Finite Group Databasebeta

← #10858752

Subgroups — #10858752

Order 768.

802 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76223C211yesmaximal, characteristic, complemented
2#73649C211yesmaximal, characteristic
3C3×#1921C211yesmaximal, characteristic, complemented
4#70638C211yesmaximal, characteristic, complemented
5#70633C211yesmaximal, characteristic, complemented
6#69629C211yesmaximal, characteristic, complemented
7#69490C211yesmaximal, characteristic, complemented
8#68893C211yesmaximal, characteristic, complemented
9#68883C211yesmaximal, characteristic
10#68881C211yesmaximal, characteristic
11#68866C211yesmaximal, characteristic, complemented
12#68846C211yesmaximal, characteristic, complemented
13#68841C211yesmaximal, characteristic, complemented
14#68678C211yesmaximal, characteristic, complemented
15#68666C211yesmaximal, characteristic, complemented
16#26409—31nomaximal, Sylow
17C22×D3×C8C2211yescharacteristic
18C2×#895C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×#892C2211yescharacteristic
21C2×#892C2211yescharacteristic
22C2×#869C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C3×#389C2211yescharacteristic
25C3×C4×C4⋊C4C2211yescharacteristic
26C2×#821C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×(C2×D3)⋊C8C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30#4994—21no—
31#4987C2211yescharacteristic
32#4984C2211yescharacteristic
33#4980—21no—
34#4977C2211yescharacteristic
35#4972C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#808C2211yescharacteristic
39#4939—21no—
40#4939C2211yescharacteristic
41C4×(C2×D3)⋊C4—21no—
42C4×(C2×D3)⋊C4C2211yescharacteristic
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44#4673—21no—
45#4670—21no—
46#4665—21no—
47#4646—21no—
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#340C2211yescharacteristic
51C3×#336C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2439—31no—
61#2363—31no—
62#2101—31no—
63#2088—31no—
64#2065—31no—
65#2023—31no—
66#1988—31no—
67#1921D311yescharacteristic, complemented
68#1893—31no—
69#1875—31no—
70#1862—31no—
71#1854—31no—
72#1850—31no—
73#1804—31no—
74#1735—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×Dic6—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
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)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×D3×C8—22no—
110C2×D3×C8—41no—
111C2×D3×C8—41no—
112#895—22no—
113#895Q814yescomplemented
114#893—22no—
115#893—41no—
116#892—22no—
117#892—22no—
118#892—41no—
119#892—41no—
120#869—22no—
121#869—41no—
122C3×#107—22no—
123C3×#95—21no—
124C3×#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#821—22no—
133#821—41no—
134(C2×D3)⋊C8—22no—
135Dic3⋊C8—41no—
136#809—41no—
137#809—41no—
138#808—41no—
139C4⋊(C3⋊C8)—41no—
140C23×C8—31no—
141C2×C4⋊Q8—31no—
142C2×#121—31no—
143C2×C4×D4—31no—
144C2×#117—31no—
145C2×#117—31no—
146C2×#107C2×D311yescharacteristic
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×#102—31no—
154C2×C22⋊C8—31no—
155#396—61no—
156#393—61no—
157#391—61no—
158#390—31no—
159#390—61no—
160#389—31no—
161#389—31no—
162#389C2×D311yescharacteristic
163#388—61no—
164#387—61no—
165#386—61no—
166#382—31no—
167#380—31no—
168C4×C4⋊C4C2×D311yescharacteristic
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#336C2×D311yescharacteristic
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×Dic6—42no—
217C2×Dic6—42no—
218C2×Dic6—81no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—42no—
232C3×C4⋊C4C2×Q812yes—
233C3×C4⋊C4#4112yes—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—81no—
237(C2×D3)⋊C4—22no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—41no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242(C2×D3)⋊C4—42no—
243(C2×D3)⋊C4—42no—
244(C2×D3)⋊C4—81no—
245(C2×D3)⋊C4—81no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—42no—
252C4⋊Dic3—81no—
253C4⋊Dic3#4112yes—
254Dic3⋊C4—81no—
255C4×Dic3—21no—
256C4×Dic3—21no—
257C4×Dic3—42no—
258C4×Dic3—42no—
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×Q8—61no—
267C22×D4—31no—
268C22×D4—61no—
269C23×C4—31no—
270C23×C4—61no—
271C22×C8—31no—
272C22×C8—31no—
273C22×C8—62no—
274C22×C8—121no—
275C22×C8—121no—
276C22×C8C22×D311yescharacteristic
277C22×C8C22×D311yescharacteristic
278C4⋊Q8—62no—
279C4⋊Q8—121no—
280#121—34no—
281#121—62no—
282C4×D4—62no—
283C4×D4—121no—
284#117—62no—
285#117—62no—
286#117—121no—
287#117—121no—
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⋊C4C22×D311yescharacteristic
296C2×C4⋊C4C22×D311yescharacteristic
297C2×C4⋊C4C22×D311yescharacteristic
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C22⋊C4—31no—
300C2×C22⋊C4—31no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4—61no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C22⋊C4—61no—
307C2×C22⋊C4—61no—
308C2×C42—21no—
309C2×C42—31no—
310C2×C42—61no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—22no—
314#106—121no—
315C4⋊C8—121no—
316C4⋊C8—121no—
317#102—62no—
318#102—121no—
319#102—121no—
320#102—121no—
321C22⋊C8—62no—
322#95—21no—
323#95—21no—
324#95—31no—
325#95—31no—
326#95—61no—
327#95—61no—
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—
398Dic6—84no—
399C3×C8—41no—
400C3×C8—41no—
401C3⋊C8—41no—
402C3⋊C8—41no—
403C24—31no—
404C24—61no—
405C2×Q8—122no—
406C2×Q8—122no—
407C2×Q8—241no—
408C2×D4—32no—
409C2×D4—64no—
410C2×D4—61no—
411C2×D4—61no—
412C2×D4—61no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—241no—
416C2×D4—241no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—31no—
422C22×C4—32no—
423C22×C4—32no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—61no—
431C22×C4—61no—
432C22×C4—61no—
433C22×C4—122no—
434C22×C4—122no—
435C22×C4—241no—
436C22×C4D3×D411yescharacteristic
437C22×C4D3×D411yescharacteristic
438C22×C4C23×D311yescharacteristic
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—61no—
449C2×C8—61no—
450C2×C8—61no—
451C2×C8—124no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—32no—
459C4⋊C4—42no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—122no—
466C4⋊C4—241no—
467C4⋊C4—241no—
468C4⋊C4#23512yes—
469C4⋊C4D3×Q812yes—
470C22⋊C4—64no—
471C22⋊C4—62no—
472C22⋊C4—121no—
473C22⋊C4—122no—
474C22⋊C4—122no—
475C22⋊C4—122no—
476C22⋊C4—122no—
477C22⋊C4—241no—
478C22⋊C4—241no—
479C42—21no—
480C42—21no—
481C42—61no—
482C42—61no—
483C42—81no—
484C42—122no—
485C42—122no—
486C22×C3C2×C22⋊Q811yescharacteristic
487C22×C3#54611yescharacteristic
488C22×C3#54611yescharacteristic
489C22×C3C2×#13511yescharacteristic
490C22×C3#57511yescharacteristic
491C22×C3#57611yescharacteristic
492C22×C3#57711yescharacteristic
493C2×D3—22no—
494C2×D3—22no—
495C2×D3—22no—
496C2×D3—41no—
497C2×D3—41no—
498C2×D3—41no—
499C2×D3—41no—
500C2×D3—42no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—81no—
504C2×D3—81no—
505C2×D3—81no—
506C2×D3—81no—
507C3×C4—22no—
508C3×C4—42no—
509C3×C4—42no—
510C3×C4—41no—
511C3×C4—81no—
512C3×C4—81no—
513C3×C4#52911yescharacteristic
514C3×C4#53811yescharacteristic
515C3×C4#55411yescharacteristic
516C3×C4#55411yescharacteristic
517Dic3—22no—
518Dic3—42no—
519Dic3—41no—
520Dic3—41no—
521Dic3—81no—
522Dic3—81no—
523C23—32no—
524C23—61no—
525C23—61no—
526C23—61no—
527C23—61no—
528C23—61no—
529C23—61no—
530C23—62no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23—121no—
537C23C2×D3×D411yescharacteristic
538Q8—244no—
539D4—62no—
540D4—62no—
541D4—121no—
542D4—121no—
543D4—244no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—32no—
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—41no—
561C2×C4—41no—
562C2×C4—61no—
563C2×C4—61no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—244no—
584C2×C4#91912yes—
585C2×C4C2×#23511yescharacteristic
586C2×C4C2×D3×Q811yescharacteristic
587C2×C4D3×#4111yescharacteristic
588C2×C4D3×#4111yescharacteristic
589C2×C4#100911yescharacteristic
590C2×C4#100911yescharacteristic
591C8—41no—
592C8—41no—
593C8—121no—
594C8—121no—
595C2×C3#245011yescharacteristic
596C2×C3#307611yescharacteristic
597C2×C3#308111yescharacteristic
598C2×C3#329711yescharacteristic
599C2×C3#330411yescharacteristic
600C2×C3#331911yescharacteristic
601C2×C3#332111yescharacteristic
602D3—42no—
603D3—82no—
604C22—62no—
605C22—62no—
606C22—62no—
607C22—121no—
608C22—121no—
609C22—121no—
610C22—121no—
611C22—122no—
612C22—122no—
613C22—122no—
614C22—241no—
615C22—241no—
616C22—241no—
617C22—241no—
618C22#555011yescharacteristic
619C22D3×C22⋊Q811yescharacteristic
620C22#565411yescharacteristic
621C22C2×#91911yescharacteristic
622C22#577011yescharacteristic
623C22#577311yescharacteristic
624C22#577411yescharacteristic
625C4—22no—
626C4—42no—
627C4—42no—
628C4—41no—
629C4—62no—
630C4—81no—
631C4—81no—
632C4—121no—
633C4—122no—
634C4—121no—
635C4—241no—
636C4—241no—
637C4#552811yescharacteristic
638C4#557911yescharacteristic
639C4#569011yescharacteristic
640C4#570411yescharacteristic
641C3#2640911yescharacteristic, complemented, Sylow
642C2—122no—
643C2—242no—
644C2#7532211yescharacteristic
645C2#7973711yescharacteristic
646C2#7988311yescharacteristic
647C2#8110711yescharacteristic
648C2#8116811yescharacteristic
649C2#8138611yescharacteristic
650C2#8148511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1921C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1921D3
hypocenterC3#26409
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1921

Complements

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

NH ≅ G/Ntype
1. #76223C2semidirect
3. C3×#1921C2semidirect
4. #70638C2semidirect
5. #70633C2semidirect
6. #69629C2semidirect
7. #69490C2semidirect
8. #68893C2semidirect
11. #68866C2semidirect
12. #68846C2semidirect
13. #68841C2semidirect
14. #68678C2semidirect
15. #68666C2semidirect
67. #1921D3semidirect
113. #895Q8semidirect
641. C3#26409semidirect

Maximal subgroups

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

subgroupindex
#762232
#736492
C3×#19212
#706382
#706332
#696292
#694902
#688932
#688832
#688812
#688662
#688462
#688412
#686782
#686662
#264093