Finite Group Databasebeta

← #10868340

Subgroups — #10868340

Order 768.

840 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76524C211yesmaximal, characteristic, complemented
2#76306C211yesmaximal, characteristic
3C3×#2055C211yesmaximal, characteristic, complemented
4#72056C211yesmaximal, characteristic
5#72047C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#70447C211yesmaximal, characteristic, complemented
8#70257C211yesmaximal, characteristic, complemented
9#70242C211yesmaximal, characteristic, complemented
10#70211C211yesmaximal, characteristic
11Dic3×#102C211yesmaximal, characteristic, complemented
12#69687C211yesmaximal, characteristic, complemented
13#69654C211yesmaximal, characteristic, complemented
14#68791C211yesmaximal, characteristic, complemented
15#68788C211yesmaximal, characteristic, complemented
16#27063—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×#215C2211yescharacteristic
19C2×#940C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C2×#67⋊C4—21no—
22C2×C4×D12C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#391C2211yescharacteristic
27C3×#390C2211yescharacteristic
28#5229C2211yescharacteristic
29#5229C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#821C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4988C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#808C2211yescharacteristic
42C2×#807C2211yescharacteristic
43#4958—21no—
44#4957—21no—
45#4956—21no—
46#4936C2211yescharacteristic
47C4×Dic3⋊C4C2211yescharacteristic
48C3⋊#401—21no—
49#4658—21no—
50#4652—21no—
51#4651—21no—
52C3×#340C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2514—31no—
61#2366—31no—
62#2092—31no—
63#2055D311yescharacteristic, complemented
64#2041—31no—
65#2014—31no—
66#1968—31no—
67#1928—31no—
68#1925—31no—
69#1919—31no—
70#1916—31no—
71#1877—31no—
72#1877—31no—
73#1804—31no—
74C4×#102—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C22×#67—21no—
78C22×#67—21no—
79C23×Dic3—21no—
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C2×C3×SD16—22no—
83C2×C3×SD16—41no—
84C2×C3×SD16—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C42C2311yescharacteristic
92C2×#217—21no—
93C2×#215—22no—
94C2×#215—22no—
95C2×#215—41no—
96#940—22no—
97#940—22no—
98D4×Dic3—22no—
99D4×Dic3—41no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4D411yescharacteristic
108C2×Dic3⋊C4D411yescharacteristic
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#67⋊C4—44no—
116C4×D12—22no—
117C4×D12—41no—
118C3×C4⋊C8—41no—
119C3×#103—41no—
120C3×#102—22no—
121C3×#95—21no—
122C3×#95—21no—
123#835—22no—
124#832—22no—
125#832—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#821—41no—
138#821—41no—
139C8×Dic3—41no—
140#809—22no—
141#808—22no—
142#807—41no—
143C22×SD16—31no—
144C22×SD16C2×D311yescharacteristic
145C2×#127—31no—
146C2×C4×D4—31no—
147C2×C4×D4—31no—
148C2×C4×D4—61no—
149C2×#107—31no—
150C2×#106—31no—
151C2×C4⋊C8C2×D311yescharacteristic
152C2×#103—31no—
153C2×#103C2×D311yescharacteristic
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102C2×D311yescharacteristic
160C2×C4×C8—31no—
161#401—61no—
162#400—61no—
163#391—31no—
164#391—31no—
165#391C2×D311yescharacteristic
166#390—31no—
167#390—61no—
168#390C2×D311yescharacteristic
169#389—31no—
170#387—61no—
171#386—61no—
172#386—61no—
173#383—61no—
174#382—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C4⋊C4—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184#336—31no—
185#336—31no—
186C24×C3—21no—
187C23×D3—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4#4112yes—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C2×#67—24no—
207C2×#67—24no—
208C2×#67—42no—
209C2×#67—42no—
210C2×#67—81no—
211C2×#67—81no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—42no—
216C22×Dic3—42no—
217C22×Dic3—81no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12#4112yes—
228C2×C4×D3—42no—
229C2×C4×D3—42no—
230C2×C4×D3—81no—
231C3×SD16—44no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C22⋊C4—42no—
243C3×C22⋊C4—81no—
244C3×C42—21no—
245C3×C42—21no—
246#217—42no—
247#215—44no—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3#4112yes—
257Dic3⋊C4—22no—
258Dic3⋊C4—22no—
259Dic3⋊C4—42no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—42no—
265C4×Dic3—42no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8#4112yes—
272C22×Q8C22×D311yescharacteristic
273C22×D4—31no—
274C22×D4—61no—
275C22×D4—61no—
276C22×D4C22×D311yescharacteristic
277C23×C4—61no—
278C23×C4—61no—
279C2×SD16—22no—
280C2×SD16—41no—
281C2×SD16—41no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—121no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#127—62no—
290#127—62no—
291C4×D4—62no—
292C4×D4—62no—
293C4×D4—121no—
294C4×D4—121no—
295C4×D4—124no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C42—31no—
311C2×C42—31no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—121no—
316#106—62no—
317C4⋊C8—41no—
318#103—41no—
319#103—62no—
320#102—22no—
321#102—62no—
322#102—62no—
323#102—121no—
324#102—121no—
325#102—121no—
326C4×C8—121no—
327#95—21no—
328#95—21no—
329#95—31no—
330#95—31no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340C23×C3—22no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3—41no—
345C23×C3—41no—
346C23×C3—41no—
347C23×C3C22×D411yescharacteristic
348C22×D3—22no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C3×Q8—41no—
356C3×Q8—41no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4—41no—
362C3×D4—41no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4—41no—
374C2×C3×C4—41no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4C2×#4111yescharacteristic
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4H(5,𝔽2)11yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384#67—48no—
385#67—84no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—21no—
392C2×Dic3—22no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—21no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—41no—
401C2×Dic3—41no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—41no—
405C2×Dic3—84no—
406C2×Dic3#13512yes—
407C2×Dic3AGL(1,ℤ/8)12yes—
408D12—22no—
409D12—22no—
410D12—41no—
411D12—41no—
412C4×D3—84no—
413C3×C8—41no—
414C3×C8—41no—
415C3⋊C8—22no—
416C3⋊C8—41no—
417C24—21no—
418C24—61no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—32no—
431C2×D4—64no—
432C2×D4—64no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—122no—
439C2×D4—122no—
440C2×D4—241no—
441C2×D4—241no—
442C2×D4#23712yes—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—31no—
447C22×C4—31no—
448C22×C4—31no—
449C22×C4—31no—
450C22×C4—61no—
451C22×C4—61no—
452C22×C4—61no—
453C22×C4—122no—
454C22×C4—122no—
455C22×C4—122no—
456C22×C4—122no—
457C22×C4—241no—
458C22×C4—241no—
459C22×C4D3×D411yescharacteristic
460C22×C4D3×D411yescharacteristic
461C22×C4C23×D311yescharacteristic
462SD16—44no—
463SD16—124no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—32no—
471C2×C8—61no—
472C2×C8—61no—
473C2×C8—61no—
474C2×C8—61no—
475C2×C8—61no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—32no—
481C4⋊C4—62no—
482C4⋊C4—62no—
483C4⋊C4—61no—
484C4⋊C4—61no—
485C4⋊C4—61no—
486C4⋊C4—61no—
487C4⋊C4—61no—
488C4⋊C4—122no—
489C22⋊C4—42no—
490C22⋊C4—81no—
491C22⋊C4—122no—
492C22⋊C4—122no—
493C22⋊C4—122no—
494C22⋊C4—241no—
495C42—21no—
496C42—21no—
497C42—61no—
498C42—61no—
499C42—61no—
500C42—61no—
501C42—122no—
502C42—122no—
503C22×C3—42no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3#52511yescharacteristic
511C22×C3#54611yescharacteristic
512C22×C3#55011yescharacteristic
513C22×C3C2×#13511yescharacteristic
514C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
515C22×C3#57711yescharacteristic
516C22×C3#57711yescharacteristic
517C2×D3—42no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—81no—
521C2×D3—81no—
522C2×D3—81no—
523C2×D3—81no—
524C3×C4—41no—
525C3×C4—41no—
526C3×C4—81no—
527C3×C4—81no—
528C3×C4—81no—
529C3×C4#52911yescharacteristic
530C3×C4#53811yescharacteristic
531C3×C4#55111yescharacteristic
532C3×C4#55311yescharacteristic
533Dic3—22no—
534Dic3—22no—
535Dic3—41no—
536Dic3—41no—
537Dic3—42no—
538Dic3—81no—
539Dic3—81no—
540C23—22no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—62no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23—121no—
552C23—121no—
553C23—121no—
554C23C2×D3×D411yescharacteristic
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559D4—22no—
560D4—22no—
561D4—41no—
562D4—41no—
563D4—62no—
564D4—62no—
565D4—128no—
566D4—121no—
567D4—121no—
568D4—244no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—32no—
576C2×C4—32no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—61no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—61no—
596C2×C4—62no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—121no—
601C2×C4—121no—
602C2×C4—121no—
603C2×C4—121no—
604C2×C4—121no—
605C2×C4—244no—
606C2×C4—244no—
607C2×C4C2×#23711yescharacteristic
608C2×C4#100411yescharacteristic
609C2×C4#100411yescharacteristic
610C2×C4D3×#4111yescharacteristic
611C2×C4D3×#4111yescharacteristic
612C2×C4D3×#4111yescharacteristic
613C8—41no—
614C8—41no—
615C8—62no—
616C8—121no—
617C2×C3—82no—
618C2×C3#246711yescharacteristic
619C2×C3#310511yescharacteristic
620C2×C3#312311yescharacteristic
621C2×C3#330411yescharacteristic
622C2×C3#331011yescharacteristic
623C2×C3#339411yescharacteristic
624C2×C3#339711yescharacteristic
625D3—82no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—122no—
634C22—122no—
635C22—122no—
636C22—241no—
637C22—241no—
638C22—241no—
639C22—241no—
640C22#559911yescharacteristic
641C22#560611yescharacteristic
642C22#561011yescharacteristic
643C22#576211yescharacteristic
644C22D3×#13511yescharacteristic
645C22D3×AGL(1,ℤ/8)11yescharacteristic
646C22#577811yescharacteristic
647C4—41no—
648C4—41no—
649C4—62no—
650C4—62no—
651C4—81no—
652C4—81no—
653C4—81no—
654C4—121no—
655C4—121no—
656C4—122no—
657C4—241no—
658C4—241no—
659C4#559311yescharacteristic
660C4#564911yescharacteristic
661C4#566111yescharacteristic
662C4#567911yescharacteristic
663C3#2706311yescharacteristic, complemented, Sylow
664C2—82no—
665C2—242no—
666C2#7692111yescharacteristic
667C2#8037611yescharacteristic
668C2#8041311yescharacteristic
669C2#8050511yescharacteristic
670C2#8058911yescharacteristic
671C2#8069111yescharacteristic
672C2#8082111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2055C2
socleC23×C3C22×D4
prime radicalC2×#940C22
hypercenter#2055D3
hypocenterC3#27063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
1. #76524C2semidirect
3. C3×#2055C2semidirect
5. #72047C2semidirect
6. #70480C2semidirect
7. #70447C2semidirect
8. #70257C2semidirect
9. #70242C2semidirect
11. Dic3×#102C2semidirect
12. #69687C2semidirect
13. #69654C2semidirect
14. #68791C2semidirect
15. #68788C2semidirect
63. #2055D3semidirect
663. C3#27063semidirect

Maximal subgroups

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

subgroupindex
#765242
#763062
C3×#20552
#720562
#720472
#704802
#704472
#702572
#702422
#702112
Dic3×#1022
#696872
#696542
#687912
#687882
#270633