Finite Group Databasebeta

← #10858934

Subgroups — #10858934

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#76000C211yesmaximal, characteristic, complemented
2#75985C211yesmaximal, characteristic
3C3×#2088C211yesmaximal, characteristic, complemented
4#70585C211yesmaximal, characteristic, complemented
5#70580C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic, complemented
7#70477C211yesmaximal, characteristic, complemented
8#69625C211yesmaximal, characteristic, complemented
9#69612C211yesmaximal, characteristic, complemented
10#68899C211yesmaximal, characteristic, complemented
11#68883C211yesmaximal, characteristic
12#68881C211yesmaximal, characteristic
13#68866C211yesmaximal, characteristic, complemented
14#68815C211yesmaximal, characteristic, complemented
15#68802C211yesmaximal, characteristic, complemented
16#26421—31nomaximal, Sylow
17C22×D3×C8C2211yescharacteristic
18C2×#895C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#890C2211yescharacteristic
22C2×C4×D12C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31#4994—21no—
32#4990—21no—
33#4988—21no—
34#4988C2211yescharacteristic
35#4987C2211yescharacteristic
36#4984—21no—
37#4984C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#809C2211yescharacteristic
41C2×#807C2211yescharacteristic
42C4×(C2×D3)⋊C4C2211yescharacteristic
43#4935C2211yescharacteristic
44#4677—21no—
45#4666—21no—
46#4665—21no—
47#4653—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2441—31no—
61#2363—31no—
62#2100—31no—
63#2088D311yescharacteristic, complemented
64#2065—31no—
65#2025—31no—
66#1989—31no—
67#1918—31no—
68#1893—31no—
69#1875—31no—
70#1862—31no—
71#1854—31no—
72#1850—31no—
73#1802—31no—
74#1735—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
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)⋊C4D411yescharacteristic
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C2×D3×C8—22no—
112C2×D3×C8—41no—
113C2×D3×C8—41no—
114#895—22no—
115#895Q814yes—
116#892—22no—
117#892—22no—
118#892—41no—
119#892—41no—
120#890—22no—
121#890—41no—
122C4×D12—22no—
123C4×D12—41no—
124C3×#106—41no—
125C3×C4⋊C8—41no—
126C3×#95—21no—
127C3×#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133#821—41no—
134#821—41no—
135(C2×D3)⋊C8—22no—
136Dic3⋊C8—41no—
137#809—22no—
138#809—41no—
139#807—22no—
140C23×C8—31no—
141C2×#125—31no—
142C2×#121—31no—
143C2×C4×D4—31no—
144C2×#117—31no—
145C2×#117—31no—
146C2×#107—31no—
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×C22⋊C8—31no—
155#399—61no—
156#396—61no—
157#390—31no—
158#390—61no—
159#389—31no—
160#389—61no—
161#389C2×D311yescharacteristic
162#389C2×D311yescharacteristic
163#388—61no—
164#387—61no—
165#386—61no—
166#384—31no—
167#380—31no—
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#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C23×D3—21no—
184C23×D3C2×D411yescharacteristic
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12—24no—
204C2×D12—81no—
205C2×D12—81no—
206C2×D12C2×Q812yes—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—81no—
214C2×C4×D3C2×Q812yes—
215C2×C4×D3#4112yes—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—42no—
229C3×C4⋊C4—81no—
230C3×C4⋊C4C2×Q812yes—
231C3×C4⋊C4#4112yes—
232C3×C42—21no—
233C3×C42—21no—
234(C2×D3)⋊C4—22no—
235(C2×D3)⋊C4—24no—
236(C2×D3)⋊C4—41no—
237(C2×D3)⋊C4—42no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—81no—
242(C2×D3)⋊C4—81no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—42no—
249C4⋊Dic3—42no—
250C4⋊Dic3—81no—
251C4⋊Dic3—81no—
252C4⋊Dic3#4112yes—
253Dic3⋊C4—42no—
254C4×Dic3—21no—
255C4×Dic3—21no—
256C4×Dic3—42no—
257C4×Dic3—42no—
258C4×Dic3—81no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265D3×C8—44no—
266C22×D4—31no—
267C22×D4—61no—
268C23×C4—31no—
269C23×C4—61no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8—62no—
273C22×C8—121no—
274C22×C8—121no—
275C22×C8C22×D311yescharacteristic
276C22×C8C22×D311yescharacteristic
277#125—62no—
278#125—121no—
279#121—34no—
280#121—62no—
281C4×D4—62no—
282C4×D4—121no—
283#117—62no—
284#117—62no—
285#117—121no—
286#117—121no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—31no—
302C2×C22⋊C4—31no—
303C2×C22⋊C4—61no—
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×C42—31no—
311C2×C42—61no—
312C2×C42—61no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—62no—
316#106—41no—
317C4⋊C8—41no—
318C4⋊C8—121no—
319#102—62no—
320#102—121no—
321#102—121no—
322#102—121no—
323C22⋊C8—62no—
324#95—21no—
325#95—21no—
326#95—31no—
327#95—31no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3C22×D411yescharacteristic
332C22×D3—21no—
333C22×D3—21no—
334C22×D3—21no—
335C22×D3—21no—
336C22×D3—21no—
337C22×D3—21no—
338C22×D3—22no—
339C22×D3—41no—
340C22×D3—41no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C22×D3C22⋊Q812yes—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4#13512yes—
364C2×C3×C4C22×Q811yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3C22⋊Q812yes—
388D12—22no—
389D12—22no—
390D12—41no—
391D12—41no—
392D12—84no—
393C4×D3—22no—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—84no—
398C3×C8—41no—
399C3×C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C24—31no—
403C24—61no—
404C2×D4—32no—
405C2×D4—64no—
406C2×D4—61no—
407C2×D4—61no—
408C2×D4—61no—
409C2×D4—61no—
410C2×D4—61no—
411C2×D4—241no—
412C2×D4—241no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—31no—
418C22×C4—32no—
419C22×C4—32no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—241no—
432C22×C4D3×D411yescharacteristic
433C22×C4D3×D411yescharacteristic
434C22×C4C23×D311yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—124no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—32no—
455C4⋊C4—42no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—81no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—241no—
466C4⋊C4—241no—
467C4⋊C4D3×Q812yes—
468C4⋊C4#23912yes—
469C22⋊C4—64no—
470C22⋊C4—62no—
471C22⋊C4—121no—
472C22⋊C4—122no—
473C22⋊C4—122no—
474C22⋊C4—122no—
475C22⋊C4—122no—
476C22⋊C4—241no—
477C22⋊C4—241no—
478C42—21no—
479C42—21no—
480C42—61no—
481C42—61no—
482C42—122no—
483C42—122no—
484C42—241no—
485C22×C3C2×C22⋊Q811yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#54611yescharacteristic
488C22×C3C2×#13511yescharacteristic
489C22×C3#57511yescharacteristic
490C22×C3#57611yescharacteristic
491C22×C3#57711yescharacteristic
492C2×D3—22no—
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—41no—
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—41no—
499C2×D3—42no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—81no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C3×C4—22no—
507C3×C4—42no—
508C3×C4—42no—
509C3×C4—41no—
510C3×C4—81no—
511C3×C4—81no—
512C3×C4#52911yescharacteristic
513C3×C4#53811yescharacteristic
514C3×C4#55411yescharacteristic
515C3×C4#55411yescharacteristic
516Dic3—22no—
517Dic3—42no—
518Dic3—41no—
519Dic3—41no—
520Dic3—81no—
521Dic3—81no—
522C23—32no—
523C23—61no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—62no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C2×D3×D411yescharacteristic
537D4—62no—
538D4—62no—
539D4—121no—
540D4—121no—
541D4—244no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
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—41no—
559C2×C4—41no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—62no—
571C2×C4—61no—
572C2×C4—61no—
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#91612yes—
583C2×C4#100411yescharacteristic
584C2×C4C2×D3×Q811yescharacteristic
585C2×C4C2×#23911yescharacteristic
586C2×C4D3×#4111yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4#100911yescharacteristic
589C8—41no—
590C8—41no—
591C8—121no—
592C8—121no—
593C2×C3#245011yescharacteristic
594C2×C3#307611yescharacteristic
595C2×C3#308111yescharacteristic
596C2×C3#329711yescharacteristic
597C2×C3#330311yescharacteristic
598C2×C3#330911yescharacteristic
599C2×C3#331711yescharacteristic
600D3—42no—
601D3—82no—
602C22—62no—
603C22—62no—
604C22—62no—
605C22—121no—
606C22—121no—
607C22—121no—
608C22—121no—
609C22—122no—
610C22—122no—
611C22—122no—
612C22—241no—
613C22—241no—
614C22—241no—
615C22—241no—
616C22#561011yescharacteristic
617C22D3×C22⋊Q811yescharacteristic
618C22#563011yescharacteristic
619C22C2×#91611yescharacteristic
620C22#576911yescharacteristic
621C22#577511yescharacteristic
622C22#577711yescharacteristic
623C4—22no—
624C4—42no—
625C4—42no—
626C4—41no—
627C4—62no—
628C4—81no—
629C4—81no—
630C4—122no—
631C4—121no—
632C4—121no—
633C4—241no—
634C4—241no—
635C4#565711yescharacteristic
636C4#569011yescharacteristic
637C4#569111yescharacteristic
638C4#573011yescharacteristic
639C3#2642111yescharacteristic, complemented, Sylow
640C2—122no—
641C2—242no—
642C2#7795311yescharacteristic
643C2#8068111yescharacteristic
644C2#8082411yescharacteristic
645C2#8101011yescharacteristic
646C2#8107611yescharacteristic
647C2#8116911yescharacteristic
648C2#8124711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2088C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2088D3
hypocenterC3#26421
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2088

Complements

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

NH ≅ G/Ntype
1. #76000C2semidirect
3. C3×#2088C2semidirect
4. #70585C2semidirect
5. #70580C2semidirect
6. #70479C2semidirect
7. #70477C2semidirect
8. #69625C2semidirect
9. #69612C2semidirect
10. #68899C2semidirect
13. #68866C2semidirect
14. #68815C2semidirect
15. #68802C2semidirect
63. #2088D3semidirect
639. C3#26421semidirect

Maximal subgroups

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

subgroupindex
#760002
#759852
C3×#20882
#705852
#705802
#704792
#704772
#696252
#696122
#688992
#688832
#688812
#688662
#688152
#688022
#264213