Finite Group Databasebeta

← #361998

Subgroups — #361998

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1983C212yesmaximal
3C2×#1843C211yesmaximal, characteristic, complemented
4C2×#1805C211yesmaximal, characteristic, complemented
5C2×#1803C211yesmaximal, characteristic, complemented
6C2×#1804C211yesmaximal, characteristic
7C2×#1802C211yesmaximal, characteristic
8#13779C211yesmaximal, characteristic, complemented
9#11077C212yesmaximal, complemented
10#11076C212yesmaximal
11#10849C212yesmaximal, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#102C2212yes—
22C22×#102C2212yes—
23#2326—24no—
24#2326C418yescomplemented
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#388—22no—
28C2×#385—22no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4C2212yes—
33C2×#376C2211yescharacteristic
34#1983—28no—
35#1843—24no—
36#1805—24no—
37#1804—24no—
38#1803—24no—
39#1802—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44#1436—22no—
45#1436—22no—
46C4⋊#95C2212yes—
47C23×Q8C2311yescharacteristic
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124—22no—
55C2×#124C2×C414yes—
56C2×#124D414yes—
57C2×#124Q814yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8C2×C414yes—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4C2×C414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—22no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—22no—
81C22×C42—21no—
82C22×C42—22no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—24no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—24no—
95C2×C4⋊C8—24no—
96C2×C4⋊C8—42no—
97C2×C4⋊C8—42no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#102—24no—
105C2×#102—24no—
106C2×#102—24no—
107C2×#102—24no—
108C2×#102—42no—
109C2×#102—42no—
110#391—22no—
111#391C2×C414yes—
112#390—22no—
113#390C2×C414yes—
114#388—216no—
115#385—216no—
116#384—22no—
117#384—22no—
118#384C2×C414yes—
119#384D414yes—
120#382—22no—
121#382D414yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—24no—
124#376C2×C414yes—
125#376D414yes—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95D412yes—
133C2×#95D412yes—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138C25—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8C22×C412yes—
151C22×Q8C2×D412yes—
152C22×Q8C2×Q812yes—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22×C412yes—
165C22×D4C2×D412yes—
166C22×D4C2×Q812yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—82no—
176C23×C4—82no—
177C23×C4C2×D412yes—
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D412yes—
181C23×C4C2411yescharacteristic
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196#124—28no—
197#124—28no—
198#124—28no—
199#124C4⋊C4116yescomplemented
200C4×Q8—28no—
201C4×Q8—28no—
202C4×Q8—44no—
203C4×Q8—44no—
204C4×D4—28no—
205C4×D4—28no—
206C4×D4—44no—
207C4×D4—44no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—82no—
250C2×C22⋊C4—82no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—24no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—44no—
259C2×C42—82no—
260C2×C42—82no—
261C2×C42C22×C412yes—
262C2×C42C22×C412yes—
263C2×C42C22×C412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×D412yes—
270C2×C42C2×Q812yes—
271C2×C42#4112yes—
272C2×C42#4112yes—
273C2×C42#4112yes—
274C2×C42#4112yes—
275C4⋊C8—48no—
276C4⋊C8—48no—
277C4⋊C8—44no—
278C4⋊C8—44no—
279#103—48no—
280#103—48no—
281#102—48no—
282#102—48no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—44no—
287#95—82no—
288#95—82no—
289#95C22⋊C418yes—
290#95C22⋊C418yes—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—41no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—42no—
299C24—42no—
300C24—42no—
301C24—42no—
302C24C22×D411yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8C2×C4⋊C414yes—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—22no—
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×D4C2×C4⋊C414yes—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
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—24no—
379C22×C4—24no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—48no—
405C22×C4—88no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—88no—
409C22×C4C2×C22⋊C414yes—
410C22×C4C2×C22⋊C414yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4#12114yes—
414C22×C4#12112yes—
415C22×C4#12112yes—
416C22×C4#12112yes—
417C22×C4#12112yes—
418C22×C4C22⋊Q814yes—
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×Q811yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4C2×#4111yescharacteristic
428C22×C4C2×#4111yescharacteristic
429C22×C4H(5,𝔽2)11yescharacteristic
430C22×C4H(5,𝔽2)11yescharacteristic
431C22×C42−1+411yescharacteristic
432C22×C42−1+411yescharacteristic
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—42no—
443C2×C8—42no—
444C2×C8—42no—
445C2×C8—42no—
446C2×C8—42no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—48no—
464C4⋊C4—88no—
465C4⋊C4—84no—
466C4⋊C4—84no—
467C22⋊C4—232no—
468C22⋊C4—48no—
469C22⋊C4—88no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—48no—
475C42—88no—
476C42C2×C4⋊C414yes—
477C42#11714yes—
478C42C4×D414yes—
479C42C4×D414yes—
480C42#12114yes—
481C42C22⋊Q814yes—
482C42#12414yes—
483C42#12714yes—
484C23—24no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—82no—
498C23—82no—
499C23—82no—
500C23—82no—
501C23—81no—
502C23—81no—
503C23—81no—
504C23—81no—
505C23—82no—
506C23—82no—
507C23C22×C22⋊C411yescharacteristic
508C23C2×#12111yescharacteristic
509C23C2×#12111yescharacteristic
510C23C2×#12111yescharacteristic
511C23C2×C22⋊Q811yescharacteristic
512C23#53711yescharacteristic
513C23#53711yescharacteristic
514C23#57612yes—
515C23#57712yes—
516C23#57712yes—
517C23#57812yes—
518Q8—24no—
519Q8—24no—
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526D4—24no—
527D4—24no—
528D4—42no—
529D4—42no—
530D4—42no—
531D4—42no—
532D4—42no—
533D4—42no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—816no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—82no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—82no—
582C2×C4—82no—
583C2×C4—82no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4#38018yes—
588C2×C4#39014yes—
589C2×C4#39014yes—
590C2×C4C22×C4⋊C411yescharacteristic
591C2×C4C2×#11711yescharacteristic
592C2×C4C2×C4×D411yescharacteristic
593C2×C4C2×C4×D411yescharacteristic
594C2×C4#52011yescharacteristic
595C2×C4#52011yescharacteristic
596C2×C4C2×#12111yescharacteristic
597C2×C4C2×C22⋊Q811yescharacteristic
598C2×C4C2×#12411yescharacteristic
599C2×C4C2×#12711yescharacteristic
600C2×C4#53511yescharacteristic
601C2×C4#53611yescharacteristic
602C2×C4D4211yescharacteristic
603C2×C4D4211yescharacteristic
604C2×C4#54711yescharacteristic
605C2×C4#54711yescharacteristic
606C2×C4#54711yescharacteristic
607C2×C4#54711yescharacteristic
608C2×C4D4×Q811yescharacteristic
609C2×C4D4×Q811yescharacteristic
610C2×C4#55011yescharacteristic
611C2×C4#55011yescharacteristic
612C2×C4#55411yescharacteristic
613C2×C4#55411yescharacteristic
614C2×C4#55511yescharacteristic
615C2×C4#55511yescharacteristic
616C2×C4#55611yescharacteristic
617C2×C4#55611yescharacteristic
618C8—82no—
619C8—82no—
620C22—44no—
621C22—48no—
622C22—82no—
623C22—82no—
624C22—82no—
625C22—82no—
626C22—84no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22C2×#38011yescharacteristic
632C22C2×#39011yescharacteristic
633C22#231111yescharacteristic
634C22#253711yescharacteristic
635C22#253711yescharacteristic
636C22#253811yescharacteristic
637C22#253811yescharacteristic
638C22#289712yes—
639C22#289812yes—
640C22#304111yescharacteristic
641C22#304211yescharacteristic
642C22#304311yescharacteristic
643C22#304411yescharacteristic
644C22#304512yes—
645C22#304512yes—
646C22#304612yes—
647C22#304612yes—
648C22#308312yes—
649C22#308412yes—
650C22#325712yes—
651C22#325812yes—
652C22#326212yes—
653C22#326312yes—
654C4—24no—
655C4—24no—
656C4—28no—
657C4—44no—
658C4—44no—
659C4—82no—
660C4—82no—
661C4—84no—
662C4—84no—
663C4D4×C4⋊C411yescharacteristic
664C4D4×C4⋊C411yescharacteristic
665C4#235411yescharacteristic
666C4#235411yescharacteristic
667C4#254111yescharacteristic
668C4#254211yescharacteristic
669C4#254711yescharacteristic
670C4#256311yescharacteristic
671C2—84no—
672C2#1390611yescharacteristic
673C2#2065212yes—
674C2#2126211yescharacteristic
675C2#2126311yescharacteristic
676C2#2189612yes—
677C2#2825711yescharacteristic
678C2#2825811yescharacteristic
679C2#2826212yes—
680C2#2828811yescharacteristic
681C2#2828911yescharacteristic
682C2#2829312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1843C2semidirect
4. C2×#1805C2semidirect
5. C2×#1803C2semidirect
8. #13779C2semidirect
9. #11077C2semidirect
11. #10849C2semidirect
24. #2326C4semidirect
199. #124C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#19832
C2×#18432
C2×#18052
C2×#18032
C2×#18042
C2×#18022
#137792
#110772
#110762
#108492