Finite Group Databasebeta

← #10858939

Subgroups — #10858939

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82637C211yesmaximal, characteristic
2#73900C211yesmaximal, characteristic, complemented
3C3×#1893C211yesmaximal, characteristic
4#71974C211yesmaximal, characteristic, complemented
5#71969C211yesmaximal, characteristic, complemented
6#71678C211yesmaximal, characteristic
7#70193C211yesmaximal, characteristic, complemented
8#70167C211yesmaximal, characteristic, complemented
9#70164C211yesmaximal, characteristic, complemented
10#70153C211yesmaximal, characteristic, complemented
11#69506C211yesmaximal, characteristic, complemented
12#68843C211yesmaximal, characteristic, complemented
13#68815C211yesmaximal, characteristic, complemented
14#68809C211yesmaximal, characteristic, complemented
15#68801C211yesmaximal, characteristic, complemented
16#26421—31nomaximal, Sylow
17C23×C3×C8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#380C2211yescharacteristic
26#5221—21no—
27#5221C2211yescharacteristic
28#5220—21no—
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C4×#217C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4984C2211yescharacteristic
36#4984C2211yescharacteristic
37#4975C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#808C2211yescharacteristic
41C2×#807C2211yescharacteristic
42#4951—21no—
43#4949—21no—
44#4948—21no—
45Dic3×C22⋊C4—21no—
46#4655—21no—
47#4647—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2441—31no—
61#2363—31no—
62#2100—31no—
63#2088—31no—
64#2065—31no—
65#2025—31no—
66#1989—31no—
67#1918—31no—
68#1893D311yescharacteristic
69#1875—31no—
70#1862—31no—
71#1854—31no—
72#1850—31no—
73#1802—31no—
74#1735—31no—
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4C2311yescharacteristic
78C23×Dic3—21no—
79C22×C3×C8—22no—
80C22×C3×C8—41no—
81C22×C3×C8—41no—
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C3×#121—22no—
85C3×#121Q814yes—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95C2×#217—21no—
96C2×#217D411yescharacteristic
97C2×#217D411yescharacteristic
98D4×Dic3—22no—
99D4×Dic3—41no—
100#926—22no—
101#926—22no—
102#926—41no—
103#926—41no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#890—22no—
123#890—41no—
124C3×#102—41no—
125C3×#102—41no—
126C3×#95—21no—
127#832—22no—
128#832—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#830—22no—
136Dic3⋊C8—41no—
137Dic3⋊C8—41no—
138#808—41no—
139#807—22no—
140C23×C8C2×D311yescharacteristic
141C2×#125—31no—
142C2×#121C2×D311yescharacteristic
143C2×C4×D4—31no—
144C2×#117—31no—
145C2×#117—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102C2×D311yescharacteristic
153C2×#102C2×D311yescharacteristic
154C2×C22⋊C8—31no—
155#399—61no—
156#396—61no—
157#390—31no—
158#390—61no—
159#389—31no—
160#389—31no—
161#389—31no—
162#389—61no—
163#388—61no—
164#387—61no—
165#386—61no—
166#384—31no—
167#380C2×D311yescharacteristic
168C4×C4⋊C4—31no—
169C4×C22⋊C4—31no—
170C4×C22⋊C4—61no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C24×C3—21no—
184C24×C3C2×D411yescharacteristic
185C2×C3×D4—24no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—81no—
192C2×C3×D4—81no—
193C2×C3×D4C2×Q812yes—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×Q812yes—
203C22×C3×C4#4112yes—
204C22×C3×C4C2411yescharacteristic
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—42no—
210C22×Dic3—42no—
211C22×Dic3—81no—
212C22×Dic3C2×D411yescharacteristic
213C22×Dic3C2×D411yescharacteristic
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—44no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4C2×Q812yes—
229C3×C22⋊C4—24no—
230C3×C22⋊C4—81no—
231#217—22no—
232#217—41no—
233#217—42no—
234#217—42no—
235#217—42no—
236#217—42no—
237#217—81no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3—81no—
246C4⋊Dic3—81no—
247C4⋊Dic3#4112yes—
248C4⋊Dic3#4112yes—
249Dic3⋊C4—42no—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—81no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—42no—
258C4×Dic3—42no—
259C4×Dic3—81no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C22×D4—21no—
267C22×D4C22×D311yescharacteristic
268C23×C4—61no—
269C23×C4C22×D311yescharacteristic
270C22×C8—22no—
271C22×C8—31no—
272C22×C8—31no—
273C22×C8—41no—
274C22×C8—41no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277#125—62no—
278#125—121no—
279#121—22no—
280#121Dic614yes—
281C4×D4—62no—
282C4×D4—121no—
283#117—62no—
284#117—62no—
285#117—121no—
286#117—121no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—31no—
304C2×C22⋊C4—31no—
305C2×C22⋊C4—61no—
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×C42—61no—
315#107—62no—
316#106—121no—
317C4⋊C8—121no—
318C4⋊C8—121no—
319#102—41no—
320#102—41no—
321#102—62no—
322#102—121no—
323C22⋊C8—62no—
324#95—21no—
325#95—31no—
326#95—31no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3—21no—
334C23×C3—21no—
335C23×C3—21no—
336C23×C3—21no—
337C23×C3—22no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3—41no—
344C23×C3C22⋊Q812yes—
345C23×C3C22×D411yescharacteristic
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—41no—
349C3×D4—41no—
350C3×D4—84no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4C22⋊Q812yes—
367C2×C3×C4C22×Q811yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—84no—
397C2×Dic3#13512yes—
398C3×C8—41no—
399C3×C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C24—21no—
403C24D3×D411yescharacteristic
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4C2×Dic612yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4C2×Dic612yes—
432C22×C4D3×D411yescharacteristic
433C22×C4#23712yes—
434C22×C4C23×D311yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—44no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—32no—
454C4⋊C4—32no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—122no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C4⋊C4—241no—
468C4⋊C4C2×Dic612yes—
469C22⋊C4—24no—
470C22⋊C4—62no—
471C22⋊C4—81no—
472C22⋊C4—121no—
473C22⋊C4—122no—
474C22⋊C4—122no—
475C22⋊C4—122no—
476C22⋊C4—122no—
477C22⋊C4—241no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—61no—
482C42—122no—
483C42—122no—
484C42—241no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—42no—
493C22×C3—42no—
494C22×C3—42no—
495C22×C3—81no—
496C22×C3—81no—
497C22×C3—81no—
498C22×C3—81no—
499C22×C3C2×C22⋊Q811yescharacteristic
500C22×C3#54611yescharacteristic
501C22×C3#54611yescharacteristic
502C22×C3C2×#13511yescharacteristic
503C22×C3#57511yescharacteristic
504C22×C3#57611yescharacteristic
505C22×C3#57711yescharacteristic
506C3×C4—22no—
507C3×C4—42no—
508C3×C4—81no—
509C3×C4—81no—
510C3×C4#52911yescharacteristic
511C3×C4#53811yescharacteristic
512C3×C4#55411yescharacteristic
513C3×C4#55411yescharacteristic
514Dic3—22no—
515Dic3—42no—
516Dic3—41no—
517Dic3—42no—
518Dic3—41no—
519Dic3—41no—
520Dic3—81no—
521Dic3—81no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—22no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23#87812yes—
536C23C2×D3×D411yescharacteristic
537D4—22no—
538D4—22no—
539D4—41no—
540D4—41no—
541D4—84no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—32no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—244no—
582C2×C4#87812yes—
583C2×C4C22×Dic611yescharacteristic
584C2×C4C2×#23711yescharacteristic
585C2×C4#100411yescharacteristic
586C2×C4#100411yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C8—41no—
590C8—41no—
591C8—121no—
592C8—121no—
593C2×C3—42no—
594C2×C3—82no—
595C2×C3#245011yescharacteristic
596C2×C3#307611yescharacteristic
597C2×C3#308111yescharacteristic
598C2×C3#329711yescharacteristic
599C2×C3#330311yescharacteristic
600C2×C3#330911yescharacteristic
601C2×C3#331711yescharacteristic
602C22—22no—
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—42no—
610C22—42no—
611C22—42no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×#87811yescharacteristic
617C22#561011yescharacteristic
618C22#561011yescharacteristic
619C22#575711yescharacteristic
620C22#576211yescharacteristic
621C22D3×#13511yescharacteristic
622C22#576811yescharacteristic
623C4—22no—
624C4—42no—
625C4—62no—
626C4—81no—
627C4—81no—
628C4—121no—
629C4—122no—
630C4—121no—
631C4—122no—
632C4—121no—
633C4—241no—
634C4—241no—
635C4#553911yescharacteristic
636C4#553911yescharacteristic
637C4#559311yescharacteristic
638C4#564911yescharacteristic
639C3#2642111yescharacteristic, complemented, Sylow
640C2—42no—
641C2—82no—
642C2#7471211yescharacteristic
643C2#7873111yescharacteristic
644C2#7882211yescharacteristic
645C2#8068211yescharacteristic
646C2#8069311yescharacteristic
647C2#8072511yescharacteristic
648C2#8073911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1893C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#1893D3
hypocenterC3#26421
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1893

Complements

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

NH ≅ G/Ntype
2. #73900C2semidirect
4. #71974C2semidirect
5. #71969C2semidirect
7. #70193C2semidirect
8. #70167C2semidirect
9. #70164C2semidirect
10. #70153C2semidirect
11. #69506C2semidirect
12. #68843C2semidirect
13. #68815C2semidirect
14. #68809C2semidirect
15. #68801C2semidirect
639. C3#26421semidirect

Maximal subgroups

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

subgroupindex
#826372
#739002
C3×#18932
#719742
#719692
#716782
#701932
#701672
#701642
#701532
#695062
#688432
#688152
#688092
#688012
#264213