Finite Group Databasebeta

← #10877257

Subgroups — #10877257

Order 768.

882 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76291C211yesmaximal, characteristic
2D3×#382C211yesmaximal, characteristic, complemented
3C3×#2092C211yesmaximal, characteristic, complemented
4#70632C211yesmaximal, characteristic, complemented
5#70616C211yesmaximal, characteristic, complemented
6#70579C211yesmaximal, characteristic, complemented
7#70564C211yesmaximal, characteristic, complemented
8#70200C211yesmaximal, characteristic
9#70200C211yesmaximal, characteristic
10#70154C211yesmaximal, characteristic, complemented
11#70154C211yesmaximal, characteristic, complemented
12#69654C211yesmaximal, characteristic, complemented
13#69542C211yesmaximal, characteristic, complemented
14#68662C211yesmaximal, characteristic, complemented
15#68644C211yesmaximal, characteristic, complemented
16#27684—31nomaximal, Sylow
17C2×#938C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×#67⋊C4—21no—
22C2×C42×D3C2211yescharacteristic
23C2×C3×#107C2211yescharacteristic
24C2×C3×#106C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2211yescharacteristic
27C3×#390C2211yescharacteristic
28C3×#382C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×(C2×D3)⋊C8C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×Dic3⋊C8C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35#4987C2211yescharacteristic
36#4977C2211yescharacteristic
37#4977C2211yescharacteristic
38Dic3×C4⋊C4C2211yescharacteristic
39C2×#808C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4956—21no—
42#4945—21no—
43Dic3×C22⋊C4—21no—
44#4936C2211yescharacteristic
45#4928C2211yescharacteristic
46D3×#95—21no—
47D3×#95—21no—
48#4650—21no—
49#4650—21no—
50C3×#336C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60C4×#121—31no—
61C2×#382—31no—
62#2101—31no—
63#2101—31no—
64#2092—31no—
65#2092D311yescharacteristic, complemented
66#1949—31no—
67#1949—31no—
68#1874—31no—
69#1874—31no—
70#1853—31no—
71#1852—31no—
72#1804—31no—
73#1804—31no—
74#1797—31no—
75C22×C3×D4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C4×D3D411yescharacteristic
81C22×C4×D3D411yescharacteristic
82C22×C4×D3C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C42C2311yescharacteristic
91C2×#217—21no—
92C2×#217—21no—
93#938—22no—
94#938Q814yes—
95D4×Dic3—22no—
96D4×Dic3—22no—
97C2×(C2×D3)⋊C4—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3D411yescharacteristic
107C2×C4×Dic3D411yescharacteristic
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111D3×C4⋊C4—41no—
112D3×C4⋊C4—41no—
113D3×C4⋊C4—41no—
114D3×C4⋊C4—41no—
115#67⋊C4—44no—
116C42×D3—22no—
117C42×D3—22no—
118C3×#107—41no—
119C3×#106—41no—
120C3×#102—41no—
121C3×#102—41no—
122C3×#95—21no—
123C3×#95—21no—
124#832—41no—
125#832—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135(C2×D3)⋊C8—22no—
136(C2×D3)⋊C8—22no—
137Dic3⋊C8—22no—
138Dic3⋊C8—22no—
139#808—41no—
140#807—41no—
141C2×#121—31no—
142C2×C4×D4—31no—
143C2×C4×D4—61no—
144C22×C4⋊C4—31no—
145C22×C4⋊C4—31no—
146C22×C42—31no—
147C2×#107—31no—
148C2×#107C2×D311yescharacteristic
149C2×#106—31no—
150C2×#106C2×D311yescharacteristic
151C2×C4⋊C8—31no—
152C2×C4⋊C8—31no—
153C2×#102—31no—
154C2×#102—31no—
155C2×#102C2×D311yescharacteristic
156C2×#102C2×D311yescharacteristic
157C2×C22⋊C8—31no—
158C2×C22⋊C8—31no—
159#390C2×D311yescharacteristic
160#389—31no—
161#388—61no—
162#386—61no—
163#382—31no—
164#382—31no—
165#382—31no—
166#382—61no—
167#382—61no—
168#382C2×D311yescharacteristic
169#380—31no—
170C4×C4⋊C4—31no—
171C4×C22⋊C4—61no—
172C2×#95—61no—
173C2×#95—61no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#340—31no—
182#336—31no—
183#336C2×D311yescharacteristic
184C24×C3—21no—
185C23×D3C2×D411yescharacteristic
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4C2×Q812yes—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2411yescharacteristic
198C2×#67—24no—
199C2×#67—42no—
200C2×#67—42no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—42no—
206C22×Dic3—42no—
207C22×Dic3—42no—
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C2×C4×D3—21no—
212C2×C4×D3—21no—
213C2×C4×D3—21no—
214C2×C4×D3—21no—
215C2×C4×D3—22no—
216C2×C4×D3—22no—
217C2×C4×D3—22no—
218C2×C4×D3—22no—
219C2×C4×D3—41no—
220C2×C4×D3—41no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3—42no—
226C2×C4×D3—42no—
227C2×C4×D3C2×Q812yes—
228C2×C4×D3#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C22⋊C4—42no—
242C3×C22⋊C4—42no—
243C3×C42—21no—
244C3×C42—21no—
245#217—24no—
246#217—42no—
247(C2×D3)⋊C4—42no—
248C4⋊Dic3—21no—
249C4⋊Dic3—21no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—21no—
255C4⋊Dic3C2×Q812yes—
256Dic3⋊C4—42no—
257Dic3⋊C4—42no—
258Dic3⋊C4—42no—
259C4×Dic3—22no—
260C4×Dic3—22no—
261C4×Dic3—42no—
262C4×Dic3#4112yes—
263C4×Dic3#4112yes—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C22×D4—61no—
271C22×D4C22×D311yescharacteristic
272C23×C4—31no—
273C23×C4—31no—
274C23×C4—31no—
275C23×C4—61no—
276C23×C4—61no—
277C23×C4—61no—
278C22×C8—31no—
279C22×C8—31no—
280C22×C8C22×D311yescharacteristic
281C22×C8C22×D311yescharacteristic
282#121—34no—
283#121—62no—
284C4×D4—62no—
285C4×D4—62no—
286C4×D4—124no—
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—121no—
295C2×C4⋊C4—121no—
296C2×C4⋊C4—121no—
297C2×C4⋊C4—121no—
298C2×C4⋊C4C22×D311yescharacteristic
299C2×C4⋊C4C22×D311yescharacteristic
300C2×C4⋊C4C22×D311yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—61no—
304C2×C22⋊C4—61no—
305C2×C22⋊C4—61no—
306C2×C42—31no—
307C2×C42—31no—
308C2×C42—31no—
309C2×C42—62no—
310C2×C42—62no—
311C2×C42—61no—
312C2×C42C22×D311yescharacteristic
313#107—41no—
314#107—121no—
315#106—41no—
316#106—121no—
317C4⋊C8—62no—
318C4⋊C8—62no—
319#102—41no—
320#102—41no—
321#102—121no—
322#102—121no—
323C22⋊C8—62no—
324C22⋊C8—62no—
325#95—21no—
326#95—21no—
327#95—61no—
328#95—61no—
329#95—61no—
330#95—61no—
331#95—61no—
332#95—61no—
333#95—61no—
334#95—61no—
335#95—61no—
336C23×C3—22no—
337C23×C3—41no—
338C23×C3—41no—
339C23×C3—41no—
340C23×C3—41no—
341C23×C3—41no—
342C23×C3—41no—
343C23×C3C22×D411yescharacteristic
344C22×D3—21no—
345C22×D3—21no—
346C22×D3—21no—
347C22×D3—21no—
348C22×D3C22⋊Q812yes—
349C22×D3#13512yes—
350C22×D3AGL(1,ℤ/8)12yes—
351C3×D4—22no—
352C3×D4—22no—
353C3×D4—41no—
354C3×D4—41no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4C22×Q811yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C4H(5,𝔽2)11yescharacteristic
376#67—48no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
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×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—84no—
396C2×Dic3C22⋊Q812yes—
397C2×Dic3#12312yes—
398C2×Dic3#12312yes—
399C4×D3—22no—
400C4×D3—22no—
401C4×D3—22no—
402C4×D3—22no—
403C4×D3—44no—
404C4×D3—44no—
405C4×D3—84no—
406C4×D3—84no—
407C3×C8—41no—
408C3×C8—41no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C24—21no—
412C24—31no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—64no—
419C2×D4—122no—
420C2×D4—122no—
421C2×D4D3×Q812yes—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—31no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—61no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—121no—
443C22×C4—121no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4—122no—
450C22×C4—122no—
451C22×C4—122no—
452C22×C4—122no—
453C22×C4D3×D411yescharacteristic
454C22×C4D3×D411yescharacteristic
455C22×C4C23×D311yescharacteristic
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8—61no—
465C2×C8—61no—
466C2×C8—61no—
467C2×C8—61no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—32no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C4⋊C4—61no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—61no—
481C4⋊C4—61no—
482C4⋊C4—122no—
483C4⋊C4—122no—
484C4⋊C4—122no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—64no—
488C22⋊C4—122no—
489C22⋊C4—122no—
490C42—21no—
491C42—21no—
492C42—32no—
493C42—32no—
494C42—62no—
495C42—62no—
496C42—122no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—81no—
501C22×C3—81no—
502C22×C3—81no—
503C22×C3—81no—
504C22×C3C2×C22⋊Q811yescharacteristic
505C22×C3C2×#12311yescharacteristic
506C22×C3#53511yescharacteristic
507C22×C3C2×#13511yescharacteristic
508C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
509C22×C3#57511yescharacteristic
510C22×C3#57511yescharacteristic
511C2×D3—22no—
512C2×D3—22no—
513C2×D3—22no—
514C2×D3—22no—
515C2×D3—22no—
516C2×D3—22no—
517C2×D3#47814yes—
518C3×C4—41no—
519C3×C4—41no—
520C3×C4—81no—
521C3×C4—81no—
522C3×C4—81no—
523C3×C4#55311yescharacteristic
524C3×C4#55311yescharacteristic
525C3×C4#55411yescharacteristic
526C3×C4#55411yescharacteristic
527Dic3—22no—
528Dic3—22no—
529Dic3—22no—
530Dic3—42no—
531Dic3—81no—
532Dic3—81no—
533Dic3—81no—
534C23—22no—
535C23—32no—
536C23—32no—
537C23—32no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—61no—
545C23—61no—
546C23—61no—
547C23—61no—
548C23C2×D3×D411yescharacteristic
549D4—22no—
550D4—22no—
551D4—41no—
552D4—41no—
553D4—128no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—32no—
561C2×C4—32no—
562C2×C4—32no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—62no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4—124no—
584C2×C4—124no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—121no—
590C2×C4—121no—
591C2×C4—121no—
592C2×C4—121no—
593C2×C4—121no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—244no—
597C2×C4—244no—
598C2×C4—244no—
599C2×C4#100411yescharacteristic
600C2×C4#100411yescharacteristic
601C2×C4C2×D3×Q811yescharacteristic
602C2×C4D3×#4111yescharacteristic
603C2×C4D3×#4111yescharacteristic
604C2×C4D3×#4111yescharacteristic
605C8—41no—
606C8—41no—
607C8—121no—
608C8—121no—
609C2×C3—82no—
610C2×C3#248711yescharacteristic
611C2×C3C2×#47811yescharacteristic
612C2×C3#308011yescharacteristic
613C2×C3#309111yescharacteristic
614C2×C3#309411yescharacteristic
615C2×C3#321111yescharacteristic
616C2×C3#321311yescharacteristic
617D3—24no—
618C22—34no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—62no—
623C22—62no—
624C22—62no—
625C22—62no—
626C22—62no—
627C22—62no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22D3×C22⋊Q811yescharacteristic
633C22D3×#12311yescharacteristic
634C22#571511yescharacteristic
635C22D3×#13511yescharacteristic
636C22#576811yescharacteristic
637C22D3×AGL(1,ℤ/8)11yescharacteristic
638C22#577311yescharacteristic
639C4—41no—
640C4—41no—
641C4—62no—
642C4—62no—
643C4—62no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—122no—
648C4—241no—
649C4—241no—
650C4—241no—
651C4#563611yescharacteristic
652C4#563611yescharacteristic
653C4#566111yescharacteristic
654C4#566111yescharacteristic
655C3#2768411yescharacteristic, complemented, Sylow
656C2—64no—
657C2—82no—
658C2#7802911yescharacteristic
659C2D3×#47811yescharacteristic
660C2#8116711yescharacteristic
661C2#8137111yescharacteristic
662C2#8137311yescharacteristic
663C2#8182511yescharacteristic
664C2#8184911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2092C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2092D3
hypocenterC3#27684
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2092

Complements

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

NH ≅ G/Ntype
2. D3×#382C2semidirect
3. C3×#2092C2semidirect
4. #70632C2semidirect
5. #70616C2semidirect
6. #70579C2semidirect
7. #70564C2semidirect
10. #70154C2semidirect
11. #70154C2semidirect
12. #69654C2semidirect
13. #69542C2semidirect
14. #68662C2semidirect
15. #68644C2semidirect
65. #2092D3semidirect
655. C3#27684semidirect

Maximal subgroups

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

subgroupindex
#762912
D3×#3822
C3×#20922
#706322
#706162
#705792
#705642
#702002
#701542
#696542
#695422
#686622
#686442
#276843