
80C22×C3×D4C2311yescharacteristic
81C23×Dic3—21no—
82C22×Dic6C2311yescharacteristic
83C3×AGL(1,ℤ/8)—22no—
84C3×AGL(1,ℤ/8)—22no—
85C2×C3×SD16—21no—
86C2×C3×D8—21no—
87C2×C3×M16C2311yescharacteristic
88C2×C3×M16C2311yescharacteristic
89C2×C3×M16C2311yescharacteristic
90C22×C3×C8C2311yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×#217—21no—
93#951—22no—
94#951—22no—
95Q8×Dic3D411yescharacteristic
96Q8×Dic3D411yescharacteristic
97C2×#216—21no—
98D4×Dic3—21no—
99D4×Dic3—21no—
100D4×Dic3—21no—
101D4×Dic3—21no—
102D4×Dic3—21no—
103D4×Dic3—21no—
104D4×Dic3D411yescharacteristic
105D4×Dic3D411yescharacteristic
106C2×#214—21no—
107C2×#214—22no—
108C2×#214—22no—
109C2×#214—22no—
110#926—21no—
111#926C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×C4×Dic3—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3C2311yescharacteristic
117C2×#208C2311yescharacteristic
118C2×#208C2311yescharacteristic
119C2×#208C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121#887—21no—
122#887—21no—
123#887—21no—
124#887—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#824—22no—
138#824—22no—
139#819—22no—
140#818—21no—
141#817—21no—
142#816—22no—
143#816—21no—
144#816—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#807—21no—
157#807—22no—
158C2×#137—31no—
159C2×AGL(1,ℤ/8)C2×D311yescharacteristic
160C22×SD16—31no—
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×#108C2×D311yescharacteristic
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#417C2×D311yescharacteristic
187C2×#103—31no—
188C2×#102C2×D311yescharacteristic
189C2×#101—31no—
190C2×#101—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—
206C3×#41—21no—
207C3×#41—21no—
208C3×#41—21no—
209C3×#41—21no—
210C2×C3×Q8C2×D411yescharacteristic
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—21no—
214C2×C3×D4—21no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4C2×D411yescharacteristic
218C2×C3×D4#4112yes—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4C2411yescharacteristic
222C22×Dic3—21no—
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×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6#4112yes—
237C3×SD16—24no—
238C3×D8—24no—
239C3×M16—21no—
240C3×M16—21no—
241C3×M16—21no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16#4112yes—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8C2×D412yes—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253#217—21no—
254#217—21no—
255#217—21no—
256#217—41no—
257#217—41no—
258#216—24no—
259#214—28no—
260#214—24no—
261C4⋊Dic3—21no—
262C4⋊Dic3—21no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3C2×D411yescharacteristic
266C4⋊Dic3C2×D411yescharacteristic
267Dic3⋊C4—41no—
268Dic3⋊C4—41no—
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#4112yes—
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×Q8—31no—
292C22×D4C22×D311yescharacteristic
293C23×C4—61no—
294#137—62no—
295#137—62no—
296AGL(1,ℤ/8)—22no—
297AGL(1,ℤ/8)—22no—
298C2×Q16—61no—
299C2×SD16—21no—
300C2×SD16—62no—
301C2×SD16—62no—
302C2×SD16—61no—
303C2×SD16—62no—
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—61no—
331C2×C4⋊C4C22×D311yescharacteristic
332C2×C22⋊C4—61no—
333C2×C42—31no—
334C2×C42—61no—
335C2×C42—61no—
336#108—22no—
337#108—62no—
338#107—61no—
339#107—62no—
340#107—61no—
341#106—61no—
342#106—61no—
343C4⋊C8—62no—
344C4⋊C8—62no—
345#103—21no—
346#103—61no—
347#103—61no—
348#103—61no—
349#103—62no—
350#102—21no—
351#102—22no—
352#102—61no—
353#102—61no—
354#102—61no—
355#101—62no—
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—21no—
377C3×Q8—21no—
378C3×D4—22no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—22no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C3×D4—41no—
386C2×C3×C4—21no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—81no—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C22×D411yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—81no—
406C2×Dic3—81no—
407C2×Dic3—82no—
408C2×Dic3—82no—
409C2×Dic3—81no—
410C2×Dic3#12111yescharacteristic
411C2×Dic3#12111yescharacteristic
412C2×Dic3#12111yescharacteristic
413C2×Dic3#12111yescharacteristic
414Dic6—22no—
415Dic6—22no—
416Dic6—41no—
417Dic6—41no—
418C3×C8—22no—
419C3×C8—22no—
420C3×C8—41no—
421C3×C8—41no—
422C3⋊C8—22no—
423C3⋊C8—22no—
424C3⋊C8—41no—
425C3⋊C8—41no—
426C24—21no—
427#41—21no—
428#41—21no—
429#41—21no—
430#41—21no—
431C2×Q8—32no—
432C2×Q8—61no—
433C2×Q8—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8D3×D411yescharacteristic
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—64no—
459SD16—24no—
460SD16—68no—
461SD16—64no—
462D8—24no—
463M16—21no—
464M16—21no—
465M16—21no—
466M16—21no—
467M16—21no—
468M16—32no—
469M16—61no—
470M16—61no—
471M16—61no—
472M16—61no—
473M16—61no—
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—21no—
488C4⋊C4—21no—
489C4⋊C4—31no—
490C4⋊C4—31no—
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—81no—
526C3×C4#52511yescharacteristic
527C3×C4#52511yescharacteristic
528C3×C4#52511yescharacteristic
529C3×C4#53211yescharacteristic
530Dic3—21no—
531Dic3—21no—
532Dic3—21no—
533Dic3—21no—
534Dic3—42no—
535Dic3—41no—
536Dic3—41no—
537Dic3—81no—
538C23—21no—
539C23—22no—
540C23—41no—
541C23—41no—
542C23—81no—
543C23—81no—
544C23D3×#4111yescharacteristic
545Q8—21no—
546Q8—21no—
547Q8—62no—
548Q8—62no—
549Q8—121no—
550Q8—121no—
551D4—21no—
552D4—21no—
553D4—22no—
554D4—22no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—31no—
561C2×C4—31no—
562C2×C4—31no—
563C2×C4—31no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—62no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—81no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—241no—
577C2×C4—241no—
578C2×C4—242no—
579C2×C4—242no—
580C2×C4—241no—
581C2×C4C2×D3×D411yescharacteristic
582C2×C4C2×D3×D411yescharacteristic
583C2×C4C2×#23711yescharacteristic
584C2×C4C2×#23711yescharacteristic
585C2×C4D3×#4111yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C8—22no—
588C8—22no—
589C8—41no—
590C8—41no—
591C8—62no—
592C8—62no—
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—
612C22D3×#12111yescharacteristic
613C22#564911yescharacteristic
614C22#568011yescharacteristic
615C4—41no—
616C4—61no—
617C4—61no—
618C4—61no—
619C4—61no—
620C4—81no—
621C4—121no—
622C4—121no—
623C4—122no—
624C4—241no—
625C4#558611yescharacteristic
626C4#559611yescharacteristic
627C4#560611yescharacteristic
628C4#565611yescharacteristic
629C3#2372111yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—81no—
633C2—82no—
634C2#7689011yescharacteristic
635C2#8093111yescharacteristic
636C2#8093811yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2051C2
socleC22×C3C2×#121
prime radicalC2×C3×AGL(1,ℤ/8)C22
hypercenter#2051D3
hypocenterC3#23721
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
2. #78867C2semidirect
3. C2×#4757C2semidirect
4. #78649C2semidirect
7. #70717C2semidirect
8. #70692C2semidirect
9. #70301C2semidirect
10. #70197C2semidirect
11. #70085C2semidirect
12. Dic3×#100C2semidirect
13. #70004C2semidirect
14. #68980C2semidirect
15. #68973C2semidirect
629. C3#23721semidirect

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
#788672
C2×#47572
#786492
C3×#20512
#720972
#707172
#706922
#703012
#701972
#700852
Dic3×#1002
#700042
#689802
#689732
#237213