Finite Group Databasebeta

← #10840272

Subgroups — #10840272

Order 768.

646 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
1#84001C211yesmaximal, characteristic
2C4×#950C211yesmaximal, characteristic, complemented
3C3×#1906C211yesmaximal, characteristic, complemented
4#72115C211yesmaximal, characteristic, complemented
5#71644C211yesmaximal, characteristic, complemented
6#70404C211yesmaximal, characteristic, complemented
7#70397C211yesmaximal, characteristic, complemented
8#70391C211yesmaximal, characteristic, complemented
9#70369C211yesmaximal, characteristic, complemented
10#70043C211yesmaximal, characteristic, complemented
11#70041C211yesmaximal, characteristic, complemented
12#69043C211yesmaximal, characteristic, complemented
13#69013C211yesmaximal, characteristic, complemented
14#68747C211yesmaximal, characteristic, complemented
15#68737C211yesmaximal, characteristic, complemented
16#24985—31nomaximal, Sylow
17C3×#575C2211yescharacteristic
18C3×C4×#41C2211yescharacteristic
19#5822C2211yescharacteristic
20C2×#950C2211yescharacteristic
21C2×#948C2211yescharacteristic
22C4×#235C2211yescharacteristic
23C3×#421C2211yescharacteristic
24C2×C3×#104C2211yescharacteristic
25C3×#410C2211yescharacteristic
26#5240C2211yescharacteristic
27#5240C2211yescharacteristic
28#5237C2211yescharacteristic
29#5233C2211yescharacteristic
30#5233C2211yescharacteristic
31#5204—21no—
32#5203—21no—
33#5200—21no—
34#5199—21no—
35#5139C2211yescharacteristic
36#5132C2211yescharacteristic
37C2×#822C2211yescharacteristic
38#5122C2211yescharacteristic
39#5119C2211yescharacteristic
40C4×#216—21no—
41C4×#215—21no—
42C4×#214—21no—
43C4×#213—21no—
44#5006C2211yescharacteristic
45#5002C2211yescharacteristic
46#4999C2211yescharacteristic
47#4998C2211yescharacteristic
48#4998C2211yescharacteristic
49C2×C4×C3⋊C8C2211yescharacteristic
50C3×#344C2211yescharacteristic
51C3×#339C2211yescharacteristic
52#4558C2211yescharacteristic
53#4557C2211yescharacteristic
54#4556C2211yescharacteristic
55#4531C2211yescharacteristic
56#4531C2211yescharacteristic
57#4531C2211yescharacteristic
58#4523C2211yescharacteristic
59#4523C2211yescharacteristic
60#3067—31no—
61C4×#135—31no—
62#2080—31no—
63#2080—31no—
64#2053—31no—
65#2053—31no—
66#1976—31no—
67#1945—31no—
68#1906—31no—
69#1906D311yescharacteristic, complemented
70#1833—31no—
71#1827—31no—
72#1827—31no—
73#1812—31no—
74#1764—31no—
75C2×C3×#41C2311yescharacteristic
76C2×C3×#41C2311yescharacteristic
77C2×#235C2311yescharacteristic
78C3×#137—21no—
79C3×#137—21no—
80C3×AGL(1,ℤ/8)—21no—
81C3×AGL(1,ℤ/8)—21no—
82C3×#135—41no—
83C2×C3×M16C2311yescharacteristic
84C2×C3×M16C2311yescharacteristic
85C3×C4×Q8—21no—
86C3×C4×D4—21no—
87C3×C4×D4—41no—
88C3×#117—21no—
89C3×#117C2311yescharacteristic
90C3×#117C2311yescharacteristic
91C2×C3×C42—21no—
92C2×C3×C42C2311yescharacteristic
93#951—21no—
94#950—21no—
95#950—21no—
96#950—21no—
97#950—21no—
98#950—41no—
99#949—21no—
100#948—21no—
101#948—21no—
102#948—21no—
103#948—21no—
104C2×#216D411yescharacteristic
105#942—21no—
106C2×#215D411yescharacteristic
107C2×#214D411yescharacteristic
108#932—21no—
109C2×#213D411yescharacteristic
110C4×#67—41no—
111#926C2311yescharacteristic
112#926C2311yescharacteristic
113#926C2311yescharacteristic
114C2×#208—21no—
115C2×#208C2311yescharacteristic
116C2×#208C2311yescharacteristic
117C22×C3⋊C8—21no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C4×D12—21no—
121#872—21no—
122C42×D3—21no—
123C4×Dic6—21no—
124C3×#104—21no—
125C3×#104—21no—
126C3×#104—41no—
127C3×#99—41no—
128#836—21no—
129#835—21no—
130#835—21no—
131#834—41no—
132#834—41no—
133#833—21no—
134#832—21no—
135#832—21no—
136#825—41no—
137#824—21no—
138#823—21no—
139#822—21no—
140#822—21no—
141#815—21no—
142Dic3⋊C8—41no—
143C8×Dic3—21no—
144#810—21no—
145#809—21no—
146#808—21no—
147#808—21no—
148#807—21no—
149#807—21no—
150#806—41no—
151C4⋊(C3⋊C8)—41no—
152C4×C3⋊C8—21no—
153C4×C3⋊C8—21no—
154#575—31no—
155#575C2×D311yescharacteristic
156C2×#135—31no—
157C2×#131—31no—
158C4×#41—31no—
159C4×#41C2×D311yescharacteristic
160#473—61no—
161#472—61no—
162#471—61no—
163#471—61no—
164C4×Q16—61no—
165C4×SD16—61no—
166C4×SD16—61no—
167C4×D8—61no—
168C2×#108—31no—
169#427—31no—
170#427—31no—
171#424—31no—
172#424—31no—
173#421—31no—
174#421C2×D311yescharacteristic
175C2×#104C2×D311yescharacteristic
176#416—31no—
177#416—31no—
178#416—31no—
179#413—31no—
180#413—31no—
181#410—31no—
182#410—31no—
183#410—31no—
184#410C2×D311yescharacteristic
185#405—31no—
186C2×C4×C8—31no—
187#344—31no—
188#344C2×D311yescharacteristic
189#341—31no—
190#339—31no—
191#339—31no—
192#339C2×D311yescharacteristic
193H(3,ℤ/4)—31no—
194H(3,ℤ/4)—31no—
195H(3,ℤ/4)—31no—
196H(3,ℤ/4)—31no—
197C3×#41—21no—
198C3×#41—21no—
199C3×#41—21no—
200C3×#41—21no—
201C3×#41—21no—
202C3×#41—21no—
203C3×#41—21no—
204C3×#41—21no—
205C2×C3×Q8C2×D411yescharacteristic
206C2×C3×Q8#4111yescharacteristic
207C2×C3×D4—21no—
208C2×C3×D4—21no—
209C2×C3×D4C2×D411yescharacteristic
210C2×C3×D4#4111yescharacteristic
211C22×C3×C4—21no—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—41no—
215C22×C3×C4C2411yescharacteristic
216C2×#67—21no—
217#235—21no—
218#235—21no—
219#235—21no—
220#235—21no—
221C2×D12C2×D411yescharacteristic
222C2×C4×D3—21no—
223C2×C4×D3—41no—
224C2×Dic6C2×D411yescharacteristic
225C3×Q16—41no—
226C3×SD16—41no—
227C3×SD16—41no—
228C3×D8—41no—
229C3×M16—21no—
230C3×M16—21no—
231C3×M16—21no—
232C3×M16—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—41no—
238C3×C22⋊C4—21no—
239C3×C22⋊C4—21no—
240C3×C22⋊C4—41no—
241C3×C42—21no—
242C3×C42—21no—
243C3×C42—21no—
244C3×C42—41no—
245C3×C42#4111yescharacteristic
246C3×C42#4111yescharacteristic
247#217—21no—
248#217—21no—
249#217—21no—
250#216—21no—
251#216—21no—
252#216—41no—
253#215—21no—
254#215—21no—
255#215—41no—
256#214—21no—
257#214—21no—
258#214—41no—
259#213—21no—
260#213—21no—
261#213—41no—
262(C2×D3)⋊C4—41no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3—21no—
266Dic3⋊C4—41no—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—41no—
271#208—21no—
272#208—21no—
273#208—21no—
274#208—21no—
275#208—41no—
276#208—41no—
277C2×C3⋊C8—21no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—41no—
283C2×C3⋊C8—41no—
284C2×C3⋊C8C2×D411yescharacteristic
285C2×C3⋊C8C2×D411yescharacteristic
286C2×C3⋊C8#4111yescharacteristic
287C2×C3⋊C8#4111yescharacteristic
288C2×C3⋊C8#4111yescharacteristic
289C2×C3⋊C8#4111yescharacteristic
290C2×#41—31no—
291C2×#41C22×D311yescharacteristic
292C2×#41C22×D311yescharacteristic
293#137—21no—
294#137—21no—
295#137—61no—
296#137—61no—
297AGL(1,ℤ/8)—21no—
298AGL(1,ℤ/8)—21no—
299AGL(1,ℤ/8)—61no—
300AGL(1,ℤ/8)—61no—
301#135—41no—
302#135—61no—
303#135—61no—
304#135—61no—
305#135—61no—
306#135—121no—
307C2×Q16—31no—
308C2×SD16—31no—
309C2×SD16—31no—
310C2×D8—31no—
311#131—61no—
312#131—61no—
313#131—61no—
314#131—61no—
315C2×M16—31no—
316C2×M16—31no—
317C2×M16—61no—
318C2×M16C22×D311yescharacteristic
319C2×M16C22×D311yescharacteristic
320C22×C8—31no—
321C22×C8—31no—
322C22×C8—61no—
323C4×Q8—21no—
324C4×Q8—61no—
325C4×D4—21no—
326C4×D4—41no—
327C4×D4—61no—
328C4×D4—121no—
329#117—21no—
330#117—31no—
331#117—31no—
332#117—31no—
333#117—61no—
334#117C22×D311yescharacteristic
335#117C22×D311yescharacteristic
336C2×C42—21no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#108—61no—
340#108—61no—
341#107—61no—
342#107—61no—
343#106—61no—
344#106—61no—
345C4⋊C8—121no—
346C4⋊C8—121no—
347#104—21no—
348#104—21no—
349#104—41no—
350#104—121no—
351#103—61no—
352#103—61no—
353#103—61no—
354#102—61no—
355#102—61no—
356#102—61no—
357#101—61no—
358#101—61no—
359#100—61no—
360#100—61no—
361#99—41no—
362#99—121no—
363#99—121no—
364#99—121no—
365#97—61no—
366C4×C8—61no—
367C4×C8—61no—
368C4×C8—61no—
369C23×C3—21no—
370C23×C3—21no—
371C23×C3H(5,𝔽2)11yescharacteristic
372C22×D3—21no—
373C3×Q8—21no—
374C3×Q8—21no—
375C3×Q8—21no—
376C3×Q8—21no—
377C3×D4—21no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—21no—
381C3×D4—41no—
382C3×D4—41no—
383C3×D4—41no—
384C3×D4—41no—
385C2×C3×C4—21no—
386C2×C3×C4—21no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—42no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—82no—
397C2×C3×C4C22×D411yescharacteristic
398C2×C3×C4C2×#4111yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403#67—41no—
404#67—41no—
405C2×Dic3—21no—
406C2×Dic3—41no—
407C2×Dic3—41no—
408C2×Dic3—41no—
409D12—21no—
410D12—21no—
411C4×D3—41no—
412C4×D3—41no—
413C4×D3—82no—
414Dic6—21no—
415Dic6—21no—
416C3×C8—41no—
417C3×C8—41no—
418C3⋊C8—41no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421C3⋊C8#12311yescharacteristic
422C3⋊C8#12311yescharacteristic
423C3⋊C8#12311yescharacteristic
424C3⋊C8#12311yescharacteristic
425#41—21no—
426#41—21no—
427#41—21no—
428#41—21no—
429#41—21no—
430#41—21no—
431#41—21no—
432#41—21no—
433#41—61no—
434#41—61no—
435#41—61no—
436#41—61no—
437C2×Q8—31no—
438C2×Q8#23511yescharacteristic
439C2×Q8D3×D411yescharacteristic
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—31no—
443C2×D4—61no—
444C2×D4#23511yescharacteristic
445C2×D4D3×D411yescharacteristic
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—41no—
450C22×C4—61no—
451C22×C4—121no—
452C22×C4C23×D311yescharacteristic
453Q16—41no—
454Q16—61no—
455Q16—61no—
456Q16—121no—
457SD16—41no—
458SD16—41no—
459SD16—61no—
460SD16—61no—
461SD16—61no—
462SD16—61no—
463SD16—121no—
464SD16—121no—
465D8—41no—
466D8—61no—
467D8—61no—
468D8—121no—
469M16—21no—
470M16—21no—
471M16—21no—
472M16—21no—
473M16—61no—
474M16—61no—
475M16—61no—
476M16—61no—
477M16—121no—
478M16—121no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—31no—
482C2×C8—31no—
483C2×C8—31no—
484C2×C8—31no—
485C2×C8—31no—
486C2×C8—31no—
487C2×C8—61no—
488C2×C8—61no—
489C2×C8—61no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—121no—
493C2×C8—121no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—41no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4—61no—
500C4⋊C4—121no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—41no—
504C22⋊C4—61no—
505C22⋊C4—61no—
506C22⋊C4—61no—
507C22⋊C4—121no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—61no—
513C42—61no—
514C42—61no—
515C42—121no—
516C42#23711yescharacteristic
517C42#23711yescharacteristic
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—41no—
521C22×C3—81no—
522C22×C3—81no—
523C22×C3#54011yescharacteristic
524C22×C3#54611yescharacteristic
525C22×C3#55311yescharacteristic
526C2×D3—41no—
527C2×D3—81no—
528C3×C4—22no—
529C3×C4—41no—
530C3×C4—41no—
531C3×C4—41no—
532C3×C4—81no—
533C3×C4—81no—
534C3×C4C2×#12311yescharacteristic
535C3×C4#52911yescharacteristic
536C3×C4#54611yescharacteristic
537C3×C4#55511yescharacteristic
538Dic3—41no—
539Dic3—81no—
540Dic3—81no—
541Dic3—81no—
542C23—21no—
543C23—21no—
544C23—61no—
545C23#100411yescharacteristic
546Q8—21no—
547Q8—21no—
548Q8—21no—
549Q8—21no—
550Q8—61no—
551Q8—61no—
552D4—21no—
553D4—21no—
554D4—21no—
555D4—21no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—61no—
561D4—61no—
562D4—121no—
563D4—121no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—61no—
576C2×C4—82no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—242no—
583C2×C4C2×#23511yescharacteristic
584C2×C4C2×D3×D411yescharacteristic
585C2×C4C2×#23711yescharacteristic
586C2×C4#100411yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C8—31no—
590C8—31no—
591C8—31no—
592C8—31no—
593C8—41no—
594C8—41no—
595C8—121no—
596C8—121no—
597C8—121no—
598C2×C3—21no—
599C2×C3—21no—
600C2×C3—21no—
601C2×C3—81no—
602C2×C3—81no—
603C2×C3#242011yescharacteristic
604D3—81no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—81no—
609C22—81no—
610C22—121no—
611C22—241no—
612C22#554011yescharacteristic
613C22#559111yescharacteristic
614C22#561511yescharacteristic
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—121no—
622C4—241no—
623C4—241no—
624C4—241no—
625C4#549011yescharacteristic
626C4#553811yescharacteristic
627C4#561611yescharacteristic
628C4D3×#12311yescharacteristic
629C3#2498511yescharacteristic, complemented, Sylow
630C2—21no—
631C2—21no—
632C2—21no—
633C2—81no—
634C2—81no—
635C2—241no—
636C2#7469511yescharacteristic
subgroupG/H
centerC4#5616
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1906C2
socleC2×C3#2420
prime radical#84001C2
hypercenter#1906D3
hypocenterC3#24985
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1906

Complements

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

NH ≅ G/Ntype
2. C4×#950C2semidirect
3. C3×#1906C2semidirect
4. #72115C2semidirect
5. #71644C2semidirect
6. #70404C2semidirect
7. #70397C2semidirect
8. #70391C2semidirect
9. #70369C2semidirect
10. #70043C2semidirect
11. #70041C2semidirect
12. #69043C2semidirect
13. #69013C2semidirect
14. #68747C2semidirect
15. #68737C2semidirect
69. #1906D3semidirect
629. C3#24985semidirect

Maximal subgroups

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

subgroupindex
#840012
C4×#9502
C3×#19062
#721152
#716442
#704042
#703972
#703912
#703692
#700432
#700412
#690432
#690132
#687472
#687372
#249853