Finite Group Databasebeta

← #409893

Subgroups — #409893

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2101C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic, complemented
4C2×#1936C211yesmaximal, characteristic
5#15244C211yesmaximal, characteristic
6#15143C211yesmaximal, characteristic, complemented
7#11372C211yesmaximal, characteristic, complemented
8#11371C211yesmaximal, characteristic, complemented
9#11049C211yesmaximal, characteristic
10#11049C211yesmaximal, characteristic
11#10927C211yesmaximal, characteristic
12#10925C211yesmaximal, characteristic
13#10843C211yesmaximal, characteristic, complemented
14#10841C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#102C2211yescharacteristic
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C2×#395—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2211yescharacteristic
28C2×#389C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#382C2211yescharacteristic
34#2101—24no—
35#2023—24no—
36#1993—24no—
37#1936—24no—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#340C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52C2×#336C2211yescharacteristic
53#1444—22no—
54#1439—22no—
55#1439—22no—
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4×#95C2211yescharacteristic
59C23×Q8—21no—
60C23×D4C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—41no—
70C2×C4⋊Q8—41no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×#102—22no—
91C2×#102—22no—
92C2×#102—22no—
93C2×#102—22no—
94C2×#102—22no—
95C2×#102—22no—
96C2×#102—22no—
97C2×#102—41no—
98C2×#102—41no—
99C2×#102—41no—
100C2×#102—41no—
101C2×#102Q814yescomplemented
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—41no—
105C2×#97—41no—
106C2×#97—41no—
107C2×#97D414yes—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—41no—
111#395—48no—
112#390—22no—
113#390—22no—
114#390—22no—
115#390—22no—
116#390—22no—
117#390—22no—
118#389—22no—
119#389—22no—
120#384—22no—
121#384—22no—
122#384—22no—
123#384—22no—
124#384—41no—
125#384—41no—
126#382—22no—
127#382—22no—
128#382—22no—
129#382—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95D412yes—
137C2×#95D412yes—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#340—22no—
146#340—22no—
147#340—22no—
148#340—22no—
149#340—22no—
150#336—22no—
151#336—22no—
152#336D414yes—
153C25—21no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—44no—
157C22×Q8—42no—
158C22×Q8—82no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4C2×Q812yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C2×D412yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2411yescharacteristic
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×D412yes—
210C22×C8C2×D412yes—
211C22×C8C2×Q812yes—
212C22×C8#4112yes—
213C4⋊Q8—28no—
214C4⋊Q8—44no—
215C4⋊Q8—44no—
216C4⋊Q8—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4—82no—
268C2×C4⋊C4—82no—
269C2×C4⋊C4—82no—
270C2×C4⋊C4C2×Q812yes—
271C2×C4⋊C4#4112yes—
272C2×C4⋊C4#4112yes—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—82no—
280C2×C22⋊C4—82no—
281C2×C22⋊C4—82no—
282C2×C22⋊C4—82no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42#4112yes—
298C2×C42#4112yes—
299C2×C42#4112yes—
300C2×C42#4112yes—
301#102—28no—
302#102—44no—
303#102—44no—
304#102—44no—
305#97—44no—
306#97—44no—
307C4×C8—44no—
308#95—24no—
309#95—24no—
310#95—44no—
311#95—44no—
312#95—42no—
313#95—42no—
314#95—42no—
315#95—44no—
316C24—22no—
317C24—41no—
318C24—41no—
319C24—41no—
320C24—41no—
321C24—41no—
322C24—41no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24C22×D411yescharacteristic
328C2×Q8—48no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×Q8—88no—
332C2×Q8—88no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—41no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4#12414yes—
415C22×C4#12414yes—
416C22×C4#12712yes—
417C22×C4#12712yes—
418C22×C4AGL(1,ℤ/8)14yes—
419C22×C4#13714yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q811yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4C2×#4111yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C4H(5,𝔽2)11yescharacteristic
432C22×C42−1+411yescharacteristic
433C22×C42−1+411yescharacteristic
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8C22⋊Q814yes—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—48no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—41no—
488C4⋊C4—41no—
489C4⋊C4—41no—
490C4⋊C4—41no—
491C4⋊C4—88no—
492C4⋊C4—84no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C22⋊C4—88no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C42—22no—
500C42—22no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—82no—
517C23—82no—
518C23—82no—
519C23—81no—
520C23—82no—
521C23—82no—
522C23—82no—
523C23—82no—
524C23—81no—
525C23—81no—
526C23—81no—
527C23—81no—
528C23—81no—
529C23—81no—
530C23—81no—
531C23—82no—
532C23—82no—
533C23—82no—
534C23C2×#12411yescharacteristic
535C23C2×#12411yescharacteristic
536C23C2×#12711yescharacteristic
537C23#54611yescharacteristic
538C23#54611yescharacteristic
539C23#54711yescharacteristic
540C23D4×Q811yescharacteristic
541C23C2×AGL(1,ℤ/8)12yes—
542C23C2×#13712yes—
543C23#57612yes—
544C23#57712yes—
545Q8—816no—
546D4—42no—
547D4—42no—
548D4—42no—
549D4—42no—
550D4—81no—
551D4—81no—
552D4—81no—
553D4—81no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—82no—
584C2×C4—81no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—81no—
593C2×C4—81no—
594C2×C4—81no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4—82no—
598C2×C4—82no—
599C2×C4—82no—
600C2×C4—82no—
601C2×C4—81no—
602C2×C4—81no—
603C2×C4—81no—
604C2×C4—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4#48918yes—
613C2×C4C2×C22⋊Q811yescharacteristic
614C2×C4#52511yescharacteristic
615C2×C4#52911yescharacteristic
616C2×C4#52911yescharacteristic
617C2×C4#53211yescharacteristic
618C2×C4#53211yescharacteristic
619C2×C4#53211yescharacteristic
620C2×C4#53211yescharacteristic
621C2×C4#53311yescharacteristic
622C2×C4#53311yescharacteristic
623C2×C4#53411yescharacteristic
624C2×C4#53611yescharacteristic
625C2×C4#53911yescharacteristic
626C2×C4#53911yescharacteristic
627C2×C4#54211yescharacteristic
628C2×C4#54211yescharacteristic
629C2×C4#54611yescharacteristic
630C2×C4#54611yescharacteristic
631C2×C4#54711yescharacteristic
632C2×C4D4×Q811yescharacteristic
633C2×C4#55111yescharacteristic
634C2×C4#55111yescharacteristic
635C2×C4#55211yescharacteristic
636C2×C4#55211yescharacteristic
637C2×C4#55411yescharacteristic
638C2×C4#55411yescharacteristic
639C2×C4#55611yescharacteristic
640C2×C4#55611yescharacteristic
641C8—24no—
642C8—42no—
643C8—42no—
644C8—42no—
645C22—82no—
646C22—82no—
647C22—84no—
648C22—84no—
649C22—82no—
650C22—162no—
651C22—162no—
652C22—161no—
653C22—161no—
654C22—161no—
655C22—161no—
656C22#248811yescharacteristic
657C22#248811yescharacteristic
658C22#249211yescharacteristic
659C22#249211yescharacteristic
660C22#253911yescharacteristic
661C22#254211yescharacteristic
662C22#285811yescharacteristic
663C22C2×#48912yes—
664C22#313612yes—
665C22#313612yes—
666C22#314112yes—
667C22#315111yescharacteristic
668C22#315211yescharacteristic
669C22#315411yescharacteristic
670C22#315511yescharacteristic
671C22#331812yes—
672C22#332012yes—
673C22#332512yes—
674C22#332912yes—
675C22#334512yes—
676C22#334612yes—
677C22#338412yes—
678C22#338512yes—
679C4—28no—
680C4—44no—
681C4—42no—
682C4—42no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—161no—
687C4—161no—
688C4—161no—
689C4—161no—
690C4#239611yescharacteristic
691C4#252111yescharacteristic
692C4#252111yescharacteristic
693C4#255611yescharacteristic
694C4#255611yescharacteristic
695C4#256411yescharacteristic
696C4#260411yescharacteristic
697C4#266311yescharacteristic
698C2—162no—
699C2#1506511yescharacteristic
700C2#2770412yes—
701C2#2771112yes—
702C2#2782112yes—
703C2#2782512yes—
704C2#2835311yescharacteristic
705C2#2836711yescharacteristic
706C2#2854311yescharacteristic
707C2#2854811yescharacteristic
708C2#3322811yescharacteristic
709C2#3323011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2101C2semidirect
3. C2×#1993C2semidirect
6. #15143C2semidirect
7. #11372C2semidirect
8. #11371C2semidirect
13. #10843C2semidirect
14. #10841C2semidirect
15. #10696C2semidirect
101. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#21012
C2×#20232
C2×#19932
C2×#19362
#152442
#151432
#113722
#113712
#110492
#109272
#109252
#108432
#108412
#106962