Finite Group Databasebeta

← #10868454

Subgroups — #10868454

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82955C211yesmaximal, characteristic, complemented
2#76339C211yesmaximal, characteristic, complemented
3C3×#2012C211yesmaximal, characteristic
4#72049C211yesmaximal, characteristic, complemented
5#72048C211yesmaximal, characteristic, complemented
6#72040C211yesmaximal, characteristic
7#72037C211yesmaximal, characteristic
8#71722C211yesmaximal, characteristic, complemented
9#70241C211yesmaximal, characteristic, complemented
10#70221C211yesmaximal, characteristic, complemented
11#70184C211yesmaximal, characteristic, complemented
12Dic3×#102C211yesmaximal, characteristic, complemented
13#69637C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68610C211yesmaximal, characteristic, complemented
16#27071—31nomaximal, Sylow
17C22×C3×SD16C2211yescharacteristic
18C22×C3×D8C2211yescharacteristic
19C2×C3×#124C2211yescharacteristic
20C2×Q8×Dic3C2211yescharacteristic
21C2×D4×Dic3C2211yescharacteristic
22C2×#879—21no—
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#390C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31#5221C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#817C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41C2×#808C2211yescharacteristic
42#4952—21no—
43#4951—21no—
44#4951—21no—
45#4950—21no—
46#4948—21no—
47#4945—21no—
48Dic3×C22⋊C4—21no—
49C4×C4⋊Dic3C2211yescharacteristic
50C4×C4⋊Dic3C2211yescharacteristic
51#4928C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61#2367—31no—
62#2094—31no—
63#2054—31no—
64#2042—31no—
65#2012D311yescharacteristic
66#1968—31no—
67#1927—31no—
68#1926—31no—
69#1920—31no—
70#1916—31no—
71#1880—31no—
72#1879—31no—
73#1803—31no—
74C4×#102—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C22×C3×D4C2311yescharacteristic
78C23×Dic3—21no—
79C2×C3×SD16—22no—
80C2×C3×SD16—22no—
81C2×C3×SD16—41no—
82C2×C3×D8—22no—
83C2×C3×D8—41no—
84C2×C3×D8—41no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#124—22no—
88C3×#124—22no—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42C2311yescharacteristic
95C2×#217—21no—
96C2×#217—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102Q8×Dic3—22no—
103Q8×Dic3—41no—
104D4×Dic3—22no—
105D4×Dic3—41no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×Dic3⋊C4—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3C2311yescharacteristic
120C2×C4×Dic3C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122C22×C3⋊C8C2311yescharacteristic
123#879—44no—
124C3×#103—41no—
125C3×#102—22no—
126#835—22no—
127#835—41no—
128#832—22no—
129#832—22no—
130#832—41no—
131#832—41no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95D411yescharacteristic
138C3⋊#95D411yescharacteristic
139#817—22no—
140Dic3⋊C8—41no—
141C8×Dic3—41no—
142#808—41no—
143C22×SD16C2×D311yescharacteristic
144C22×D8C2×D311yescharacteristic
145C2×#126—61no—
146C2×#124C2×D311yescharacteristic
147C2×C4×Q8—31no—
148C2×C4×D4—31no—
149C2×#106—31no—
150C2×#106—31no—
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103C2×D311yescharacteristic
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102C2×D311yescharacteristic
160C2×C4×C8—31no—
161#397—61no—
162#396—61no—
163#391—31no—
164#390—31no—
165#390—31no—
166#390—31no—
167#390C2×D311yescharacteristic
168#389—31no—
169#388—61no—
170#388—61no—
171#388—61no—
172#385—61no—
173#384—31no—
174C4×C4⋊C4—31no—
175C4×C4⋊C4—31no—
176C4×C4⋊C4—31no—
177C4×C22⋊C4—61no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C24×C3—21no—
187C24×C3—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8—21no—
190C2×C3×Q8—21no—
191C2×C3×Q8—21no—
192C2×C3×Q8—21no—
193C2×C3×Q8#4112yes—
194C2×C3×D4—21no—
195C2×C3×D4—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4—21no—
200C2×C3×D4—21no—
201C2×C3×D4—21no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—21no—
205C2×C3×D4#4112yes—
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—21no—
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
222C3×SD16—44no—
223C3×D8—44no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8#4112yes—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C22⋊C4—24no—
233C3×C22⋊C4—24no—
234C3×C22⋊C4—42no—
235C3×C22⋊C4—81no—
236C3×C42—21no—
237C3×C42—21no—
238#217—42no—
239#217—42no—
240#217—42no—
241#217—42no—
242#217—42no—
243#217—81no—
244#217—81no—
245C4⋊Dic3—21no—
246C4⋊Dic3—22no—
247C4⋊Dic3—22no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—42no—
255C4⋊Dic3#4112yes—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—81no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—21no—
263C4×Dic3—42no—
264C4×Dic3—42no—
265C4×Dic3—81no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C22×Q8C22×D311yescharacteristic
273C22×D4C22×D311yescharacteristic
274C22×D4C22×D311yescharacteristic
275C23×C4—61no—
276C2×SD16—22no—
277C2×SD16—22no—
278C2×SD16—41no—
279C2×D8—22no—
280C2×D8—41no—
281C2×D8—41no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286#126—124no—
287#124—22no—
288#124—22no—
289C4×Q8—62no—
290C4×Q8—121no—
291C4×D4—62no—
292C4×D4—121no—
293C2×C4⋊C4—31no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—61no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321#106—62no—
322#106—121no—
323C4⋊C8—121no—
324#103—41no—
325#103—62no—
326#103—121no—
327#102—22no—
328#102—62no—
329#102—62no—
330#102—121no—
331#102—121no—
332C4×C8—121no—
333#95—31no—
334#95—31no—
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×C3—41no—
348C23×C3—41no—
349C23×C3—41no—
350C23×C3—41no—
351C23×C3—41no—
352C23×C3—41no—
353C23×C3—41no—
354C23×C3C22×D411yescharacteristic
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8—41no—
358C3×Q8—41no—
359C3×D4—22no—
360C3×D4—22no—
361C3×D4—41no—
362C3×D4—41no—
363C3×D4—41no—
364C3×D4—41no—
365C3×D4—41no—
366C3×D4—41no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—22no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
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×C4—41no—
379C2×C3×C4C2×#4111yescharacteristic
380C2×C3×C4C2×#4111yescharacteristic
381C2×C3×C4C2×#4111yescharacteristic
382C2×C3×C4C2×#4111yescharacteristic
383C2×C3×C4H(5,𝔽2)11yescharacteristic
384C2×C3×C4H(5,𝔽2)11yescharacteristic
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—21no—
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3—21no—
394C2×Dic3—21no—
395C2×Dic3—21no—
396C2×Dic3—41no—
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—41no—
406C2×Dic3—41no—
407C2×Dic3—84no—
408C2×Dic3#13512yes—
409C2×Dic3AGL(1,ℤ/8)12yes—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C24—21no—
415C24—21no—
416C2×Q8—21no—
417C2×Q8—21no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8#23712yes—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4—21no—
428C2×D4—21no—
429C2×D4—21no—
430C2×D4—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4#23712yes—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—31no—
437C22×C4—31no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—61no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—241no—
447C22×C4D3×D411yescharacteristic
448C22×C4D3×D411yescharacteristic
449C22×C4C23×D311yescharacteristic
450SD16—44no—
451D8—44no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8#23712yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—32no—
467C4⋊C4—62no—
468C4⋊C4—62no—
469C4⋊C4—61no—
470C4⋊C4—61no—
471C4⋊C4—61no—
472C4⋊C4—61no—
473C4⋊C4—61no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—122no—
479C4⋊C4—241no—
480C22⋊C4—24no—
481C22⋊C4—24no—
482C22⋊C4—42no—
483C22⋊C4—81no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—122no—
489C22⋊C4—241no—
490C22⋊C4—241no—
491C42—21no—
492C42—21no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42—61no—
497C42—122no—
498C42—122no—
499C42—241no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—42no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3—81no—
506C22×C3—81no—
507C22×C3—81no—
508C22×C3—81no—
509C22×C3—81no—
510C22×C3—81no—
511C22×C3—81no—
512C22×C3—81no—
513C22×C3—81no—
514C22×C3#52511yescharacteristic
515C22×C3#54611yescharacteristic
516C22×C3#55011yescharacteristic
517C22×C3C2×#13511yescharacteristic
518C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
519C22×C3#57711yescharacteristic
520C22×C3#57711yescharacteristic
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—42no—
524C3×C4—81no—
525C3×C4#52911yescharacteristic
526C3×C4#53811yescharacteristic
527C3×C4#55111yescharacteristic
528C3×C4#55311yescharacteristic
529Dic3—22no—
530Dic3—22no—
531Dic3—41no—
532Dic3—41no—
533Dic3—42no—
534Dic3—81no—
535Dic3—81no—
536Dic3—81no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C2×D3×D411yescharacteristic
552Q8—22no—
553Q8—22no—
554Q8—41no—
555Q8—41no—
556D4—22no—
557D4—22no—
558D4—41no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—32no—
572C2×C4—32no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—61no—
579C2×C4—61no—
580C2×C4—61no—
581C2×C4—61no—
582C2×C4—61no—
583C2×C4—61no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—62no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—244no—
601C2×C4C2×#23711yescharacteristic
602C2×C4C2×#23711yescharacteristic
603C2×C4C2×#23711yescharacteristic
604C2×C4#100411yescharacteristic
605C2×C4#100411yescharacteristic
606C2×C4D3×#4111yescharacteristic
607C8—22no—
608C8—41no—
609C8—121no—
610C8—121no—
611C2×C3—82no—
612C2×C3—161no—
613C2×C3#246711yescharacteristic
614C2×C3#310611yescharacteristic
615C2×C3#312511yescharacteristic
616C2×C3#330111yescharacteristic
617C2×C3#330611yescharacteristic
618C2×C3#339411yescharacteristic
619C2×C3#339711yescharacteristic
620C22—42no—
621C22—42no—
622C22—42no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22#558611yescharacteristic
635C22#561211yescharacteristic
636C22#571811yescharacteristic
637C22D3×#13511yescharacteristic
638C22#577011yescharacteristic
639C22D3×AGL(1,ℤ/8)11yescharacteristic
640C22#577511yescharacteristic
641C4—42no—
642C4—41no—
643C4—41no—
644C4—62no—
645C4—62no—
646C4—81no—
647C4—121no—
648C4—121no—
649C4—122no—
650C4—241no—
651C4—241no—
652C4—241no—
653C4#558711yescharacteristic
654C4#559311yescharacteristic
655C4#560011yescharacteristic
656C4#560111yescharacteristic
657C3#2707111yescharacteristic, complemented, Sylow
658C2—82no—
659C2—161no—
660C2#7668411yescharacteristic
661C2#8006911yescharacteristic
662C2#8007511yescharacteristic
663C2#8056011yescharacteristic
664C2#8056111yescharacteristic
665C2#8181411yescharacteristic
666C2#8183911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2012C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
hypercenter#2012D3
hypocenterC3#27071
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2012

Complements

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

NH ≅ G/Ntype
1. #82955C2semidirect
2. #76339C2semidirect
4. #72049C2semidirect
5. #72048C2semidirect
8. #71722C2semidirect
9. #70241C2semidirect
10. #70221C2semidirect
11. #70184C2semidirect
12. Dic3×#102C2semidirect
13. #69637C2semidirect
14. #68623C2semidirect
15. #68610C2semidirect
657. C3#27071semidirect

Maximal subgroups

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

subgroupindex
#829552
#763392
C3×#20122
#720492
#720482
#720402
#720372
#717222
#702412
#702212
#701842
Dic3×#1022
#696372
#686232
#686102
#270713