Finite Group Databasebeta

← #10868456

Subgroups — #10868456

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#76047C211yesmaximal, characteristic, complemented
2#73654C211yesmaximal, characteristic, complemented
3C3×#1920C211yesmaximal, characteristic, complemented
4#70587C211yesmaximal, characteristic, complemented
5#70574C211yesmaximal, characteristic, complemented
6#69628C211yesmaximal, characteristic, complemented
7#69463C211yesmaximal, characteristic, complemented
8#68909C211yesmaximal, characteristic, complemented
9#68904C211yesmaximal, characteristic, complemented
10#68897C211yesmaximal, characteristic
11#68862C211yesmaximal, characteristic
12#68793C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic, complemented
14#68675C211yesmaximal, characteristic
15#68610C211yesmaximal, characteristic, complemented
16#27071—31nomaximal, Sylow
17C22×D24C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#898—21no—
20C2×#893C2211yescharacteristic
21C2×#887C2211yescharacteristic
22C2×#875C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#816C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4994—21no—
31#4988C2211yescharacteristic
32#4988C2211yescharacteristic
33#4988C2211yescharacteristic
34#4975C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36#4973C2211yescharacteristic
37C2×#810C2211yescharacteristic
38C2×#810C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44#4940—21no—
45#4939C2211yescharacteristic
46C4×(C2×D3)⋊C4—21no—
47C4×C4⋊Dic3C2211yescharacteristic
48#4675—21no—
49#4670—21no—
50#4669—21no—
51#4668—21no—
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#340C2211yescharacteristic
55C3×#336C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61#2367—31no—
62#2094—31no—
63#2054—31no—
64#2042—31no—
65#2012—31no—
66#1968—31no—
67#1927—31no—
68#1926—31no—
69#1920D311yescharacteristic, complemented
70#1916—31no—
71#1880—31no—
72#1879—31no—
73#1803—31no—
74C4×#102—31no—
75C22×D12C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3D411yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3C2311yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×D24—22no—
112C2×D24—41no—
113C2×D24—41no—
114C2×#204—22no—
115C2×#204—22no—
116C2×#204—41no—
117#898—44no—
118#893—22no—
119#893—41no—
120#887—22no—
121#887—41no—
122#875—22no—
123#875—22no—
124C3×#106—22no—
125C3×#95—21no—
126C3×#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#821—22no—
133#816—41no—
134Dic3⋊C8—41no—
135C8×Dic3—41no—
136#810—22no—
137#810—41no—
138#809—22no—
139#809—22no—
140#809—41no—
141#809—41no—
142#808—41no—
143C22×SD16—31no—
144C22×D8—31no—
145C2×#126—61no—
146C2×#124—31no—
147C2×C4×Q8—31no—
148C2×C4×D4—31no—
149C2×#106—31no—
150C2×#106C2×D311yescharacteristic
151C2×C4⋊C8—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#103—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158C2×#102—31no—
159C2×#102—31no—
160C2×C4×C8—31no—
161#397—61no—
162#396—61no—
163#391—31no—
164#390—31no—
165#390—31no—
166#390—31no—
167#390—31no—
168#389C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
177C4×C22⋊C4—61no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#340C2×D311yescharacteristic
182#336—31no—
183#336—31no—
184#336—31no—
185#336C2×D311yescharacteristic
186C23×D3—21no—
187C23×D3—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2411yescharacteristic
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3C2×D411yescharacteristic
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12#4112yes—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—21no—
219C2×Dic6—21no—
220C2×Dic6—21no—
221C2×Dic6#4112yes—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8—21no—
227C2×C3×C8#4112yes—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—21no—
235C3×C4⋊C4—42no—
236C3×C4⋊C4#4112yes—
237C3×C42—21no—
238C3×C42—21no—
239C3×C42—81no—
240(C2×D3)⋊C4—24no—
241(C2×D3)⋊C4—24no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—42no—
245(C2×D3)⋊C4—42no—
246(C2×D3)⋊C4—42no—
247(C2×D3)⋊C4—42no—
248(C2×D3)⋊C4—81no—
249(C2×D3)⋊C4—81no—
250(C2×D3)⋊C4—81no—
251C4⋊Dic3—22no—
252C4⋊Dic3—22no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—21no—
262C4×Dic3—42no—
263C4×Dic3—42no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270D24—44no—
271#204—44no—
272C22×Q8—31no—
273C22×D4—31no—
274C22×D4—31no—
275C23×C4—61no—
276C2×SD16—62no—
277C2×SD16—62no—
278C2×SD16—121no—
279C2×D8—62no—
280C2×D8—121no—
281C2×D8—121no—
282C22×C8—31no—
283C22×C8—31no—
284C22×C8C22×D311yescharacteristic
285C22×C8C22×D311yescharacteristic
286#126—124no—
287#124—62no—
288#124—62no—
289C4×Q8—62no—
290C4×Q8—121no—
291C4×D4—62no—
292C4×D4—121no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—31no—
295C2×C4⋊C4—31no—
296C2×C4⋊C4—31no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C4⋊C4C22×D311yescharacteristic
303C2×C4⋊C4C22×D311yescharacteristic
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
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—21no—
316C2×C42—31no—
317C2×C42—31no—
318C2×C42—61no—
319C2×C42—61no—
320C2×C42C22×D311yescharacteristic
321#106—22no—
322#106—121no—
323C4⋊C8—121no—
324#103—62no—
325#103—121no—
326#103—121no—
327#102—62no—
328#102—62no—
329#102—62no—
330#102—121no—
331#102—121no—
332C4×C8—121no—
333#95—21no—
334#95—21no—
335#95—31no—
336#95—31no—
337#95—61no—
338#95—61no—
339#95—61no—
340C23×C3C22×D411yescharacteristic
341C22×D3—22no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3—41no—
346C22×D3—41no—
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C22×D3—41no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—22no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—41no—
371C2×C3×C4—41no—
372C2×C3×C4—41no—
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4C2×#4111yescharacteristic
377C2×C3×C4H(5,𝔽2)11yescharacteristic
378C2×C3×C4H(5,𝔽2)11yescharacteristic
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3#13512yes—
396C2×Dic3AGL(1,ℤ/8)12yes—
397D12—22no—
398D12—22no—
399D12—41no—
400D12—41no—
401D12—41no—
402D12—41no—
403D12—41no—
404D12—41no—
405C4×D3—84no—
406Dic6—22no—
407Dic6—22no—
408Dic6—41no—
409Dic6—41no—
410C3×C8—22no—
411C3×C8—41no—
412C3⋊C8—41no—
413C3⋊C8—41no—
414C24—61no—
415C24—61no—
416C2×Q8—32no—
417C2×Q8—61no—
418C2×Q8—61no—
419C2×Q8—61no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×D4—32no—
423C2×D4—61no—
424C2×D4—61no—
425C2×D4—61no—
426C2×D4—61no—
427C2×D4—61no—
428C2×D4—61no—
429C2×D4—61no—
430C2×D4—61no—
431C2×D4—61no—
432C2×D4—61no—
433C2×D4—61no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—31no—
439C22×C4—31no—
440C22×C4—31no—
441C22×C4—31no—
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—124no—
451D8—124no—
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#23512yes—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—62no—
473C4⋊C4—62no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—122no—
477C4⋊C4—122no—
478C4⋊C4—241no—
479C4⋊C4#23912yes—
480C22⋊C4—64no—
481C22⋊C4—64no—
482C22⋊C4—122no—
483C22⋊C4—122no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—241no—
489C22⋊C4—241no—
490C22⋊C4—241no—
491C42—21no—
492C42—21no—
493C42—61no—
494C42—61no—
495C42—61no—
496C42—61no—
497C42—81no—
498C42—122no—
499C42—122no—
500C22×C3#52511yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3#55011yescharacteristic
503C22×C3C2×#13511yescharacteristic
504C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
505C22×C3#57711yescharacteristic
506C22×C3#57711yescharacteristic
507C2×D3—42no—
508C2×D3—42no—
509C2×D3—42no—
510C2×D3—81no—
511C2×D3—81no—
512C2×D3—81no—
513C2×D3—81no—
514C2×D3—81no—
515C2×D3—81no—
516C2×D3—81no—
517C2×D3—81no—
518C2×D3—81no—
519C2×D3—81no—
520C2×D3—81no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—42no—
524C3×C4—81no—
525C3×C4—81no—
526C3×C4—81no—
527C3×C4#52911yescharacteristic
528C3×C4#53811yescharacteristic
529C3×C4#55111yescharacteristic
530C3×C4#55311yescharacteristic
531Dic3—22no—
532Dic3—22no—
533Dic3—42no—
534Dic3—41no—
535Dic3—41no—
536Dic3—81no—
537C23—62no—
538C23—121no—
539C23—121no—
540C23—121no—
541C23—121no—
542C23—121no—
543C23—121no—
544C23—121no—
545C23—121no—
546C23—121no—
547C23—121no—
548C23—121no—
549C23—121no—
550C23—121no—
551C23C2×D3×D411yescharacteristic
552Q8—62no—
553Q8—62no—
554Q8—121no—
555Q8—121no—
556D4—62no—
557D4—62no—
558D4—121no—
559D4—121no—
560D4—121no—
561D4—121no—
562D4—121no—
563D4—121no—
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—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—61no—
585C2×C4—61no—
586C2×C4—61no—
587C2×C4—61no—
588C2×C4—62no—
589C2×C4—61no—
590C2×C4—61no—
591C2×C4—61no—
592C2×C4—61no—
593C2×C4—61no—
594C2×C4—61no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—121no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—244no—
601C2×C4C2×#23511yescharacteristic
602C2×C4C2×#23911yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C2×C4#100911yescharacteristic
606C2×C4#100911yescharacteristic
607C8—22no—
608C8—41no—
609C8—121no—
610C8—121no—
611C2×C3#246711yescharacteristic
612C2×C3#310611yescharacteristic
613C2×C3#312511yescharacteristic
614C2×C3#330111yescharacteristic
615C2×C3#330611yescharacteristic
616C2×C3#339411yescharacteristic
617C2×C3#339711yescharacteristic
618D3—82no—
619D3—161no—
620C22—122no—
621C22—122no—
622C22—122no—
623C22—241no—
624C22—241no—
625C22—241no—
626C22—241no—
627C22—241no—
628C22—241no—
629C22—241no—
630C22—241no—
631C22—241no—
632C22—241no—
633C22—241no—
634C22#555311yescharacteristic
635C22#563011yescharacteristic
636C22#565611yescharacteristic
637C22#576211yescharacteristic
638C22D3×#13511yescharacteristic
639C22D3×AGL(1,ℤ/8)11yescharacteristic
640C22#577811yescharacteristic
641C4—41no—
642C4—42no—
643C4—41no—
644C4—62no—
645C4—62no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—121no—
650C4—121no—
651C4—122no—
652C4—241no—
653C4#552711yescharacteristic
654C4#558011yescharacteristic
655C4#569511yescharacteristic
656C4#570411yescharacteristic
657C3#2707111yescharacteristic, complemented, Sylow
658C2—242no—
659C2—481no—
660C2#7500211yescharacteristic
661C2#7963911yescharacteristic
662C2#7976311yescharacteristic
663C2#8107711yescharacteristic
664C2#8124611yescharacteristic
665C2#8134911yescharacteristic
666C2#8141711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1920C2
socleC23×C3C22×D4
prime radicalC22×D24C22
hypercenter#1920D3
hypocenterC3#27071
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
1. #76047C2semidirect
2. #73654C2semidirect
3. C3×#1920C2semidirect
4. #70587C2semidirect
5. #70574C2semidirect
6. #69628C2semidirect
7. #69463C2semidirect
8. #68909C2semidirect
9. #68904C2semidirect
12. #68793C2semidirect
13. #68780C2semidirect
15. #68610C2semidirect
69. #1920D3semidirect
657. C3#27071semidirect

Maximal subgroups

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

subgroupindex
#760472
#736542
C3×#19202
#705872
#705742
#696282
#694632
#689092
#689042
#688972
#688622
#687932
#687802
#686752
#686102
#270713