Finite Group Databasebeta

← #10765300

Subgroups — #10765300

Order 768.

678 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79590C211yesmaximal, characteristic
2#79431C211yesmaximal, characteristic, complemented
3#79426C211yesmaximal, characteristic, complemented
4C3×C8×D8C211yesmaximal, characteristic, complemented
5#70879C211yesmaximal, characteristic, complemented
6#70840C211yesmaximal, characteristic, complemented
7#70840C211yesmaximal, characteristic, complemented
8#68573C211yesmaximal, characteristic, complemented
9#68559C211yesmaximal, characteristic, complemented
10#68175C211yesmaximal, characteristic, complemented
11#68171C211yesmaximal, characteristic, complemented
12#68158C211yesmaximal, characteristic, complemented
13#68157C211yesmaximal, characteristic, complemented
14C8×D24C211yesmaximal, characteristic, complemented
15#68048C211yesmaximal, characteristic, complemented
16#17922—31nomaximal, Sylow
17#5548C2211yescharacteristic
18C4×D3×D4C2211yescharacteristic
19C3×C4×D8C2211yescharacteristic
20C3×C8×D4C2211yescharacteristic
21C3×C8×D4C2211yescharacteristic, complemented
22#5156C411yescharacteristic
23#5150C411yescharacteristic
24#5111—21no—
25#5110—21no—
26C8×#67—21no—
27C2×(C2×D3)⋊C8—21no—
28C4×#213C2211yescharacteristic
29C4×#213C2211yescharacteristic
30#5012C2211yescharacteristic, complemented
31D4×C3⋊C8C2211yescharacteristic
32#4890C411yescharacteristic
33#4882C411yescharacteristic
34#4838C2211yescharacteristic, complemented
35#4835C2211yescharacteristic
36#4834C2211yescharacteristic
37#4833C2211yescharacteristic
38#4786C411yescharacteristic
39#4774C411yescharacteristic
40#4773C411yescharacteristic
41#4759C411yescharacteristic
42#4728—21no—
43#4727—21no—
44#67⋊C8—21no—
45D3×C22⋊C8—21no—
46C4×D24C2211yescharacteristic
47C4×ΓL(1,𝔽25)C2211yescharacteristic
48C8×D12C2211yescharacteristic
49C8×D12C2211yescharacteristic, complemented
50#4685C2211yescharacteristic
51#4685C2211yescharacteristic
52C3×#335C2211yescharacteristic, complemented
53C3×D4⋊C8C2211yescharacteristic
54C3×D4⋊C8C2211yescharacteristic, complemented
55C3×C82C2211yescharacteristic, complemented
56D4⋊(C3⋊C8)C2211yescharacteristic
57D4⋊(C3⋊C8)C2211yescharacteristic, complemented
58#4483C2211yescharacteristic
59#4483C2211yescharacteristic, complemented
60#4480C2211yescharacteristic, complemented
61#4480C2211yescharacteristic, complemented
62#4459C2211yescharacteristic
63#4459C2211yescharacteristic, complemented
64#4458C2211yescharacteristic, complemented
65#4455C2211yescharacteristic, complemented
66#4454C2211yescharacteristic, complemented
67#4454C2211yescharacteristic, complemented
68#3100—31no—
69#2981—31no—
70#2972—31no—
71#1586—31no—
72#1586—31no—
73#1581—31no—
74#1581—31no—
75#1580—31no—
76#1580—31no—
77C8×D8—31no—
78C8×D8D311yescharacteristic, complemented
79#1572—31no—
80#1490—31no—
81#1484—31no—
82#1453—31no—
83C2×D3×D4C2×C411yescharacteristic
84C2×#235—21no—
85C22×C4×D3—21no—
86C2×C3×D8C2×C411yescharacteristic
87C22×C3×C8—21no—
88C22×C3×C8—21no—
89C3×C4×D4C2311yescharacteristic
90C3×C4×D4C2311yescharacteristic
91#940C2×C411yescharacteristic
92#938C2×C411yescharacteristic
93D4×Dic3C2×C411yescharacteristic
94C2×#213C2×C411yescharacteristic
95C2×#213C2×C411yescharacteristic
96C4×#67—21no—
97C4×#67—21no—
98C2×#208—21no—
99C22×C3⋊C8—21no—
100C2×D24C2×C411yescharacteristic
101C2×ΓL(1,𝔽25)—21no—
102C2×ΓL(1,𝔽25)—21no—
103C2×D3×C8—21no—
104C2×D3×C8—21no—
105C2×D3×C8—21no—
106#895C2×C411yescharacteristic
107#893C2×C411yescharacteristic
108D3×C4⋊C4C2×C411yescharacteristic
109#888C2×C411yescharacteristic
110#885—21no—
111#884—21no—
112#67⋊C4—21no—
113D3×C22⋊C4—21no—
114C4×D12C2311yescharacteristic
115C4×D12C2311yescharacteristic
116C42×D3C2311yescharacteristic
117C3×#107C2×C411yescharacteristic
118C3×C4⋊C8C2311yescharacteristic
119C3×C4⋊C8C2311yescharacteristic
120C3×#102C2×C411yescharacteristic
121C3×#102C2×C411yescharacteristic
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×C4×C8C2311yescharacteristic
125C3×C4×C8C2311yescharacteristic
126C3×C4×C8C2311yescharacteristic
127#832C2×C411yescharacteristic
128#832C2×C411yescharacteristic
129#830—21no—
130#830—21no—
131#821C2×C411yescharacteristic
132#821C2×C411yescharacteristic
133(C2×D3)⋊C8—21no—
134(C2×D3)⋊C8—21no—
135(C2×D3)⋊C8—21no—
136(C2×D3)⋊C8—21no—
137(C2×D3)⋊C8—21no—
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8C2×C411yescharacteristic
140(C2×D3)⋊C8C2×C411yescharacteristic
141(C2×D3)⋊C8D411yescharacteristic
142(C2×D3)⋊C8D411yescharacteristic
143#818C2×C411yescharacteristic
144#815—21no—
145#815—21no—
146Dic3⋊C8C2×C411yescharacteristic
147Dic3⋊C8C2×C411yescharacteristic
148Dic3⋊C8D411yescharacteristic
149Dic3⋊C8D411yescharacteristic
150C8×Dic3—21no—
151#809C2×C411yescharacteristic
152#809C2×C411yescharacteristic
153#807C2×C411yescharacteristic
154#807C2×C411yescharacteristic
155C4⋊(C3⋊C8)C2311yescharacteristic
156C4⋊(C3⋊C8)C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#532—31no—
161C2×C4×D4—31no—
162#480—31no—
163#480—31no—
164#474—31no—
165#474—31no—
166#459—31no—
167#459—31no—
168#447—31no—
169#447—31no—
170C4×D8—31no—
171C4×D8—31no—
172C4×D8—31no—
173C4×D8C2×D311yescharacteristic
174#436—31no—
175#436—31no—
176#435—61no—
177#435—61no—
178C8×D4—31no—
179C8×D4—31no—
180C8×D4—31no—
181C8×D4—31no—
182C8×D4—61no—
183C8×D4—61no—
184C8×D4C2×D311yescharacteristic
185C8×D4C2×D311yescharacteristic, complemented
186#432—31no—
187#431—31no—
188#431—31no—
189#431—31no—
190#408—61no—
191#408—61no—
192C2×C22⋊C8—61no—
193C2×C22⋊C8—61no—
194C4×M16—31no—
195#335—31no—
196#335C2×D311yescharacteristic, complemented
197#334—31no—
198#334—31no—
199D4⋊C8—31no—
200D4⋊C8—31no—
201D4⋊C8—31no—
202D4⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8C2×D311yescharacteristic
206D4⋊C8C2×D311yescharacteristic, complemented
207#322—31no—
208#322—31no—
209#322—31no—
210C82C2×D311yescharacteristic, complemented
211C23×D3—21no—
212C2×C3×D4C22×C411yescharacteristic
213C2×C3×D4C22×C411yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C2×#67—21no—
217C2×#67—21no—
218C2×#67—21no—
219C2×#67—21no—
220C22×Dic3—21no—
221D3×D4—22no—
222D3×D4—22no—
223#235—81no—
224#235—81no—
225C2×D12—21no—
226C2×D12C22×C411yescharacteristic
227C2×D12C22×C411yescharacteristic
228C2×C4×D3—21no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—81no—
233C2×C4×D3—81no—
234C2×C4×D3C22×C411yescharacteristic
235C2×C4×D3C2×D411yescharacteristic
236C2×C4×D3#4111yescharacteristic
237C2×Dic6—21no—
238C3×D8—22no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—81no—
243C2×C3×C8—81no—
244C2×C3×C8C22×C411yescharacteristic
245C2×C3×C8C22×C411yescharacteristic
246C2×C3×C8C2×D411yescharacteristic
247C2×C3×C8C2×D411yescharacteristic
248C3×C4⋊C4C22×C411yescharacteristic
249C3×C4⋊C4C22×C411yescharacteristic
250C3×C22⋊C4—21no—
251C3×C22⋊C4—21no—
252C3×C42C2411yescharacteristic
253#217—21no—
254#217—21no—
255#213—22no—
256#213—22no—
257(C2×D3)⋊C4—21no—
258(C2×D3)⋊C4—21no—
259(C2×D3)⋊C4—21no—
260(C2×D3)⋊C4—21no—
261C4⋊Dic3C22×C411yescharacteristic
262C4⋊Dic3C22×C411yescharacteristic
263Dic3⋊C4—21no—
264Dic3⋊C4—21no—
265C4×Dic3C22×C411yescharacteristic
266C4×Dic3C2×D411yescharacteristic
267C4×Dic3#4111yescharacteristic
268#208—81no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8—81no—
273C2×C3⋊C8C22×C411yescharacteristic
274C2×C3⋊C8C22×C411yescharacteristic
275C2×C3⋊C8C2×D411yescharacteristic
276C2×C3⋊C8C2×D411yescharacteristic
277D24—22no—
278ΓL(1,𝔽25)—42no—
279ΓL(1,𝔽25)—81no—
280D3×C8—81no—
281D3×C8—81no—
282D3×C8—81no—
283C2×#41—61no—
284C22×D4—31no—
285C23×C4—61no—
286C2×D8—31no—
287C2×D8—31no—
288C2×D8—31no—
289C2×D8C4×D311yescharacteristic
290C2×M16—61no—
291C2×M16—61no—
292C2×M16—61no—
293C22×C8—21no—
294C22×C8—21no—
295C22×C8—61no—
296C22×C8—61no—
297C22×C8—61no—
298C22×C8—61no—
299C4⋊Q8—31no—
300#127—31no—
301#124—61no—
302#121—31no—
303#121—31no—
304#121—61no—
305C4×D4—31no—
306C4×D4—31no—
307C4×D4—31no—
308C4×D4—31no—
309C4×D4—61no—
310C4×D4—61no—
311C4×D4—61no—
312C4×D4C22×D311yescharacteristic
313C4×D4C22×D311yescharacteristic
314C2×C4⋊C4—31no—
315C2×C22⋊C4—61no—
316C2×C42—31no—
317#107—31no—
318#107—31no—
319#107—31no—
320#107C4×D311yescharacteristic
321C4⋊C8—31no—
322C4⋊C8—31no—
323C4⋊C8—31no—
324C4⋊C8—31no—
325C4⋊C8—31no—
326C4⋊C8—31no—
327C4⋊C8C22×D311yescharacteristic
328C4⋊C8C22×D311yescharacteristic
329#102—31no—
330#102—31no—
331#102—31no—
332#102—31no—
333#102—31no—
334#102—31no—
335#102C4×D311yescharacteristic
336#102C4×D311yescharacteristic
337C22⋊C8—21no—
338C22⋊C8—21no—
339C22⋊C8—31no—
340C22⋊C8—31no—
341C22⋊C8—31no—
342C22⋊C8—31no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8—61no—
346C22⋊C8—61no—
347C22⋊C8—61no—
348C22⋊C8—61no—
349C22⋊C8—61no—
350C22⋊C8—61no—
351#97—61no—
352#97—61no—
353C4×C8—31no—
354C4×C8—31no—
355C4×C8—31no—
356C4×C8—61no—
357C4×C8C22×D311yescharacteristic
358C4×C8C22×D311yescharacteristic
359C4×C8C22×D311yescharacteristic
360C23×C3—21no—
361C23×C3—21no—
362C22×D3—21no—
363C22×D3—24no—
364C22×D3—21no—
365C22×D3—81no—
366C22×D3—81no—
367C22×D3C4×D411yescharacteristic
368C3×D4—21no—
369C3×D4—21no—
370C3×D4—21no—
371C3×D4#13112yes—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4C23×C411yescharacteristic
379C2×C3×C4C22×D411yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381#67—24no—
382#67—81no—
383#67—81no—
384#67—81no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—81no—
388C2×Dic3C4×D411yescharacteristic
389C2×Dic3C4×D411yescharacteristic
390C2×Dic3C4×D411yescharacteristic
391D12—21no—
392D12—21no—
393D12—21no—
394D12—81no—
395D12#13112yes—
396C4×D3—21no—
397C4×D3—21no—
398C4×D3—21no—
399C4×D3—21no—
400C4×D3—41no—
401C4×D3—41no—
402C4×D3—41no—
403C4×D3—41no—
404C4×D3—41no—
405C4×D3—81no—
406C4×D3—81no—
407Dic6—81no—
408C3×C8—22no—
409C3×C8—21no—
410C3×C8—21no—
411C3×C8—21no—
412C3×C8—21no—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—21no—
416C3⋊C8—21no—
417C3⋊C8—21no—
418C3⋊C8—21no—
419C3⋊C8—41no—
420C3⋊C8—41no—
421C3⋊C8—41no—
422C24—61no—
423#41—241no—
424#41—241no—
425C2×Q8—61no—
426C2×D4—31no—
427C2×D4—31no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—62no—
433C2×D4—62no—
434C2×D4—61no—
435C2×D4C2×C4×D311yescharacteristic
436C2×D4C2×C4×D311yescharacteristic
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—241no—
448C22×C4—241no—
449D8—22no—
450D8—62no—
451D8—62no—
452D8—62no—
453M16—122no—
454M16—241no—
455M16—241no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—31no—
460C2×C8—31no—
461C2×C8—31no—
462C2×C8—31no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—81no—
467C2×C8—81no—
468C2×C8—241no—
469C2×C8—241no—
470C2×C8—241no—
471C2×C8—241no—
472C2×C8C2×C4×D311yescharacteristic
473C2×C8C2×C4×D311yescharacteristic
474C2×C8D3×D411yescharacteristic
475C2×C8D3×D411yescharacteristic
476C4⋊C4—31no—
477C4⋊C4—31no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4C2×C4×D311yescharacteristic
481C4⋊C4C2×C4×D311yescharacteristic
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—61no—
485C22⋊C4—61no—
486C22⋊C4—61no—
487C22⋊C4—61no—
488C22⋊C4—61no—
489C22⋊C4—61no—
490C42—31no—
491C42—31no—
492C42—31no—
493C42C23×D311yescharacteristic
494C22×C3—22no—
495C22×C3—41no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—41no—
499C22×C3—41no—
500C22×C3C2×C4×D411yescharacteristic
501C2×D3—21no—
502C2×D3—22no—
503C2×D3—21no—
504C2×D3—21no—
505C2×D3—41no—
506C2×D3—41no—
507C2×D3—41no—
508C2×D3—41no—
509C2×D3—41no—
510C2×D3—84no—
511C3×C4—21no—
512C3×C4—41no—
513C3×C4—41no—
514C3×C4C2×#13111yescharacteristic
515C3×C4C2×#13111yescharacteristic
516C3×C4#57611yescharacteristic
517C3×C4#57611yescharacteristic
518Dic3—21no—
519Dic3—21no—
520Dic3—21no—
521Dic3—41no—
522Dic3—41no—
523C23—21no—
524C23—21no—
525C23—31no—
526C23—64no—
527C23—61no—
528C23—61no—
529C23—241no—
530C23—241no—
531Q8—241no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—32no—
536D4—64no—
537D4—61no—
538D4—61no—
539D4—61no—
540D4—241no—
541D4—241no—
542D4—241no—
543D4—241no—
544D4#90112yes—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—31no—
548C2×C4—31no—
549C2×C4—31no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—241no—
566C2×C4—241no—
567C2×C4—241no—
568C2×C4C22×C4×D311yescharacteristic
569C2×C4C2×D3×D411yescharacteristic
570C2×C4D3×#4111yescharacteristic
571C8—22no—
572C8—21no—
573C8—21no—
574C8—21no—
575C8—21no—
576C8—41no—
577C8—41no—
578C8—61no—
579C8—61no—
580C8—61no—
581C8—61no—
582C8—121no—
583C8—121no—
584C8—121no—
585C2×C3—42no—
586C2×C3—81no—
587C2×C3#292911yescharacteristic
588C2×C3#294811yescharacteristic
589C2×C3#295611yescharacteristic
590D3—41no—
591D3—42no—
592D3—81no—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—61no—
600C22—61no—
601C22—62no—
602C22—61no—
603C22—121no—
604C22—121no—
605C22—121no—
606C22—121no—
607C22—121no—
608C22—244no—
609C22C4×D3×D411yescharacteristic
610C4—21no—
611C4—41no—
612C4—41no—
613C4—61no—
614C4—61no—
615C4—61no—
616C4—121no—
617C4—121no—
618C4C2×#90111yescharacteristic
619C4D3×#13111yescharacteristic
620C4#575711yescharacteristic
621C4#576911yescharacteristic
622C3#1792211yescharacteristic, complemented, Sylow
623C2—42no—
624C2—81no—
625C2—121no—
626C2—122no—
627C2—241no—
628C2#7942111yescharacteristic
629C2#7958811yescharacteristic
630C2#7988911yescharacteristic
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×C8×D8C2
socleC22×C3C2×C4×D4
prime radical#79590C2
cosocleC3×C42C24
hypercenterC8×D8D3
hypocenterC3#17922
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×D8

Complements

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

NH ≅ G/Ntype
2. #79431C2semidirect
3. #79426C2semidirect
4. C3×C8×D8C2semidirect
5. #70879C2semidirect
6. #70840C2semidirect
7. #70840C2semidirect
8. #68573C2semidirect
9. #68559C2semidirect
10. #68175C2semidirect
11. #68171C2semidirect
12. #68158C2semidirect
13. #68157C2semidirect
14. C8×D24C2semidirect
15. #68048C2semidirect
21. C3×C8×D4C22semidirect
30. #5012C22semidirect
34. #4838C22semidirect
49. C8×D12C22semidirect
52. C3×#335C22semidirect
54. C3×D4⋊C8C22semidirect
55. C3×C82C22semidirect
57. D4⋊(C3⋊C8)C22semidirect
59. #4483C22semidirect
60. #4480C22semidirect
61. #4480C22semidirect
63. #4459C22semidirect
64. #4458C22semidirect
65. #4455C22semidirect
66. #4454C22semidirect
67. #4454C22semidirect
78. C8×D8D3semidirect
185. C8×D4C2×D3semidirect
196. #335C2×D3semidirect
206. D4⋊C8C2×D3semidirect
210. C82C2×D3semidirect
622. C3#17922semidirect

Maximal subgroups

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

subgroupindex
#795902
#794312
#794262
C3×C8×D82
#708792
#708402
#685732
#685592
#681752
#681712
#681582
#681572
C8×D242
#680482
#179223