Finite Group Databasebeta

← #10875960

Subgroups — #10875960

Order 768.

836 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76337C211yesmaximal, characteristic, complemented
2#76202C211yesmaximal, characteristic, complemented
3C3×#2067C211yesmaximal, characteristic, complemented
4#70638C211yesmaximal, characteristic, complemented
5#70610C211yesmaximal, characteristic, complemented
6#70479C211yesmaximal, characteristic, complemented
7#70448C211yesmaximal, characteristic, complemented
8#70208C211yesmaximal, characteristic
9#70208C211yesmaximal, characteristic
10#70187C211yesmaximal, characteristic, complemented
11#70165C211yesmaximal, characteristic, complemented
12#69657C211yesmaximal, characteristic
13#69612C211yesmaximal, characteristic, complemented
14#68811C211yesmaximal, characteristic, complemented
15#68782C211yesmaximal, characteristic, complemented
16#27571—31nomaximal, Sylow
17C22×#213C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#893C2211yescharacteristic
20C2×D3×C4⋊C4—21no—
21C2×#879—21no—
22C2×C4×D12C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#389C2211yescharacteristic
29#5221C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#815C2211yescharacteristic
35#4989—21no—
36#4984C2211yescharacteristic
37#4975C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39#4972C2211yescharacteristic
40C2×#809C2211yescharacteristic
41#4953—21no—
42#4951—21no—
43#4948—21no—
44#4935C2211yescharacteristic
45C4×Dic3⋊C4C2211yescharacteristic
46#4668—21no—
47#4667—21no—
48C3×#340C2211yescharacteristic
49#4551C2211yescharacteristic
50#4551C2211yescharacteristic
51#4550C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2512—31no—
61D4×C4⋊C4—31no—
62#2100—31no—
63#2100—31no—
64#2071—31no—
65#2067D311yescharacteristic, complemented
66#1936—31no—
67#1921—31no—
68#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1862—31no—
73#1802—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—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×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#213—22no—
94C2×#213—22no—
95C2×#213—22no—
96C2×(C2×D3)⋊C4—21no—
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4D411yescharacteristic
109C2×Dic3⋊C4D411yescharacteristic
110C2×C4×Dic3—21no—
111C2×C4×Dic3C2311yescharacteristic
112C2×C4×Dic3C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115#893—22no—
116#893—22no—
117#893—22no—
118#893—41no—
119D3×C4⋊C4—44no—
120#879—44no—
121C4×D12—22no—
122C4×D12—41no—
123C3×#107—22no—
124C3×C4⋊C8—41no—
125C3×#102—22no—
126C3×#102—41no—
127C3×#95—21no—
128#832—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134#821—22no—
135#821—22no—
136#815—41no—
137#809—22no—
138C22×D8—31no—
139C2×#126—61no—
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143C22×C4⋊C4—61no—
144C2×#107C2×D311yescharacteristic
145C2×C4⋊C8C2×D311yescharacteristic
146C2×#102—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#102C2×D311yescharacteristic
151C2×#102C2×D311yescharacteristic
152C2×#97—31no—
153#397—61no—
154#396—61no—
155#390—31no—
156#390—31no—
157#390C2×D311yescharacteristic
158#389—31no—
159#389—31no—
160#389C2×D311yescharacteristic
161#388—61no—
162#385—61no—
163#385—61no—
164#384—31no—
165#384—31no—
166#380—61no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340C2×D311yescharacteristic
177#336—31no—
178#336—31no—
179#336—31no—
180#336—31no—
181C24×C3—21no—
182C23×D3—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—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×Dic3—21no—
198C22×Dic3C2×D411yescharacteristic
199C22×Dic3C2×D411yescharacteristic
200C22×Dic3C2×D411yescharacteristic
201C22×Dic3C2×D411yescharacteristic
202C2×D12—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12#4112yes—
206C2×D12#4112yes—
207C2×D12#4112yes—
208C2×C4×D3—24no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—42no—
212C2×C4×D3—42no—
213C2×C4×D3—42no—
214C2×C4×D3—42no—
215C2×C4×D3—81no—
216C2×C4×D3—81no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—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#4112yes—
230C3×C22⋊C4—42no—
231C3×C22⋊C4—42no—
232C3×C42—21no—
233C3×C42—21no—
234#217—42no—
235#217—42no—
236#217—81no—
237#217—81no—
238#213—44no—
239(C2×D3)⋊C4—24no—
240(C2×D3)⋊C4—42no—
241(C2×D3)⋊C4—42no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—21no—
247C4⋊Dic3—21no—
248C4⋊Dic3—42no—
249C4⋊Dic3—42no—
250C4⋊Dic3—81no—
251Dic3⋊C4—22no—
252Dic3⋊C4—22no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—81no—
256C4×Dic3—21no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—42no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C22×D4—31no—
268C22×D4C22×D311yescharacteristic
269C23×C4—61no—
270C23×C4—61no—
271C23×C4—61no—
272C2×D8—62no—
273C2×D8—62no—
274C2×D8—62no—
275C22×C8—31no—
276C22×C8—31no—
277C22×C8C22×D311yescharacteristic
278C22×C8C22×D311yescharacteristic
279#126—124no—
280C4×D4—62no—
281C4×D4—62no—
282C4×D4—62no—
283C4×D4—62no—
284C4×D4—121no—
285C4×D4—121no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—31no—
288C2×C4⋊C4—31no—
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⋊C4—61no—
298C2×C4⋊C4—124no—
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C4⋊C4C22×D311yescharacteristic
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
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×C42—31no—
312C2×C42—31no—
313C2×C42—61no—
314C2×C42C22×D311yescharacteristic
315#107—22no—
316C4⋊C8—41no—
317#102—22no—
318#102—41no—
319#102—62no—
320#102—62no—
321#102—62no—
322#102—121no—
323#97—121no—
324#95—21no—
325#95—31no—
326#95—31no—
327#95—61no—
328#95—61no—
329#95—61no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3—41no—
336C23×C3—41no—
337C23×C3C22×D411yescharacteristic
338C22×D3—22no—
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—41no—
344C22×D3—41no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4—41no—
348C3×D4—41no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
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×C4AGL(1,ℤ/8)12yes—
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C42−1+411yescharacteristic
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3AGL(1,ℤ/8)12yes—
391D12—22no—
392D12—22no—
393D12—22no—
394D12#12614yescomplemented
395C4×D3—48no—
396C4×D3—84no—
397C4×D3—84no—
398C3×C8—41no—
399C3×C8—41no—
400C3⋊C8—41no—
401C3⋊C8—41no—
402C24—21no—
403C24—61no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—32no—
411C2×D4—32no—
412C2×D4—32no—
413C2×D4—61no—
414C2×D4—61no—
415C2×D4—61no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—31no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—64no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—122no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—122no—
431C22×C4—122no—
432C22×C4—122no—
433C22×C4—241no—
434C22×C4—241no—
435C22×C4D3×D411yescharacteristic
436C22×C4D3×D411yescharacteristic
437C22×C4C23×D311yescharacteristic
438D8—124no—
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—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—62no—
458C4⋊C4—62no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—61no—
463C4⋊C4—61no—
464C4⋊C4—61no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—122no—
469C4⋊C4—241no—
470C4⋊C4—241no—
471C4⋊C4#23712yes—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—64no—
475C22⋊C4—122no—
476C22⋊C4—122no—
477C22⋊C4—122no—
478C22⋊C4—122no—
479C22⋊C4—241no—
480C22⋊C4—241no—
481C42—21no—
482C42—21no—
483C42—61no—
484C42—61no—
485C42—61no—
486C42—61no—
487C42—122no—
488C22×C3—81no—
489C22×C3—81no—
490C22×C3—81no—
491C22×C3—81no—
492C22×C3—81no—
493C22×C3—81no—
494C22×C3—81no—
495C22×C3#52511yescharacteristic
496C22×C3#54611yescharacteristic
497C22×C3#54811yescharacteristic
498C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
499C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
500C22×C3#57611yescharacteristic
501C22×C3#57811yescharacteristic
502C2×D3—24no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—42no—
506C2×D3—42no—
507C2×D3—42no—
508C2×D3—42no—
509C3×C4—22no—
510C3×C4—41no—
511C3×C4—42no—
512C3×C4—81no—
513C3×C4—81no—
514C3×C4C2×#12611yescharacteristic
515C3×C4#53911yescharacteristic
516C3×C4#55211yescharacteristic
517C3×C4#55311yescharacteristic
518Dic3—22no—
519Dic3—41no—
520Dic3—41no—
521Dic3—41no—
522Dic3—81no—
523Dic3—81no—
524Dic3—81no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—62no—
533C23—62no—
534C23—62no—
535C23—121no—
536C23—121no—
537C23—121no—
538C23—121no—
539C23C2×D3×D411yescharacteristic
540D4—34no—
541D4—41no—
542D4—41no—
543D4—41no—
544D4—41no—
545D4—62no—
546D4—62no—
547D4—62no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—32no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—61no—
564C2×C4—61no—
565C2×C4—61no—
566C2×C4—61no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—61no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—128no—
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—121no—
584C2×C4—244no—
585C2×C4—244no—
586C2×C4#91112yes—
587C2×C4C2×#23711yescharacteristic
588C2×C4#100411yescharacteristic
589C2×C4#100711yescharacteristic
590C2×C4D3×#4111yescharacteristic
591C2×C4D3×#4111yescharacteristic
592C2×C4D3×#4111yescharacteristic
593C8—41no—
594C8—41no—
595C8—121no—
596C8—121no—
597C2×C3—161no—
598C2×C3#248111yescharacteristic
599C2×C3#310911yescharacteristic
600C2×C3#312411yescharacteristic
601C2×C3#331811yescharacteristic
602C2×C3#333011yescharacteristic
603C2×C3#336611yescharacteristic
604C2×C3#336711yescharacteristic
605D3—44no—
606C22—64no—
607C22—81no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—122no—
615C22—122no—
616C22—122no—
617C22—122no—
618C22—122no—
619C22—122no—
620C22#560611yescharacteristic
621C22#561011yescharacteristic
622C22#563311yescharacteristic
623C22C2×#91111yescharacteristic
624C22#576911yescharacteristic
625C22D3×AGL(1,ℤ/8)11yescharacteristic
626C22#577911yescharacteristic
627C4—22no—
628C4—42no—
629C4—41no—
630C4—62no—
631C4—81no—
632C4—81no—
633C4—121no—
634C4—121no—
635C4—121no—
636C4—241no—
637C4—241no—
638C4—241no—
639C4#561411yescharacteristic
640C4#563811yescharacteristic
641C4#563911yescharacteristic
642C4D3×#12611yescharacteristic
643C3#2757111yescharacteristic, complemented, Sylow
644C2—124no—
645C2—161no—
646C2#7730411yescharacteristic
647C2#8031411yescharacteristic
648C2#8057411yescharacteristic
649C2#8073211yescharacteristic
650C2#8084211yescharacteristic
651C2#8099311yescharacteristic
652C2#8115211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2067C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2067D3
hypocenterC3#27571
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2067

Complements

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

NH ≅ G/Ntype
1. #76337C2semidirect
2. #76202C2semidirect
3. C3×#2067C2semidirect
4. #70638C2semidirect
5. #70610C2semidirect
6. #70479C2semidirect
7. #70448C2semidirect
10. #70187C2semidirect
11. #70165C2semidirect
13. #69612C2semidirect
14. #68811C2semidirect
15. #68782C2semidirect
65. #2067D3semidirect
394. D12#126semidirect
643. C3#27571semidirect

Maximal subgroups

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

subgroupindex
#763372
#762022
C3×#20672
#706382
#706102
#704792
#704482
#702082
#701872
#701652
#696572
#696122
#688112
#687822
#275713