Finite Group Databasebeta

← #10858970

Subgroups — #10858970

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#82341C211yesmaximal, characteristic
2#74327C211yesmaximal, characteristic, complemented
3C3×#1802C211yesmaximal, characteristic, complemented
4#71607C211yesmaximal, characteristic, complemented
5#70477C211yesmaximal, characteristic, complemented
6#70463C211yesmaximal, characteristic, complemented
7#70202C211yesmaximal, characteristic
8#70201C211yesmaximal, characteristic
9#70180C211yesmaximal, characteristic, complemented
10#70164C211yesmaximal, characteristic, complemented
11#69571C211yesmaximal, characteristic, complemented
12#68877C211yesmaximal, characteristic, complemented
13#68866C211yesmaximal, characteristic, complemented
14#68802C211yesmaximal, characteristic, complemented
15#68789C211yesmaximal, characteristic, complemented
16#26423—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#938C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×#384C2211yescharacteristic
27#5221C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5213—21no—
31C4×#217—21no—
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4994—21no—
36#4993—21no—
37#4987C2211yescharacteristic
38#4980—21no—
39#4976—21no—
40Dic3×C4⋊C4C2211yescharacteristic
41#4972C2211yescharacteristic
42C2×#808C2211yescharacteristic
43#4957—21no—
44#4944—21no—
45C4×(C2×D3)⋊C4C2211yescharacteristic
46#4928C2211yescharacteristic
47#4928C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50#4551C2211yescharacteristic
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#2099—31no—
63#2087—31no—
64#2067—31no—
65#2025—31no—
66#1989—31no—
67#1920—31no—
68#1893—31no—
69#1875—31no—
70#1864—31no—
71#1854—31no—
72#1850—31no—
73#1802D311yescharacteristic, complemented
74#1735—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C22×#67—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×C4×D4—22no—
82C3×C4×D4—41no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93#938—22no—
94#938Q814yes—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4D411yescharacteristic
99C2×(C2×D3)⋊C4D411yescharacteristic
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×D3×C8—22no—
116C2×D3×C8—41no—
117C2×D3×C8—41no—
118#892—22no—
119#892—22no—
120#892—41no—
121#892—41no—
122#890—22no—
123#890—41no—
124C3×C4⋊C8—41no—
125C3×#102—41no—
126C3×#102—41no—
127#832—22no—
128#832—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95D411yescharacteristic
135C3⋊#95D411yescharacteristic
136(C2×D3)⋊C8—22no—
137#818—41no—
138Dic3⋊C8—41no—
139#808—22no—
140C23×C8—31no—
141C2×#125—31no—
142C2×#121—31no—
143C2×C4×D4C2×D311yescharacteristic
144C2×#117—31no—
145C2×#117—31no—
146C2×#107—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
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#384C2×D311yescharacteristic
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#340—31no—
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336—31no—
183C24×C3—21no—
184C23×D3C2×D411yescharacteristic
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4C2×Q812yes—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—81no—
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—81no—
202C2×#67—81no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
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—42no—
228C3×C4⋊C4—81no—
229C3×C4⋊C4#4112yes—
230C3×C22⋊C4—42no—
231C3×C42—21no—
232C3×C42—21no—
233#217—24no—
234#217—42no—
235#217—81no—
236(C2×D3)⋊C4—22no—
237(C2×D3)⋊C4—41no—
238(C2×D3)⋊C4—42no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—81no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—42no—
248C4⋊Dic3C2×Q812yes—
249C4⋊Dic3#4112yes—
250Dic3⋊C4—42no—
251Dic3⋊C4—42no—
252Dic3⋊C4—81no—
253Dic3⋊C4—81no—
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—61no—
267C22×D4C22×D311yescharacteristic
268C23×C4—21no—
269C23×C4—31no—
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—22no—
282C4×D4—41no—
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—31no—
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—31no—
303C2×C22⋊C4—31no—
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—121no—
316#106—62no—
317C4⋊C8—41no—
318C4⋊C8—121no—
319#102—41no—
320#102—41no—
321#102—62no—
322#102—121no—
323C22⋊C8—62no—
324#95—31no—
325#95—31no—
326#95—61no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331C23×C3—22no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3—41no—
338C23×C3C22×D411yescharacteristic
339C22×D3—21no—
340C22×D3—21no—
341C22×D3—21no—
342C22×D3—21no—
343C22×D3—21no—
344C22×D3—21no—
345C22×D3C22⋊Q812yes—
346C3×D4—22no—
347C3×D4—22no—
348C3×D4—41no—
349C3×D4—41no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
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—41no—
364C2×C3×C4—84no—
365C2×C3×C4#13512yes—
366C2×C3×C4C22×Q811yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×C3×C4H(5,𝔽2)11yescharacteristic
372#67—84no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—22no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—22no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3C22⋊Q812yes—
394C4×D3—22no—
395C4×D3—22no—
396C4×D3—22no—
397C4×D3—22no—
398C3×C8—41no—
399C3×C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C24—21no—
403C24—31no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—64no—
410C2×D4—241no—
411C2×D4—241no—
412C2×D4D3×Q812yes—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—32no—
418C22×C4—32no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—81no—
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—32no—
454C4⋊C4—32no—
455C4⋊C4—42no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—81no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—122no—
466C4⋊C4—241no—
467C4⋊C4—241no—
468C4⋊C4#23512yes—
469C22⋊C4—42no—
470C22⋊C4—64no—
471C22⋊C4—62no—
472C22⋊C4—121no—
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×C3—42no—
486C22×C3—42no—
487C22×C3—42no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3C2×C22⋊Q811yescharacteristic
493C22×C3#54611yescharacteristic
494C22×C3#54611yescharacteristic
495C22×C3C2×#13511yescharacteristic
496C22×C3#57511yescharacteristic
497C22×C3#57611yescharacteristic
498C22×C3#57711yescharacteristic
499C2×D3—22no—
500C2×D3—22no—
501C2×D3—22no—
502C2×D3—41no—
503C2×D3—41no—
504C2×D3—41no—
505C2×D3—41no—
506C3×C4—22no—
507C3×C4—42no—
508C3×C4—41no—
509C3×C4—81no—
510C3×C4—81no—
511C3×C4#52911yescharacteristic
512C3×C4#53811yescharacteristic
513C3×C4#55411yescharacteristic
514C3×C4#55411yescharacteristic
515Dic3—22no—
516Dic3—42no—
517Dic3—41no—
518Dic3—41no—
519Dic3—42no—
520Dic3—81no—
521Dic3—81no—
522C23—22no—
523C23—32no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—61no—
531C23—61no—
532C23—61no—
533C23—61no—
534C23—61no—
535C23—61no—
536C23C2×D3×D411yescharacteristic
537D4—22no—
538D4—22no—
539D4—41no—
540D4—41no—
541D4—244no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—32no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
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—84no—
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—121no—
582C2×C4#91212yes—
583C2×C4C2×#23511yescharacteristic
584C2×C4#100411yescharacteristic
585C2×C4#100411yescharacteristic
586C2×C4C2×D3×Q811yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C8—41no—
590C8—41no—
591C8—121no—
592C8—121no—
593C2×C3—82no—
594C2×C3#245011yescharacteristic
595C2×C3#307611yescharacteristic
596C2×C3#308111yescharacteristic
597C2×C3#329611yescharacteristic
598C2×C3#330511yescharacteristic
599C2×C3#330711yescharacteristic
600C2×C3#330911yescharacteristic
601D3—42no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—62no—
606C22—62no—
607C22—62no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—121no—
613C22—121no—
614C22—121no—
615C22—121no—
616C22#549011yescharacteristic
617C22#561211yescharacteristic
618C22D3×C22⋊Q811yescharacteristic
619C22C2×#91211yescharacteristic
620C22#576211yescharacteristic
621C22#577411yescharacteristic
622C22#577711yescharacteristic
623C4—22no—
624C4—41no—
625C4—42no—
626C4—62no—
627C4—81no—
628C4—81no—
629C4—121no—
630C4—122no—
631C4—122no—
632C4—121no—
633C4—241no—
634C4—241no—
635C4#551111yescharacteristic
636C4#552811yescharacteristic
637C4#561311yescharacteristic
638C4#563611yescharacteristic
639C3#2642311yescharacteristic, complemented, Sylow
640C2—82no—
641C2—122no—
642C2#7402311yescharacteristic
643C2#7875911yescharacteristic
644C2#7889711yescharacteristic
645C2#8035311yescharacteristic
646C2#8045211yescharacteristic
647C2#8096411yescharacteristic
648C2#8106211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1802C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#1802D3
hypocenterC3#26423
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1802

Complements

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

NH ≅ G/Ntype
2. #74327C2semidirect
3. C3×#1802C2semidirect
4. #71607C2semidirect
5. #70477C2semidirect
6. #70463C2semidirect
9. #70180C2semidirect
10. #70164C2semidirect
11. #69571C2semidirect
12. #68877C2semidirect
13. #68866C2semidirect
14. #68802C2semidirect
15. #68789C2semidirect
73. #1802D3semidirect
639. C3#26423semidirect

Maximal subgroups

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

subgroupindex
#823412
#743272
C3×#18022
#716072
#704772
#704632
#702022
#702012
#701802
#701642
#695712
#688772
#688662
#688022
#687892
#264233